Discrete Financial Mathematics 2004

Discrete Financial Mathematics

Wim Schoutens

Leuven, 2004-2005 Lecture Notes to the Course (G0Q20a) Discrete Financial Mathematics

Abstract The aim of the course is to give a rigorous yet accessible introduction to the modern theory of discrete financial mathematics. The student should already be comfortable with calculus and probability theory. Prior knowledge of basic notions of finance is useful. We start with providing some background on the financial markets and the instruments traded. We will look at different kinds of derivative securities, the main group of underlying assets, the markets where derivative securities are traded and the financial agents involved in these activities. The fundamental problem in the mathematics of financial derivatives is that of pricing and hedging. The pricing is based on the no-arbitrage assumptions. We start by discussing option pricing in the simplest idealised case: the Single-Period Market. Next, we turn to Binomial tree models. Under these models we price European and American options and discuss pricing methods for the more involved exotic options. Monte-Carlo issues come into play here. Finally, we set up general discrete-time models and look in detail at the mathematical counterpart of the economic principle of no-arbitrage: the existence of equivalent martingale measures. We look when the models are complete, i.e. claims can be hedged perfectly. We discuss the Fundamental theorem of asset pricing in a discrete setting.

To conclude the course, we make a bridge to continuous-time models. We look at them as limiting cases of discrete models. The discrete models will guide us in the analysis of continuous-time models in the Continuous Mathematical Finance Course.

2

Contents 1 Derivative Background 1.1

1

Financial Markets and Instruments . . . . . . . . . . . . . . . . .

1

1.1.1

Basic Instruments . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

The Bank Account . . . . . . . . . . . . . . . . . . . . . .

6

1.1.3

Derivative Instruments . . . . . . . . . . . . . . . . . . . .

9

1.1.4

Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.1.5

Contract Specifications

. . . . . . . . . . . . . . . . . . .

14

1.1.6

Types of Traders . . . . . . . . . . . . . . . . . . . . . . .

15

1.1.7

Modelling Assumptions . . . . . . . . . . . . . . . . . . .

17

1.2

Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.3

Arbitrage Relationships . . . . . . . . . . . . . . . . . . . . . . .

23

1.3.1

The Put-Call Parity . . . . . . . . . . . . . . . . . . . . .

23

1.3.2

The Forward Contract . . . . . . . . . . . . . . . . . . . .

26

1.3.3

Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

1.3.4

Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

1

1.3.5

Commodities . . . . . . . . . . . . . . . . . . . . . . . . .

32

1.3.6

The Cost of Carry . . . . . . . . . . . . . . . . . . . . . .

32

2 Binomial Trees

34

2.1

Single Period Market Models . . . . . . . . . . . . . . . . . . . .

34

2.2

Two-Step Binomial Trees . . . . . . . . . . . . . . . . . . . . . .

43

2.2.1

European Call . . . . . . . . . . . . . . . . . . . . . . . .

43

2.2.2

Matching Volatility with u and d . . . . . . . . . . . . . .

46

Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.3.1

European Call and Put Options

. . . . . . . . . . . . . .

49

2.3.2

American Options . . . . . . . . . . . . . . . . . . . . . .

53

Moving towards The Black-Scholes Model . . . . . . . . . . . . .

59

2.3

2.4

3 Mathematical Finance in Discrete Time

62

3.1

Information and Trading Strategies . . . . . . . . . . . . . . . . .

63

3.2

No-Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . .

67

3.3

Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.4

Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.5

The Fundamental Theorem of Asset Pricing . . . . . . . . . . . .

75

3.5.1

77

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Exotic Options

82

4.1

Monte Carlo Pricing . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.2

Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.3

Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

2

4.4

Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 The Black-Scholes Option Price Model 5.1

94 100

Continuous-Time Stochastic Processes . . . . . . . . . . . . . . . 101 5.1.1

Information and Filtration . . . . . . . . . . . . . . . . . . 101

5.1.2

Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2

Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3

Itˆ o’s Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.1

Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . 107

5.3.2

Itˆ o’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4

Stochastic Differential Equations . . . . . . . . . . . . . . . . . . 110

5.5

Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . 112

5.6

The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7

Equivalent Martingale Measures and Risk-Neutral Pricing . . . . 117 5.7.1

The Pricing of Options under the Black-Scholes Model . . 120

5.7.2

Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.8

The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.9

Drawbacks of the Black-Scholes Model . . . . . . . . . . . . . . . 129

6 Miscellaneous

132

6.1

Decomposing Options into Vanilla Position . . . . . . . . . . . . 132

6.2

Variance Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3

Chapter 1

Derivative Background 1.1

Financial Markets and Instruments

A market typically consist out of a riskfree Bank account and some other risky assets. On these basic instruments other financial contracts are written; these financial contracts are so-called derivative securities. This text is on the (riskneutral) pricing of derivative securities. This section provides the institutional background on the main group of underlying assets, the related derivative securities, the markets where derivatives securities are traded and the financial agents involved in these activities.

1.1.1

Basic Instruments

Next, we highlight some of the most common underlying securities.

1

Stocks – Equity The basis of modern economic life are companies owned by their shareholders; the shares provide partial ownership of the company, pro rata with investment. Shares have value, reflecting both the value of the company’s real assets and the earning power of the company’s dividends. With publicly quoted companies, shares are quoted and traded on the Stock Exchange. Stock is the generic term for assets held in the form of shares. Stock markets date back to at least 1531, when one was started in Antwerp, Belgium. Today there are over 150 stock exchanges.

Interest Rates – Fixed-Income The value of some financial assets depends solely on the level of interest rates, e.g. Treasury notes, municipal and corporate bonds. These are fixed-income securities by which national, state and local governments and large companies partially finance their economic activity. Fixed-income securities require the payment of interest in the form of a fixed amount of money at predetermined points in time, as well as the repayment of the principal at maturity of the security. Interest rates themselves are notional assets, which cannot be delivered. A special fixed-income product is the bank account, which we typically assume to be riskfree. We go into more detail on possible bank account models in Section 1.1.2.

2

Currencies – Foreign Exchange A currency is the denomination of the national units of payment (money) and as such is a financial asset. Companies may wish to hedge adverse movements of foreign currencies and in doing so use derivative instruments. A foreign currency has the property that the holder of the currency can earn interest at the risk-free interest rate prevailing in the foreign country. We thus have to kind of interest rates, the domestic and the foreign interest rate.

Commodities Commodities are a kind of physical products like gold, oil, cattle, fruit juice. Trade in these assets can be for different purposes: for using them in the production process or for speculation. Derivative instruments on these asset can be used for hedging and speculation. Special care has to be taken with commodities because of storage costs (see Section 1.3.5) In Figure 1.1, one sees the some prices (of the market on the 16th of October 2004) of (futures on) energy, metal, livestock and other commodities.

3

Figure 1.1: Commodities future prices on 16-10-2004

Miscellaneous Indexes An index tracks the value of a basket of stocks (FTSE100, S&P500, Dow Jones Industrial, NASDAQ Composite, BEL20, EUROSTOXX50, ...), bonds, and so on.

Derivative instruments on indices may be used for hedging if no

4

derivative instruments on a particular asset in question are available and if the correlation in movement between the index and the asset is significant. Furthermore, institutional funds (such as pension funds), which manage large diversified stock portfolios, try to mimic particular stock indices and use derivative on stock indices as a portfolio management tool. On the other hand, a speculator may wish to bet on a certain overall development in a market without exposing him/herself to a particular asset. In Figure 1.2, one sees the Belgian Bel-20 Index over a period of more than 4 years.

Figure 1.2: BEL-20 A new kind of index was generated with the Index of Catastrophe Losses (CAT-index) by the Chicago Board of Trade (CBOT) lately. The growing number of huge natural disasters (such as hurricanes and earthquakes) has led the insurance industry to try to find new ways of increasing its capacity to carry risks. Currently investors are offered options on the CAT-index, thereby taking 5

in effect the position of traditional reinsurance. Credit Risk Market Credit risk captures the risk on a financial loss that an institution incurs when it lends money to another institution or person. This financial loss realizes whenever the borrower does not meet all of its obligations under its borrowing contract. Because credit risk is so important for financial institutions the banking world has developed instruments that allows them to evacuate credit risk rather easily. The most commonly known and used example is the credit default swap. These instruments can best be considered as tradable insurance contracts. This means that today I can buy protection on a bond and tomorrow I can sell that same protection as easily as I bought it. Credit default swaps work just as an insurance contract. The protection buyer (the insurance taker) pays a fee and in exchange he gets reimbursed his losses if the company on which he bought protection defaults. In Figure 1.3, one sees credit default swap bid and offer rates for major aerospace/transport and auto (parts) companies.

1.1.2

The Bank Account

In the next Chapter we will start building models for the stock or asset price process. Here we focus on one instrument which we typically assume to be available in all the later on encountered market models: the bank account. There are two (quite related) regimes under which we will work: discrete or continuous compounding. This has all to do with when and how frequently the interest

6

Figure 1.3: Credit Default Swap rates

gained on the invested money is paid out. Typically the discrete compounding will be only used in discrete time models; the continuous compound can be used in almost any situation. Consider an amount A is invested for n years at an interest rate of R per annum. If the rate is compounded once per annum, the terminal value of the investment is A(1 + R)n . If it is compounded m times per annum, the terminal value of the investment is 

R A 1+ m

mn

.

We note that there is a difference. Indeed, take A = 100 euro and n = 1, when 7

the interest rate is 10 percent a year. The first regime leads to 110 euros after one year. However, with quarterly payments (m = 4), i.e. with payments every three month, we have after one year 100 × (1.0025)4 = 110.38 euros. In Figure 1.4 one sees the effect of increasing the compounding frequency from a yearly compounding to a daily compounding. discrete compound interest rates (A=100, n=1, r=0.10) 110.6

110.5

end value

110.4

110.3

110.2

110.1

110

0

50

100

150 200 compounding frequency (m)

250

300

350

Figure 1.4: Discrete compounding

The limit as m tends to infinity is known as continuous compounding. We have 

R lim A 1 + m→∞ m

mn

= A exp(nR).

With such a continuous compounding the invested amount A grows to A exp(nR) after n years. Note that it is because of the above discussion, important to state the units/frequency in which the interest rate is measured/compounded. For example, an interest rate of 10 percent continuous compounding is the same as 8

(1 − exp(0.10))/100 = 10.517 percent annual compounding. Throughout the text we will make use of a bank account on which we can put money and borrow money on a fixed continuously compounded interest rate r. This means that 1 euro on the bank account at time 0 will give rise to ert euro on time t > 0. Similarly, if we borrow 1 euro now, we have to pay back ert euro at time t > 0. Or equivalently, if we borrow now e−rt euro we have to pay back 1 euro at time t. One euro on the bank account will grow over time; at some time t we denote its value by B(t). Note thus that we set B(0) = 1. We call B = {B(t), t ≥ 0} the bank account price process or bond price process. Related to all this is the time value of money. An investor will prefer 100 euro in his pocket today to 100 euro in his pocket one year from now. The interest paid on the riskless bank account expresses this. Using continuous compounding with a rate r = 0.10, 100 euro is equivalent with 110.517 euros in one year. If we receive a cash-flow X at some future time T , the equivalent now is equal to exp(−rT )X. 100 euro in one year is equivalent with 90.484 euros now. This procedure is called discounting and exp(−rT ) is the discounting factor.

1.1.3

Derivative Instruments

In practitioner’s terms a ’derivative security’ is a security whose value depends on the value of other more basic underlying securities. We adopt the more precise definition: A derivative security, or contingent claim, is a financial contract whose value

9

at expiration date T (more briefly, expiry) is determined exactly by the price process of the underlying financial assets (or instruments) up to time T. Derivative securities can be grouped under three general headings: Options, Forwards and Futures, and Swaps. During this text we will mainly deal with options although our pricing techniques may be readily applied to forwards, futures and swaps as well.

Options An option is a financial instrument giving one the right but not the obligation to make a specified transaction at (or by) a specified date at a specified price. A lot of different type of options exists. We give here the basic types. Call options give one the right to buy. Put options give one the right to sell. European options give one the right to buy/sell on the specified date, the expiry date, on when the option expires or matures. American options give one the right to buy/sell at any time prior to or at expiry. Asian options depend on the average price over a period. Lookback options depend on the maximum or minimum price over a period and barrier options, depend on some price level being attained or not. The price at which the transaction to buy/sell the underlying assets (or simply the underlying), on/by the expiry date (if exercised), is made is called the exercise price or strike price. We usually use K for strike price, time t = 0, for the initial time (when the contract between the buyer and the seller of the

10

option is struck), time t = T for the expiry or final time. Consider, say, an European call option, with strike price K; write St for the value (or price) of the underlying at time t. If St > K, the option is in the money, if St = K, the option is said to be at the money and if St < K, the option is out the money. This terminology is of course motivated by the payoff, the value of the option at maturity, from the option which is ST − K if ST > K and 0 otherwise (more briefly written as (ST − K)+ ). This payoff function for K = 100 is visualized in Figure 1.5 Payoff of European Call (K=100) 20

18

16

14

Payoff

12

10

8

6

4

2

0 80

85

90

95

100 105 stock price at maturity

110

115

120

Figure 1.5: Payoff of Call Option (K=100)

There are two sides to every option contract. On one side there is the person who has bought the option (the long position); on the other side you have the

11

person who has sold or written the option (the short poistion). The writer receives cash up front but has potential liabilities later. In Figure 1.6, one can see that by investing in an option one can make huge gains, but also if markets goes the opposite direction as anticipate, it is possible to loose all money one has invested.

Figure 1.6: Stock Prices and European Call Option at time t = 0 and t = T .

In 1973, the Chicago Board Options Exchange (CBOE) began trading in options on some stocks. Since then, the growth of options has been explosive. Risk Magazine (12/1997) estimated $35 trillion as the gross figure for worldwide derivatives markets in 1996. In Figure 1.7, one sees some of the prices of call options written on the SP500index. The main aim of this text is to give a basic introduction to models for determining these kind of option prices.

12

Forwards, Futures A forward contract is an agreement to buy or sell an asset at a certain future date T for a certain price K. It is usually between two large and sophisticated financial agents (banks, institutional investors, large corporations, and brokerage firms) and not traded on an exchange. The agents who agrees to buy the underlying asset is said to have a long position, the other agent assumes a short position. The payoff from a long position in a forward contract on one unit of an asset with price ST at the maturity time T of the contract is ST − K. Compared with a call option with the same maturity and strike price K we see that the investor now faces a downside risk, too. He has the obligation to buy the asset for price K. A futures contract, like a forward contract, is an agreement to buy or sell an asset at a certain future date for a certain price. The difference is that futures are traded. As such, the default risk is removed from the parties to the contract and borne by the clearing house.

Swaps A swap is an agreement whereby two parties undertake to exchange, at known dates in the future, various financial assets (or cash flows) according to a prearranged formula that depends on the value of one or more underlying assets. Examples are currency swaps (exchange currencies) and interest-rate swaps (ex-

13

change of fixed for floating set of interest payments) and the nowadays popular credit default swaps as in Figure 1.3.

1.1.4

Markets

Financial derivatives are basically traded in two ways: on organized exchanges and over-the-counter (OTC). Organized exchanges are subject to regulatory rules, require a certain degree of standardization of the traded instruments (strike price, maturity dates, size of contract, etc.). Examples are the Chicago Board Options Exchange (CBOE), the London International Financial Futures Exchange (LIFFE). The exchange clearinghouse is an adjunct of the exchange and acts as an intermediary in the transactions. It garantuess the performanjce of the parties to each transaction. Its main task is to keep track of all the transactions that take place during a day so it can calculate the net poistions of each of its members. OTC trading takes between various commercial and investments banks such as Goldman Sachs, Citibank, Deutsche Bank.

1.1.5

Contract Specifications

It is very important that the financial contract specifies in detail the exact nature of the agreement between the two parties. It must specify the contract size (how much of the asset will be delivered under one contract), where delivery will be made, when exactly the delivery is made, etc. When the contract is traded at

14

an exchange, it should be made clear how prices will be quoted, when trade is allowed, etc. Financial assets in derivatives are generally well defined and unambiguous, e.g. it is clear what a Japanese Yen is. When the asset is a commodity, there may be quite a variation in the quality and it is important that the exchange stipulates the grade or grades of the commodity that are acceptable. Most contracts are refered to by its delivery month and year. The contract must specify in detail the period of that month when delivery can be made. For some future the delivery period is the entire month. For other contract delivery must be at a special day, hour, etc. Some contracts are in terms of a so-called settlement price. For example derivatives on indices (like the SP-500). The settlement price is calculated by the exchange by a very detailed algorithm. It can be e.g. averages of the index taken every five minutes during one hour, but also just the closing price of the asset. Other specification by the exchange deal with movement limits. Trade will be halted if these limits are exceeded. The purpose of price limits is to prevent large movements from occurring because of speculative excesses, extremal situation (11th of September), ...

1.1.6

Types of Traders

We can classify the traders of derivatives securities in three different classes.

15

Hedgers Successful companies concentrate on economic activities in which they to best. They use the market to insure themselves against adverse movements of prices, currencies, interest rates etc. Hedging is an attempt to reduce exposure to risk. Hedgers prefer to forgo the chance to make exceptional profits when future uncertainty works to their advantage by protecting themselves against exceptional loss.

Speculators Speculators want to take a position in the market – they take the opposite position to hedgers. Indeed, speculation is needed to make hedging possible, in that a hedger, wishing to lay off risk, cannot do so unless someone is willing to take it on. In speculation, available funds are invested opportunistically in the hope of making a profit: the underlying itself is irrelevant to the speculator, who is only interested in the potential for possible profit that trade involving it may present.

Arbitrageurs Arbitrageurs try to lock in riskless profit by simultaneously entering into transactions in two or more markets. An arbitrage opportunity exists, for example, if a security can be bought in New York at one price and sold at a slightly higher price in London. The underlying concept of the here presented theory is the absence of arbitrage opportunities.

16

1.1.7

Modelling Assumptions

We will discuss contingent claim pricing in an idealized case. We will not allow market frictions; there is no default risk, agents are rational and there is no arbitrage. More concrete this means • no transaction costs • no bid/ask spread • no taxes • no margin requirements • no restrictions on short sales • no transaction delays • same interest for borrowing and lending • market participants act as price takers • market participants prefer more to less We develop the theory of an ideal – frictionless – market so as to focus irreducible essentials of the theory and as a first-order approximation to reality. Understanding frictionless markets is also a necessary step to understand markets with frictions. The risk of failure of a company – bankruptcy – is inescapably present in its economic activity: death is part of life.

Moreover those risks also appear

at the national level: quite apart from war, recent decades have seen default 17

of interest payments of international debt, or the threat of it (see for example the 1998 Russian crisis). We ignore default risk for simplicity while developing understanding of the principal aspects. We assume financial agents to be price takers, not price makers. This implies that even large amounts of trading in a security by one agent does not influence the security’s price. Hence agents can buy or sell as much of any security as they wish without changing the security’s price. To assume that market participants prefer more to less is a very weak assumption on the preferences of market participants. Apart from this we will develop a preference-free theory. The relaxation of all these assumptions is subject to ongoing research. We want to mention the special character of the no-arbitrage assumption. It is the basis for the arbitrage pricing technique that we shall develop, and we discuss it in more detail below.

1.2

Arbitrage

The essence of arbitrage is that it should not be possible to guarantee a profit without exposure to risk. Were it possible to do so, arbitrageurs would do so, in unlimited quantity, using the market as a money-pump to extract arbitrarily large quantities of riskless profit. This would, for instance, make it impossible for the market to be in equilibrium. We shall see that arbitrage arguments suffice to determine prices - the arbitrage pricing technique.

18

To explain the fundamental arguments of the arbitrage pricing technique we use the following: Example: Consider an investor who acts in a market in which only three financial assets are traded: (riskless) bonds B (bank account), stocks S and European Call options C with strike K = 100 on the stock S. The investor may invest today, time t = 0, in all three assets, leave his investment until time t = T and gets his returns back then. We assume the option C expires at time t = T . We assume the current prices (in euro, say) of the financial assets are given by B(0) = 1, S(0) = 100, C(0) = 20 and that at t = T there can be only two states of the world: an up-state with euro prices B(T, u) = 1.25, S(T, u) = 175, and therefore C(T, u) = 75, and a down-state with euro prices B(T, d) = 1.25, S(T, d) = 75, and therefore C(T, d) = 0. Now our investor has a starting capital of 25000 euro from which he buy the following portfolio,

Portfolio I:

Asset

Number

Total amount in euro

Bond

10000

10000

Stock

100

10000

Call option

250

5000

19

Depending of the state of the world at time t = T the value of his portfolio will differ: In the up state the total value of his portfolio is 48750 euro: Asset

Number × Price

Total amount in euro

Bond

10000 × 1.25

12500

Stock

100 × 175

17500

Call option

250 × 75

18750

TOTAL

48750

whether in the down-state his portfolio has a value of 20000 euro: Asset

Number × Price

Total amount in euro

Bond

10000 × 1.25

12500

Stock

100 × 75

7500

Call option

250 × 0

0

TOTAL

20000

Can the investor do better ? Let us consider the restructured portfolio with initial investment of 24600 euro:

Portfolio II:

Asset

Number

Total amount in euro

Bond

11800

11800

Stock

70

7000

Call option

290

5800

We compute its return in the different possible states. In the up-state the total

20

value of his portfolio is again 48750 euro: Asset

Number × Price

Total amount in euro

Bond

11800 × 1.25

14750

Stock

70 × 175

12250

Call option

290 × 75

21750

TOTAL

48750

and in the down-state his portfolio has again a value of 20000 euro: Asset

Number × Price

Total amount in euro

Bond

11800 × 1.25

14750

Stock

70 × 75

5250

Call option

290 × 0

0

TOTAL

20000

We see that this portfolio generates the same time t = T return while costing only 24600 euro now, a saving of 400 euro against the first portfolio. So the investor should use the second portfolio and have a free lunch today! In the above example the investor was able to restructure his portfolio, reducing the current (t = 0) expenses without changing the return at the future date t = T in both possible states of the world. So there is an arbitrage possibility in the above market situation, and the prices quoted are not arbitrage prices. If we regard (as we shall do) the prices of the bond and the stock (our underlying) as given, the option must be mispriced. Let us have a closer look between the differences between Portfolio II, consisting of 11800 bonds, 70 stocks and 29 call options, in short (11800, 70, 290) and Portfolio I, of the form 21

(10000, 100, 250) The difference is the portfolio, Portfolio III say, of the form (11800, 70, 290) − (10000, 100, 250) = (1800, −30, 40). Asset

Number

Total amount in euro

Bond

1800

1800

Stock

-30

-3000

Call option

40

800

So if you sell short 30 stocks, you will receive 3000 euro from which you buy 40 options, put 1800 euro in your bank account and have a gastronomic lunch of 400 euro. But what is the effect of doing that ? Let us consider the consequences in the possible states of the world. We see in both cases that the effects of the different positions of Portfolio III offset themselves: In the up-state: Asset

Number × Price

Total amount in euro

Bond

1800 × 1.25

2250

Stock

-30 × 175

-5250

Call option

40 × 75

3000

TOTAL

0

In the down state: Asset

Number × Price

Total amount in euro

Bond

1800 × 1.25

2250

Stock

-30 × 75

-2250

Call option

40 × 75

0

TOTAL

0

22

But clearly the portfolio generates an income at t = 0 of which you had a free lunch, and a good one. Therefore it is itself an arbitrage opportunity. If we only look at the position in bonds and stocks, we can say that this position covers us against possible price movements of the option, i.e. having 1800 euro in your bank account and being 30 stocks short has the opposite time t = T value as owning 40 call options. We say that the bond/stock position is a hedge against the position in options. Let us emphasize that the above arguments were independent of the preferences and plans of the investor.

1.3

Arbitrage Relationships

We will in this section use arbitrage-based arguments to develop general bounds on the value of options. In our analysis here we use non-dividend paying stocks as the underlying, with price process S = {St , t ≥ 0}. We assume we have a risk-free bank account available which uses continuously compounding with a fixed interest rate r.

1.3.1

The Put-Call Parity

Next, we will deduce a fundamental relation between put and call options, the so-called put-call parity. Suppose there is a stock (with value St at time t), with European call and put options on it, with value Ct and Pt respectively at time t, with expiry time T and strike-price K. Consider a portfolio consisting of one stock, one put and a short position in one call (the holder of the portfolio has 23

written the call); write Πt for the time t value of this portfolio. So Πt = S t + P t − C t . Recall that the payoffs at expiry are for the call :

CT = max{ST − K, 0} = (ST − K)+ ,

for the put :

PT = max{K − ST , 0} = (K − ST )+ .

For the above portfolio we hence get at time T the payoff if ST

≥ K : ΠT = ST + 0 − (St − K) = K,

if ST

≤ K : ΠT = ST + (K − St ) − 0 = K.

This portfolio thus guarantees a payoff K at time T . How much is it worth at time t? The riskless way to guarantee a payoff K at time T is to deposit Ke−r(T −t) in the bank at time t and do nothing (we assume continuously compounded interest here). Under the assumption that the market is arbitrage-free the value of the portfolio at time t must thus be Ke−r(T −t) , for it acts as a synthetic bank account and any other price will offer arbitrage opportunities. Let us explore these arbitrage opportunities. If the portfolio is offered for sale at time t too cheaply–at price Πt < Ke−r(T −t) – we can buy it, borrow Ke−r(T −t) from the bank, and pocket a positive profit Ke−r(T −t) − Πt > 0. At time T our portfolio yields K, while our bank debt has grown to K. We clear our cash account – use the one to pay off the other – thus locking in our earlier profit, which is riskless. If on the other hand the portfolio is priced at time t at a too high price – at price Πt > Ke−r(T −t) – we can do the exact opposite. We sell the portfolio 24

short – that is,we buy its negative: buy one call, write one put, sell short one stock, for Πt and invest Ke−r(T −t) in the bank account, pocketing a positive profit Πt − Ke−r(T −t) > 0. At time T , our bank deposit has grown to K, and again we clear our cash account – use this to meet our obligation K on the portfolio we sold short, again locking in our earlier riskless profit. We illustrate the above with so-called arbitrage tables. In such a table we simply enter the current value of a given portfolio and then compute its value in all possible states of the world when the portfolio is cashed in. In the case Πt < Ke−r(T −t) :

Transactions

Current cash flow

Value at expiry ST < K

ST ≥ K

buy 1 stock

−St

ST

ST

buy 1 put

−Pt

K − ST

0

write 1 call

Ct

0

−ST + K

borrow

Ke−r(T −t)

−K

−K

0

0

TOTAL

Ke−r(T −t) − St −Pt + Ct > 0

Thus the rational price for the portfolio at time t is exactly Ke−r(T −t). Any other price presents arbitrageurs with an arbitrage opportunity (to make and lock in a riskless profit) – which they will take ! Therefore Proposition 1 We have the following put-call parity between the prices of the underlying asset and its European call and put options with the same strike price 25

and maturity on stocks that pay no dividends: St + Pt − Ct = Ke−r(T −t) . The value of the portfolio above is the discounted value of the riskless equivalent. This is a first glimpse at the central principle, or insight, of the entire subject of option pricing. Arbitrage arguments allow one to calculate precisely the rational price – or arbitrage price – of a portfolio. The put-call parity argument above is the simplest example of the arbitrage pricing technique.

1.3.2

The Forward Contract

Next, we will deduce a fair price (based on the no-arbitrage assumption) for the following forward contract: The contract states that party A (the buyer) must buy from party B (the seller) the (non-dividend paying) stock at time T at the price K (the strike price). We claim that F = S0 − exp(−rT )K is the correct initial price of this derivative which party A will pay to party B at time t = 0. Indeed, suppose you are party B, so you sold the forward contract and has received at time t = 0, the amount F . At time zero, you do the following: • buy 1 stock for the price S0 ; • borrow exp(−rT )K. To buy the stock you need S0 . You already have from the forward F = S0 − exp(−rT )K and receives from your loan exp(−rT )K. So you spent all the available money and at time t = 0 you have the following portfolio: 26

• long 1 stock. • short 1 forward; • short exp(−rT )K bonds. Look what happens at time T . You must deliver the stock to party A. You give away your stock in your portfolio, for this you receive K. The forward contract ends and you pay back your bank. You have to pay back the amount exp(rT ) exp(−rT )K = K. This you can do exactly with the money you received from party A. In the end everything is settled, you have no gain, no lost. Note that your initial investment is also zero. Note that any other price for the forward would have led to an arbitrage situation. Indeed, suppose you received Fˆ > F . Then by following the above strategy you pocket at time t = 0 the difference Fˆ − F > 0 which you can freely spent. At time T you just close all the position as described above. Without any initial investment and risk, you have then spent at time 0, Fˆ − F > 0. This is clearly an arbitrage opportunity (for party B). In case Fˆ < F , party A con set up a portfolio will always leads to an arbitrage opportunity (check this !). Forward/future contracts in practice are almost always struck at the price K, such that F = S0 − exp(−rT )K = 0, i.e. K = exp(rT )S0 . By doing this entering (in both ways: long or short) a forward contract is at zero cost. For this reason, exp(rT )S0 is called the time T forward price of the stock. 27

Finally, note that the put-call parity can be simply rewritten in terms of the call price, the put price and the forward contract price as: C − P = F (all derivatives have the same strike and time to maturity). From this one can see that at the forward price of the stock, i.e. in case K = exp(rT )S0 and hence F = 0, call and puts have the same value: C = P .

1.3.3

Dividends

Up to now, we have assumed that the risky asset pays no dividends, however in reality stocks can pay some dividend to the stock holders at some moments. We assume that the amount and timing of the dividends during the live of an option can be predicted with certainty. Moreover, we will assume that the stock pays a continuous compound dividend yield at rate q per annum. Other ways of dividend payments can be considered and techniques are described in the literature to deal with this. Continuous payment of a dividend yield at rate q means that our stock is following a process of the form: St = exp(−qt)S¯t , where S¯ is describing the stock prices behavior not taking into account dividends. A stock which pays continuously dividends and an identical stock that not pays dividends should provide the same overall return, i.e. dividends plus capital gains. The payment of dividends causes the growth of the stock price to be less than it would otherwise be by an amount q. In other words, if, with a continuous dividend yield of q, the stock price grows from S0 to ST at time T , 28

then in absence of dividends it would grow from S0 to exp(qt)ST . Alternatively, in absence of dividends it would grow from exp(−qt)S0 to ST . This argumentation brings us to the fact that we get the same probability distribution for the stock price at time T in the following cases: (1) The stock starts at S0 and pays a continuous dividend yield at rate q and (2) the stock starts at price exp(−qt)S0 and pays no dividend yield. The put-call parity for a stock with dividend yield q can be obtained from the put-call parity for non-dividend-paying stocks. With no dividends we obtained : St + Pt − Ct = K exp(−r(T − t)). If we now take into account dividends, the change comes down to replacing St with St exp(−q(T − t)). We have: exp(−q(T − t))St + Pt − Ct = K exp(−r(T − t)). This relation can be proven by considering the portfolio consisting of exp(−q(T − t)) number of stocks, one put option and minus one call option. We reinvest the dividends on the shares instantaneously in additional shares, i.e. at some future time point t ≤ s ≤ T , we have exp(−q(T − s)) number of stock; at the expiry date of the option we own one stock, one put and minus one call. The value of the portfolio at that time thus always equals K. By the no-arbitrage argument the time T value of the portfolio must equal K exp(−r(T − t)), the time t-value of a future payment (at time T ) of K. If our asset is an index, the dividend yield is the (weighted) average of the

29

dividends yields on the stocks composing the index.In practice, the dividend yield can be determined from the forward price of the asset. It is the agreement to buy or sell an asset at a certain future time for a certain price, the delivery price. At the time the contract is entered into, the delivery price is chosen so that the value of the forward is zero. This means that it costs nothing to buy or sell the contract. For an asset paying a continuous yield at rate q, the delivery price of a forward contract expiring at time T , is given by (proof this yourself !) F = S0 exp((r − q)T ).

(1.1)

Assuming the short rate r and the delivery price of the forward as given, q can easily be obtained.

1.3.4

Currencies

If the underlying is not a stock but a currency, we must take into account the domestic as well as the foreign interest rate. Let us continuous compounding and denote these interest rates by rd and rf , respectively. We are in Europe so our domestic currency is the euro. Consider a forward contract on the USD: you must buy N USD at some point in the future T for the price of K EUR/USD. Assume the currence exchange rate is S0 EUR/USD. What is the value of this future contract and for what value of K such that the contract has a zero value (the forward price of the USD). It will turn out now K = exp((rd − rf )T )S0 .

30

Indeed, suppose K > exp((rd − rf )T )S0 . An investor can then do the following (at time 0). • Borrow N × S0 exp(−rf T ) EUR at rate rd . • Use this cash to buy N × exp(−rf T ) USD and put this on an USDbankaccount at rate rf • Short the forward contract. Then the holding of the foreign currency grows to N because of the interest (rf ) earned. Under the terms of the contract this holding is exchanged for N × K at time T . An amount exp(rd T )N × S0 exp(−rf T ) is required to repay the borrowing. Hence a net profit of N × (K − S0 exp((rd − rf )T )) > 0 is, therefor, made at time T . In case K < exp((rd − rf )T )S0 you can the following • Borrow N × exp(−rf T ) USD at rate rf . • Use this cash to buy N × S0 exp(−rf T ) EURD and put this on an EURbankaccount at rate rd • Take a long position in the forward contract. then the domestic currency grows to N × S0 exp((rd − rf )T )), you pay N × K to receive N USD and uses these dollars to pay the loan. In total you earned N × (S0 exp((rd − rf )T )) − K) which in this case was assumed to be positive.

31

1.3.5

Commodities

We now consider the cas of commodities. Important here is the impact of storage costs. If the storage costs incurred at any time are proportional to the price of the commodity, they can be regarded as providing a negative dividend yield. In this case from equation (1.1), F = S0 exp((r + u)T ), where u is the storage costs per annum as a proportion of the spot price.

1.3.6

The Cost of Carry

The relationship between all above future/forward prices and spot prices can be summerized in terms of what is known as the cost of carry. This measures the storage cost plus the interest that is paid to finance the asset less the income earned on the asset. For a non-dividend paying stock, the cost of carry is r since there are no storage costs and no income is earned; for a stock index, it is r − q since income is earned at rate q on the asset; for a currency it is rd − rf ; for a commodity with storage costs that are a proportion u of the price, it is r + u; and so on. Define the cost of carry as c. For an investment asset, the future price is F = S0 exp(cT ).

32

Figure 1.7: Call options on S&P 500 Index

33

Chapter 2

Binomial Trees 2.1

Single Period Market Models

Our aim here is to show in the simplest possible non-trivial model how the theory based on the principle of no-arbitrage works.

Example Let our financial market consist of two financial assets, a riskless bank account (or bond) B and a risky stock S, with today’s price S0 = 20 euro. We look at a single-period model and assume that starting from today (t = 0) the world can only be in one of two states at time t = T : the stock price will either be ST = 22 euro or ST = 18 euro. We are interested in valuing a European call option to buy the stock for 21 euro at time t = T . At time t = T , this option can have only two possible values. It will have value 1 euro, if the stock price

34

Figure 2.1: One-period binomial tree example is 22 euro; if the stock price turns out to be 18 euro at time t = T , the value of the option will be zero. The situtation is illustrated in Figure 2.1. It turns out that we can price the option by the assumption that no arbitrage opportunities exist. We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the time of expiry, t = T . We then argue that, because the portfolio has no risk, the return earned on it must equal the risk-free interest rate of the bank account. This enables us to work out the cost of setting up the portfolio and, therefore, the option’s price. Consider a portfolio consisting of a long postion in ∆ shares of the stock and a short position in one call option. We calculate the value of ∆ that makes the portfolio riskless. If the stock price moves up from 20 to 22 euro, the value of the shares is 22∆ and the value of the option is 1 euro, so that the total value of the portfolio is 22∆ − 1 euro. If the stock price moves down from 20 to 18 euro, the value of the shares is 18∆ euro and the value of the option is zero, so that the total value of the portfolio is 18∆ euro. The portfolio is riskless if the

35

value of ∆ is chosen so that the final value of the portfolio is the same for both alternatives. This means 22∆ − 1 = 18∆ or ∆ = 0.25 A riskless portfolio is, therefore, • Long 0.25 shares. • Short 1 option. If the stock price moves up to 22 euro, the value of the portfolio is 22 × 0.25 − 1 = 4.5. If the stock price moves down to 18 euro, the value of the portfolio is 18 × 0.25 = 4.5. Regardless of whether the stock price moves up or down, the value of the portfolio is always 4.5 euro at the end of the life of the option. Riskless portfolios must, in the absence of arbitrage opportunities, earn the risk free rate of interest. Suppose that in this case the risk-free rate is 12 percent per annum and that T = 0.5, i.e. six months. It follows that the value of the portfolio today must be the present value of 4.5 euro, or 4.5e−0.12×0.5 = 4.238

36

The value of the stock today is known to be 20 euro. Suppose the option price is denoted by f . The value of the portfolio today is 20 × 0.25 − f = 5 − f It follows that 5 − f = 4.238 or f = 0.762. This shows that, in the absence of arbitrage opportunities, the current value of the option must be 0.762. If the value of the option were more than 0.762 euro, the portfolio would cost less than 4.238 euro to set up and would earn more than the risk-free rate. If the value of the option were less than 0.762 euro, shorting the portfolio would provide a way of borrowing money at less than the risk-free rate. In other words, if the value of the option were more than 0.762 euro, for example 1 euro, you can borrow for example 42380 euro and buy 10000 times the above portfolio at a cost of 10000(0.25 × 20 − 1) = 40000euro. You pocket 2380 euro and after 6 months, you sell 10000 portfolio and cashes in 45000, because the value of one portfolio is always 4.5 euro. With this money you pay back the bank for the money you borrowed plus the interests on it, i.e. you pay the bank an amount of 42380 × e0.12×0.5 = 45000 euro. At the end 37

Figure 2.2: General one-period binomial tree of all this you earned 2380 euro without taking any risk and without an initial capital. If the value of the option were less than 0.762 euro, you do the opposite.

Generalization We can generalize the argument just presented by considering a stock whose price is initially S0 and an option on the stock whose current price is f . We suppose that the option lasts for time T and that during the life of the option the stock can move up from S0 to a new level, S0 u or down from S0 to a new level, S0 d (u > 1; 0 < d < 1). If the stock price moves up to S0 u, we suppose that the payoff from the option is fu ; if the stock price moves down to S0 d, we suppose the payoff from the option is fd . The situation is illustrated in Figure 2.2. As before, we imagine a portfolio constisting of a long position in ∆ shares and a short position in one option. We calculate the value of ∆ that makes the portfolio riskless. If there is an up movement in the stock price, the value of the

38

portfolio at the end of the life op the option is S0 u∆ − fu . If there is a down movement in the stock price, the value becomes S0 d∆ − fd . The two are equal when S0 u∆ − fu = S0 d∆ − fd , or ∆=

fu − f d . S0 u − S 0 d

(2.1)

In this case, the portfolio is riskless and must earn the riskless interest rate. If we denote the risk-free interest rate by r, the present value of the portfolio is (S0 u∆ − fu )e−rT = (S0 d∆ − fd )e−rT . The cost of setting up the portfolio is S0 ∆ − f. It follows that (S0 u∆ − fu )e−rT = S0 ∆ − f, or f = S0 ∆ − (S0 u∆ − fu )e−rT .

39

Substituting from equation (2.1) for ∆ and simplifying, this equation reduces to f = e−rT [pfu + (1 − p)fd ]

(2.2)

where p=

erT − d u−d

(2.3)

Remark 2 If we assume that u > erT , together with u > 1 and 0 < d < 1, one can easily show that the value of p given in (2.3) satisfies 0 < p < 1. Note that it is natural to assume that u > erT , because it means that after a time T , you can gain more (a factor u) by investing in the risky stocks, than you can earn with a riskless investment in bond (a factor erT ). If this was not the case no one would invest in stocks. Ofcourse, you can also lose money (d factior by investing in stocks. Remark 3 Equation (2.1) shows that ∆ is the ratio of the change in the option price to the change in the stock price. Remark 4 The option pricing formula in (2.2) does not involve the probabilities of the stock moving up or down. This is suprising and seems counterintuitive. The key reason is that the probabilities of future up or down movements are already incorporated into the price of the stock.

Risk-Neutral Valuation Although we do not need to make any assumptions about the probabilities of an up and down movement in order to derive Equation (2.2), it is natural to 40

interpret the variable p in Equation (2.2) as the probability of an up movement in the stock price. The variable 1−p is then the probability of a down movement, and the expression pfu + (1 − p)fd is the expected payoff from the option. With this interpretation of p, Equation (2.2) then states that the value of the option today is its expected future value discounted at the risk-free rate. We now investigate the expected return from the stock when the probability of an up movement is assumed to be p. The expected stock price at time T , Ep [ST ], is given by Ep [ST ] = pS0 u + (1 − p)S0 d = pS0 (u − d) + S0 d. Substituting from (2.3) for p, this reduces to Ep [ST ] = S0 erT

(2.4)

showing that the stock price grows, on average, at the risk-free rate. Setting the probability of an up movement equal to p is therefore, equivalent to assuming that the return on the stock equals the rsik-free rate. In a risk-neutral world the expected return on all securities is the risk-free interest rate. Equation (2.4) shows that we are assuming a risk-neutral world when we set the proability of an up movement to p. Equation (2.2) shows that the value of the option is its expected payoff in a risk-neutral world discounted at the risk-free rate.

41

This result is an example of an important genereal principle in option pricing known as risk-neutral valuation. The principle states that it is valid to assume the world is risk neutral when pricing options. The resulting option prices are correct not just in a risk-neutral world, but in the real world as well.

The Single-Period Example Revisited We now turn back to the numerical example in Figure 2.1 to illustrate that riskneutral valuation gives the same answers as no-arbitrage arguments. In Figure 2.1, the stock price is currently 20 euro and will move either up to 22 euro or down to 18 euro at the end of six months. The option considered is a European call option with strike price of 21 euro and an expiration date in six months. The risk-free interest rate is 12 percent per annum. We define p as the probability of an upward movement in the stock price in a risk-neutral world. (We know from the analysis given earlier in this section that p is given by Equation (2.3). However, for the purpose of this illustration we suppose that we do not know this.) In a risk-neutral world the expected return on the stock must be the risk-free rate of 12 percent. This means that p must satisfy 22p + 18(1 − p) = 20e0.12×0.5 or p=

20e0.12×0.5 − 18 = 0.8092 4

At the end of the six months, the call option has a 0.8092 probability of being 42

worth 1 euro and a 0.1908 probability of being worth zero. Its expected value is, therefore, 0.8092 × 1 + 0.1908 × 0 = 0.8092 In a risk-neutral world, this should be discounted at the risk-free rate. The value of the option today is, therefore, 0.8092e−0.12×0.5 = 0.7620 This is the same value as the value obtained earlier, illustrating that no-arbitrage arguments and risk-neutral valuation give the same answer.

2.2

Two-Step Binomial Trees

We can extend the analysis to a two-step binomial tree. The objective of the analysis is to calculate the option price at the initial node of the tree. This can be done by repeatedly applying the principles established earlier in the chapter.

2.2.1

European Call

We can first apply the analysis to a two-step binomial tree. Here the stock price starts at 20 euro and in each of the two time steps may go up by 10 percent or down by 10 percent. We suppose that each time step is six months long and the risk-free interest rate is 12 percent per annum. We consider a European call option with a strike price of 21 euro. Figure 2.3 shows the tree with both the stock price and the option price at each node. (The stock price is the upper number and the option price is the lower number.) 43

Figure 2.3: Two-period binomial tree example The option prices at the final nodes of the tree are easily calculated. They are the payoffs from the option. At node D, the stock price is 24.2 euro and the option price is 24.2 − 21 = 3.2 euro; at nodes E and F, the option is out of the money and its value is zero. At node C, the option price is zero, because node C leads to either node E or node F and at both nodes the option price is zero. Next, we calculate the option price at node B. Using the notation introduced earlier in the chapter, u = 1.1, d = 0.9, r = 0.12, and T = 0.5 so that p = 0.8092. Equation (2.2) gives the value of the option at node B as e−0.12×0.5 [0.8092 × 3.2 + 0.1908 × 0] = 2.4386 It remains for us to calculate the option at the initial node, A. We do so by focusing on the first step of the tree. We know that the value of the option at node B is 2.4386 and that at node C it is zero. Equation (2.2), therefore, gives

44

Figure 2.4: General two-period binomial tree the value at node A as e−0.12×0.5 [0.8092 × 2.4386 + 0.1908 × 0] = 1.8583 The value of the option is 1.8583 euro. We can generalize the case of two time steps by considering the situation in Figure 2.4. The stock price is initially S0 . During each step, it either moves up to u times its value or moves down to d times its value. The notation for the value of the option is shown on the tree. For example, after two up movements, the value of the option is fuu . We suppose that the risk-free interest rate is r and the length of the time step is ∆t years. Repeated application of Equation (2.2) gives fu

= e−r∆t [pfuu + (1 − p)fud ]

(2.5)

fd

= e−r∆t [pfud + (1 − p)fdd ]

(2.6)

= e−r∆t [pfu + (1 − p)fd ]

(2.7)

f

45

Substituting the first two equations in the last one, we get f = e−2r∆t [p2 fuu + 2p(1 − p)fud + (1 − p)2 fdd ].

(2.8)

This is constistent with the principle of risk-neutral valuation mentioned earlier. The variable p2 , 2p(1 − p), and (1 − p)2 are the probabilities that the upper, middle, and lower final nodes will be reached. The option price is equal to its expected payoff in a risk-neutral world discounted at the risk-free interest rate. As we add more steps to a binomial tree, the risk-neutral valuation principle continues to hold. The option price is always equal to the present value (discounting at the risk-free interest rate) of its expected payoff in a risk-neutral world.

2.2.2

Matching Volatility with u and d

In practice, when constructing a binomial tree to represent the movements in a stock price, we choose the parameters u and d to match the volatility of the stock price. To see how this is done, suppose that the expected return on a stock in the real world is µ: The expected stock price at the end of the first time step is S0 (1 + µ∆t). The volatility of a stock price, σ, is defined so that σ 2 ∆t is the variance of the return in a short period of time of length ∆t. Suppose from empirical data we estimated that the probability of an up movement in the real world is equal to q. In order to match the expected return on the stock, we

46

must therefore, have qS0 u + (1 − q)S0 d = S0 (1 + µ∆t), or q=

(1 + µ∆t) − d u−d

(2.9)

The variance of the stock price return is qu2 + (1 − q)d2 − [qu + (1 − q)d]2 . In order to match the real world stock price volatility we must therefore have qu2 + (1 − q)d2 − [qu + (1 − q)d]2 = σ 2 ∆t. or equivalently q(1 − q)(u − d)2 = σ 2 ∆t.

(2.10)

Substituting from Equation (2.9)into Equation (2.10) we get ((1 + µ∆t) − d) (u − (1 + µ∆t)) = σ 2 ∆t When terms in (∆t)2 and higher powers of ∆t are ignored (remember ∆t is supposed to be small), one solution to this equation is √ u = (1 + σ ∆t)

(2.11)

√ d = (1 − σ ∆t)

(2.12)

47

Indeed, ((1 + µ∆t) − d) (u − (1 + µ∆t)) = −(1 + µ∆t)2 + (1 + µ∆t)(u + d) − ud √ √ = −1 − 2µ∆t − (µ∆t)2 + 2(1 + µ∆t) − (1 + σ ∆t)(1 − σ ∆t) = −(µ∆t)2 + σ 2 ∆t

Another setting is u = eσ

√ ∆t

d = e−σ

√ ∆t

(2.13) ,

(2.14)

which is, because ∆t is supposed to be small, approximatelly the same as (2.11). These are the values proposed by Cox, Ross and Rubinstein. Note that in both cases the values of u and d are independent of µ, which implies that if we move from the real world to the risk-neutral world the volatility on the stock remains the same (at least in the limit as ∆t tends to zero). This is an illustration of an important general result known as Girsanov’s theroem. When we move from a world with one set of risk preferences to a world with another set of risk preferences, the expected growth rates change, but their volatilities remain the same. Moving from one set of risk preferences to another is sometimes referred to as changing the measure.

48

2.3

Binomial Trees

The above one- and two-steps binomial trees are very imprecise models of reality and are used only for illustrative purposes. Clearly an analyst can expect to obtain only a very rough approximation to an option price by assuming that the stock movements during the life of the option consist of one or two binomial steps. When binomial trees are used in pratice, the life of the option is typically divided into 30 or more time steps of length ∆t. In each time step there is a binomial stock movement. With 30 time steps this means that 31 terminal stock prices and 230 possible stock price paths are considered.

2.3.1

European Call and Put Options

Consider the evaluation of an option on a non-dividend-paying stock. We start by dividing the life of the option into a large number of small intervals of length ∆t. We assume that in each time interval the stock price moves from its initial value S to one of two new values Su and Sd. In general, u > 1 and 0 < d < 1. The movement from S to Su is, therefore, an ”up” movement and the movement from S to Sd is a ”down” movement. In the above sections we introduced what is known as the risk-neutral valuation principle. This states that any security which is dependent on a stock can be valued on the assumption that the world is risk neutral. It means that for the purposes of valuing an option, we can assume: • The expected return from all traded securities is the risk-free interest rate.

49

• Future cash flows can be valued by discounting their expected values at the risk-free interest rate. We make use of this when using a binomial tree. The tree is designed to represent the behavior of a stock price in a risk-neutral world. In this risk-neutral world the probability of an up movement will be denoted by p. The probability of a down movement is 1 − p; as seen above in (2.3): p=

er∆t − d . u−d

As mentioned above, a popular way of chosing the parameters u and d is u = eσ d

√ ∆t

= e−σ

√ ∆t

Figure 2.5 illustrates the tree of stock prices over 5 time periods that is considered when the binomial model is used. At time zero, the stock price S0 is known. At time ∆t there are two possible stock prices, S0 u and S0 d; at time 2∆t, there are three possible stock prices, S0 u2 , S0 ud, and S0 d2 ; and so on. In general, at time i∆t, i + 1 stock prices are considered. These are S0 uj di−j ,

j = 0, . . . , i.

European call and put options are evaluated by starting at the end of the tree (time T ) and working backward. The value of the option is known at time T . For example, a European put option is worth max{K − ST , 0} and a European call option is worth max{ST − X, 0}, where ST is the stock price at time T and 50

Figure 2.5: General binomial tree for stock price K is the strike price. Because a risk-neutral world is being assumed, the value at each node at time T − ∆t can be calculated as the expected value at time T discounted at rate r for a time period ∆t. Similarly, the value at each node at time T − 2∆t can be calculated as the expected value at time T − ∆t discounted for a time period ∆t at rate r, and so on. Eventually, by working back through all the nodes, the value of the option at time zero is obtained. This procedure is illustrated in Figure 2.6. Another way of calculating the option prices is by directly taking the discounted value of the expected payoff of the option in the risk-neutral world. For example the European put, with strike price K and maturity T has a value:   e−rT

N X  N    max{K − S0 uj dN −j , 0}pj (1 − p)N −j   j=0 j

For more complex options, but where the payoff only depends on the final stock price, i.e. the payoff is a function of ST , g(ST ) say, a similar expression can be 51

Figure 2.6: General binomial tree for stock price derived; the current value of the option is then given by:   e−rT Ep [g(ST )] = e−rT

N X  N    g(S0 uj dN −j )pj (1 − p)N −j ,   j=0 j

where Ep denotes the expectation in the risk-neutral world, i.e. with a probability p given by (2.3) of an up-move of size u , and a probability of a down-move of (1 − p), or equivalently, with a probability  

 N  j   p (1 − p)N −j   j

(2.15)

of ending with a time T stock price of S0 uj dN −j . The distribution (2.15) is called the Binomial distribution.

52

2.3.2

American Options

If the option is American, the procedure only changes slightly. It is necessary to check at each node to see whether early exercise is preferable to holding the option for a further time period ∆t. Eventually, again by working back through all the nodes the value of the option at time zero is obtained.

American put option Consider a five-month American put option on a non-dividend-paying stock when the current stock price is 50 euro, the strike price is also 50 euro, the riskfree interest rate is 10 percent per annum, and the volatility is 40 percent per annum. With our usual notation, this means that S0 = 50, K = 50, r = 0.10, σ = 0.40, and T = 152/365 = 0.416. Suppose that we divide the life of the option into five intervals of length one month (= 0.0833 year) for the purposes of constructing a binomial tree. Then ∆t = 0.0833 u = eσ

√ ∆t

d = e−σ

= 1.1224

√ ∆t

= 0.8909

p = (er∆t − d)/(u − d) = 0.5073

Figure 2.7 shows the related binomial tree. At each node there are two numbers. The top one shows the stock price at the node; the lower one shows the value of the option at the node. The 53

Figure 2.7: Binomial tree for American put option probability of an up movement is always 0.5073; the probability of a down movement is always 0.4927. The stock price at the jth node (j = 0, 1, . . . , i) at time i∆t (i = 0, 1, 2, 3, 4, 5) is calculated as S0 uj di−j . The option prices at the final nodes are calculated as max{K − ST , 0}. The option prices at the penultimate nodes are calculated from the option prices at the final nodes. First, we assume no exercise of the option at the nodes. This means that the option price is calculated as the present value of the expected option price one step later. For example at node C, the option price is calculated as (0.5073 × 0 + 0.4927 × 5.45)e−0.10×0.0833 = 2.66

54

whereas at node A it is calculated as (0.5073 × 5.45 + 0.4927 × 14.64)e−0.10×0.0833 = 9.90 We then check to see if early exercise is preferable to waiting. At node C, early exercise would give a value for the option of zero because both the stock price and the strick price are 50 euro. Clearly it is best to wait. The correct value for the option at node C is, therefore, 2.66 euro. At node A, it is a different story. If the option is exercised, it is worth 50 − 39.69 = 10.31 euro. This is more than 9.90. If node A is reached, the option should therefore, be exercised and the correct value for the option at node A is 10.31 euro. Option prices at earlier nodes are calculated in a similar way. Note that it is not always best to exercise an option early when it is in the money. Consider node B. If the option is exercised, it is worth 50 − 39.69 = 10.31 euro. However, if it is held, it is worth (0.5073 × 6.38 + 0.4927 × 14.64)e−0.10×0.0833 = 10.36 The option should, therefore, not be exercised at this node, and the correct option value at the node is 10.36 euro. Working back through the tree, we find the value of the option at the initial node to be 4.49 euro. This is our numerical estimate for the option’s current value. In practice, a smaller value of ∆t, and many more nodes, would be used. It can be shown that with 30, 50, and 100 time steps we get values for the option of 4.263, 4.272, and 4.278. In general suppose that the life of an American put option on a non-dividend55

paying stock is divided into N subintervals of length ∆t. We will refer to the jth node at time i∆t as the (i, j) node. Define fi,j as the value of the option at the (i, j) node. The stock price at the (i, j) node is S0 uj di−j . Because the value of an American put at its expiration date is max{K − ST , 0}, we know that fN,j = max{K − S0 uj dN −j , 0},

j = 0, 1, . . . , N

There is a probability, p, of moving from the (i, j) node at time i∆t to the (i + 1, j + 1) node at time (i + 1)∆t, and a probability 1 − p of moving from the (i, j) node at time i∆t to the (i + 1, j) node at time (i + 1)∆t. Assuming no early exercise, risk-neutral valuation gives fi,j = e−r∆t (pfi+1,j+1 + (1 − p)fi+1,j ) for 0 ≤ i ≤ N − 1 and 0 ≤ j ≤ i. When early exercise is taken into account, this value for fi,j must be compared with the option’s intrinsic value, and we obtain  fi,j = max K − S0 uj di−j , e−r∆t (pfi+1,j+1 + (1 − p)fi+1,j ) Note that, because the calculations start at time T and work backward, the value at time i∆t captures not only the effect of early exercise possibilities at time i∆t, but also the effect of early exercise at subsequent times. In the limit as ∆t tends to zero, an exact value for the American put is obtained. In practice, N = 30 usually gives reasonable results.

56

It is never optimal to exercise an American call option We are now going to proof that for a non-dividend paying stock the price of a European call and an American call are the same. This means that an early exercise of an American call is never optimal. To prove this striking result we first proof Proposition 5 The current price C of a European (and American) call option, with strike price K and time to expiry T , on a non-dividend paying stock with current price S satisfies : C ≥ max{S − e−rT K, 0}. Proof: That C ≥ 0 is obvious, otherwise ’buying’ the call would give a riskless profit now and no obligations later. To prove the remaining lower bound, we setup an arbitrage table (Table 2.1) to examine the cash flows of the following portfolio: sell 1 stock short, buy 1 call, invest in bank account e−rT K. Assuming the condition C ≥ S − e−rT K is violated, i.e. C < S − e−rT K we get the arbitrage Table 2.1. So in all possible states of the world at expiry we have a non-negative return for a portfolio, which has a positive current cash flow. This is clearly an arbitrage opportunity and hence our assumption was wrong. • Suppose now that the American call is exercised at some time t strictly less than expiry T , i.e. t < T . The financial agent thereby realises a cash-flow 57

Portfolio

Current cash flow

Value at expiry ST ≤ K

ST > K

Short 1 stock

S

−ST

−ST

Buy 1 call

−C

0

ST − K

Bank account

−e−rT K

K

K

Balance

S − C − e−rT K ≥ 0

K − ST ≥ 0

0

Table 2.1: Arbitrage table for bounds on calls St − K. From the above proposition we know that the value of the call must be greater or equal to St − e−r(T −t) K, which is greater than St − K. Hence selling the call would have realised a higher cash-flow and the early exercise of the call was suboptimal. In conclusion: CA = C E There are two reasons why an American call should not be exercised early. • Insurance: An investor which holds the call option does not care if the share price falls far below the strike price - he just discards the option but if he held the stock, he would. Thus the option insures the investor against such a fall in stock price, and if he exercises early, he loses this insurance. • Interest on the strike price: When the holder exercises the option, he buys the stock and pays the strike price, K. Early exercise at time t < T 58

deprives the holder of the interest on K between times t and T : the later he pays out K, the better. Notice how this changes when we consider American puts in place of calls: The insurance aspect above still holds, but the interest aspect above is reversed (the holder receives cash K at the exercise time, rather than paying it out).

2.4

Moving towards The Black-Scholes Model

By creating a tree with more and more time steps, that is by taking smaller and smaller time-steps, we can get finer and finer graduations at the final stage and thus hopefully a more accurate price. However, we have to be a little careful about how we do this in order to get the prices to converge to a meaningful value. Which limiting price we obtain will depend on how we make the tree finer - this essentially comes down to assumptions we make about the random process the asset follows. Let us try to price an option with payoff function f (ST ) and we will refine the Cox-Ross-Rubenstein model with choices u = eσ

√ ∆t

d = e−σ

√ ∆t

(2.16) .

(2.17)

Taking N time steps we have that risk-neutral probaility of moving upwards equals: pN

p exp(rT ) − exp(−σ T /N ) p p = exp(σ T /N − exp(−σ T /N)) 59

Let us now investigate the risk-neutral limiting distribution of ST :   N N √ Y X √ ST = S 0 eZj σ T /N = S0 exp σ ∆t Zj  , j=1

j=1

where Zj are independent random variables taking the values −1 and 1, with probabilities pN and 1 − pN respectively, for j = 1, . . . , N . In other words: log ST = log S0 + σ

N X p T /N Zj . j=1

Now we can apply the Central Limiting Theorem (CLT). Theorem 6 (CLT) Assume X1 , X2 , . . . is a series of independent random varoables, all with the same distribution as X of which the second moment is finite. Then PN

j=1 −N E[X] p →D N , N Var[X]

with N a standard Normal distributed variable (with mean zero and variance equal to one). We note that E[Zj ] = 2pN − 1; Var[Zj ] = 4pN (1 − pN ). Hence, a simple calculation, using

q

p N E[Zj ] T /Nσ

Var[Zj ]N

p T /Nσ

→ →

60

(r − (1/2)σ 2 )T ; √ σ T.

leads to √

 1 2 log ST →D log S0 + σ T N + r − σ T 2 

when N → +∞. The distribution of the logarithm of the stock price thus follows
a Normal distribution with mean r − 21 σ 2 T and variance σ 2 T ; the stock price

itself is thus lognormally distributed.

The price of the derivative in the limit will be given by      √ 1 2 lim exp(−rT )EpN [g(ST )] = exp(−rT )E g S0 exp(σ T N + r − σ T . N →∞ 2 In case of the European call option with strike K and time to maturity T , one can with a little effort show that its initial price is given by: C(K, T ) = S0 N(d1 ) − K exp(−rT )N(d2 ), where d1

=

ln(S0 /K) + (r + √ σ T

σ2 2 )T

d2

=

ln(S0 /K) + (r − √ σ T

σ2 2 )T

(2.18) √ = d1 − σ T

(2.19)

and N (x) is the cumulative probability distribution function for a variable that is standard normal distributed. This is the famous Black-Scholes formula. This lognormal model (the Black-Scholes model), will be studied in detail in the course ”Continuous Financial Mathematics”.

61

Chapter 3

Mathematical Finance in Discrete Time Any variable whose value changes over time in an uncertain way is said to follow a stochastic p.rocess. Stochastic processes can be classified as discretetime or continuous-time. A discrete-time stochastic process is one where the value of the variable can change only at certain fixed points in time, whereas a continuous-time stochastic process is one where changes can take place at any time. Stochastic processes can also be classified as continuous-variables or discrete-variables. In a continuous-variable process, the underlying variable can take any value within a certain range, whereas in a discrete-variable process, only certain discrete values are possible. Binomial tree models belong to the discrete-time, discrete-variable stochastic processes.

62

In this chapter we study so-called finite markets, i.e. discrete-time models of financial markets in which all relevant quantities take a finite number of values. We specify a time horizon T , which is the terminal date for all economic activities considered. For a simple option pricing model the time horizon typically corresponds to the expiry date of the option. We thus work with a finite probability space (Ω, P ), with a finite number |Ω| of possible outcomes ω, each with a positive probability: P ({ω}) > 0.

3.1

Information and Trading Strategies

Access to full, accurate, up-to-date information is clearly essential to anyone actively engaged in financial activity or trading. Indeed, information is arguably the most important determinant of success in financial life. We shall confine ourselves to the situation where agents take decisions on the basis of information in the public domain, available to all. We shall further assume that information once known remains known and can be accessed in real time. Our financial market contains two financial assets. A risk-free asset (the bond) with a deterministic price process Bi , and a risky assets with a stochastic price process Si . We assume B0 = 1 (we reckon in units of the initial value of the bond) and Bi > 0; we say it is a numeraire. 1/Bi is called the discounting factor at time i. As time passes, new information becomes available to all agents. There exists a mathematical object to model this information flow, unfolding with

63

time: filtrations. The concept filtration is not that easy to understand. The full theory will lead us too far. In order to clear this out a bit, we explain the idea of filtration in a very idealized situation. We will consider a stochastic process X which starts at some value, zero say. It will remain there until time t = 1, at which it can jump with positive probability to the value a or to a different value b. The process will stay at that value until time t = 2 at which it will jump again with positive probability to two different values: c and d say if is was at time t = 1 at a and f and g say if the process was at time t = 1 at state b. From then on the process will stay in the same value. The universum of the probability space consists of all possible paths the process can follow, i.e. all possible outcomes of the experiment. We will denote the path 0 → a → c by ω1 , similarly the paths 0 → a → d, 0 → b → f and 0 → b → g are denoted by ω2 , ω3 and ω4 respectively. So we have Ω = {ω1 , ω2 , ω3 , ω4 }. In this situation we will take the following flow of information, i.e. filtrations: Ft =

{∅, Ω}

0 ≤ t < 1;

Ft = {∅, Ω, {ω1, ω2 }, {ω3 , ω4 }} 1 ≤ t < 2; Ft =

D(Ω) = F

2 ≤ t.

We set here F = D(Ω), the set of all subsets of Ω. To each of the filtrations given above, we associate resp. the following par-

64

titions (i.e. the finest possible one) of Ω: P0 =

{Ω}

0 ≤ t < 1;

P1 =

{{ω1 , ω2 }, {ω3, ω4 }}

1 ≤ t < 2;

P2 = {{ω1 }, {ω2 }, {ω3}, {ω4 }} 2 ≤ t. At time t = 0 we only know that some event ω ∈ Ω will happen, at time t = 2 we will know which event ω ∗ ∈ Ω has happened. So at times 0 ≤ t < 1 we only know that some event ω ∗ ∈ Ω. At time point after t = 1 and strictly before t = 2, i.e. 1 ≤ t < 2, we know to which state the process has jumped at time t = 1: a or b. So at that time we will know to which set of P1 , ω ∗ will belong: it will belong to {ω1 , ω2 } if we jumped at time t = 1 to a and to {ω3 , ω4 } if we jumped to b. Finally, at time t = 2, we will know to which set of P2 , ω ∗ will belong, in other words we will know then the complete path of the process. During the flow of time we thus learn about the partitions. Having the information Ft revealed is equivalent to knowing in which set of the partition of that time, the event ω ∗ is. The partitions become finer in each step and thus information on ω ∗ becomes more detailed. We thus keep in mind that a filtration

F = (Fi , i = 0, 1, . . . , T ) exists of a

sequence of mathematical objects (σ-algebras), F0 ⊂ F1 ⊂ · · · ⊂ FT , describing the information available. At time i we have access to information in Fi . It is clear that the price of the stock Si at time i (and i − 1, i − 2, ..., 0) is contained in the information Fi . If a random variable X is known with respect to the information G we say 65

it is G-measurable. So we have that Si is Fi -measurable. A stochastic process {Xi , i = 0, 1, . . . , T } is called adapted to the filtration (or just

G = (Gi , i = 0, 1, . . . , T )

G−adapted) if at every time point i = 0, 1, . . . , T the random variable

Xi is Gi -measurable. So we have that S = {Si , i = 0, 1, . . . , T } is

F-adapted.

A trading strategy ϕ = {ϕi = (βi , ζi ), i = 1, . . . , T } is a real vector stochastic process such that each ϕi is Fi−1 -adapted. Here βi , ζi denotes the numbers of bonds ands stocks resp. held at time i and to be determined on the basis of information available strictly before time i: Fi−1 ; i.e. the investor selects his time i portfolio after observing the prices Si−1 . The components βi , ζi may assume negative values as well as positive values, reflecting the fact that we allow short sales and assume that the assets are perfectly divisible. The value of the portfolio ϕ at time i, Viϕ = Vi , is called the wealth or value process of the trading strategy: Viϕ = Vi = βi Bi + ζi Si ,

i = 1, 2, . . . , T

We will denote by V0 the initial investment or endowment of the investor. Now βi Bi−1 + ζi Si−1 reflects the market value of the portfolio just after it has been established at time i − 1, whereas βi Bi + ζi Si is the value just after time i prices are observed, but before changes are made in the portfolio. Hence βi (Bi − Bi−1 ) + ζi (Si − Si−1 ) is the change in the market value due to changes in security prices which occur between time i − 1 and i. We call Gϕ = G = {Gi , i = 1, . . . , T }, where Gϕ i = Gi =

i X j=1

(βj (Bj − Bj−1 ) + ζj (Sj − Sj−1 )) 66

the gains process. After the new prices (Bi , Si ) are quoted at time i, the investor adjusts his portfolio from ϕi = (βi , ζi ) to ϕi+1 = (βi+1 , ζi+1 ). We do not allow him bringing in or consuming any wealth, so we must have V0 = β 1 B 0 + ζ 1 S 0 ,

Vi = βi+1 Bi + ζi+1 Si ,

i = 1, . . . , T

We say our trading strategy is self-financing and denote this by ϕ ∈ Φ. To avoid negative wealth and unbounded short sales we also introduce the concept of admissible strategies. A self-financing trading strategie ϕ ∈ Φ is called admissible if Viϕ ≥ 0 for each i = 0, 1, . . . , T . We write Φa for the class of admissible trading strategies. Clearly Φa ⊂ Φ.

3.2

No-Arbitrage Condition

The central principle in the Binomial tree models was the absence of arbitrage opportunities, i.e. the absence of risk-free plans for making profits without any investment. As mentioned there this principle is central for any market model, and we now define the mathematical counterpart of this economic principle in our setting. We call a self-financing trading strategy ϕ an arbitrage opportunity if P (V0ϕ = 0) = 1 and the terminal wealth of ϕ satisfies P (VTϕ ≥ 0) = 1 and P (VTϕ > 0) > 0 So an arbitrage opportunity is a self-financing strategy with zero initial value,

67

which produces a non-negative final value with probability one and has a postive probability of a positive value. We say that our market is arbitrage-free if there are no self-financing trading strategies which are arbitrage opportunities. We will link the economic principle of an arbitrage free market to a mathematical one: the existence of an equivalent martingale. We say a probability measure P ∗ on Ω is equivalent to P , if it has the same null sets. Here it means P ∗ ({ω}) > 0. We say a probability measure Q on Ω is a martingale measure for a process X = {Xi , i = 0, 1, . . . , T }, if Xi is a Q-martingale with respect to the filtration • X is

F, i.e.

F-adapted

• EQ [Xi |Fi−1 ] = Xi−1 ,

i = 1, . . . , T

Note that in a more general context a third condition is required: EQ [|Xi |] < ∞. Because we work in a finite probability space this condition is in our setting automatically satisfied. One can show that the secound condition is equivalent to EQ [Xi |Fj ] = Xj ,

0 ≤ j ≤ i ≤ T.

We denote by P(X) the class of equivalent martingale measures for X and ˜ for the discounted version of the process X : X ˜i = will use the notation X Bi−1 Xi . For eaxmple, we will denote by S˜ the discounted stock price process : S˜i = Bi−1 Si . 68

As a kind of example of the above concepts, we show the following proposittion which we will later on need to prove one direction of the No-Arbitrage Theorem. ˜ and ϕ ∈ Φ, then V˜ is a P ∗ -martingale. Proposition 7 Let P ∗ ∈ P(S) Proof: First note that V˜iϕ = Bi−1 (βi Bi + ζi Si ) and since Bi , Si , βi , ζi ∈ Fi , we also have that V˜iϕ ∈ Fi . Hence V˜ ϕ is F-adapted. Next, we will prove EP ∗ [V˜i |Fi−1 ] = V˜i−1 , i = 1, . . . , T . We have EP ∗ [V˜i |Fi−1 ] = EP ∗ [Bi−1 (βi Bi + ζi Si )|Fi−1 ]; = βi + ζi EP ∗ [Bi−1 Si )|Fi−1 ]; −1 = βi + ζi Bi−1 Si−1 ; −1 = Bi−1 (βi Bi−1 + ζi Si−1 ); −1 = Bi−1 (βi−1 Bi−1 + ζi−1 Si−1 );

=

V˜i−1 ,

where the third line is because S˜ is a P ∗ -martingale and the fifth line is because of the self-financing property of ϕ. ♦ The next result is the key-result in discrete mathematical finance. Theorem 8 (No-Arbitrage Theorem) The market is arbitrage-free if and only if there exists an equivalent martingale measure for the discounted price ˜ 6= ∅. process of the stock S˜i = Bi−1 Si , i.e. P(S) 69

˜ 6= ∅ implies that the market is arbitrageProof : We only prove that P(S) free; the other direction can be proven using the Hahn-Banach theorem from Functional Analysis. ˜ 6= ∅ and let P ∗ ∈ P(S). ˜ For any self-financing strategy ϕ ∈ Φ, Assume P(S) we have from the above proposition that V˜ ϕ is a P ∗ -martingale. So EP ∗ [VTϕ ] = V0ϕ . Suppose ϕ is an arbitrage opportunity. Then P (V0ϕ = 0) = 1, so P ∗ (V0ϕ = 0) = 1 and thus EP ∗ [VTϕ ] = 0. We must have P ∗ (VTϕ ≥ 0) = 1 and P ∗ (VTϕ > 0) > 0. Together with P ∗ ({ω}) > 0, this leads to a contradiction. ♦ One can show that a security market which has no arbitrage opportunities in Φa , is also arbitrage-free with respect to Φ.

3.3

Risk-Neutral Pricing

We now turn to the main underlying question of this text, namely the pricing of contingent claims (i.e. financial derivatives). First we have to model these financial instruments in our current framework. This is done in the following fashion. Definition 9 A contingent claim X with maturity date T is an arbitrary nonnegative FT -measurable random variable. We denote the class of all contingent claims by X . 70

We say that the claim is attainable if there exists an (admissible) selffinancing strategy ϕ ∈ Φ such that VTϕ = X. The self-financing strategy ϕ ∈ Φ is said to be a replicating strategy. It generates the same time T cash-flow as X does. We now return to the main question of the section: given a contingent claim X, i.e. a cash-flow at time T , how can we determine its value (price) at time i < T ? For attainable contingent claims this value should be given by the value of any replicating strategy (perfect hedge) at time i, i.e. there should be a unique value process (say ViX ) representing the time i value of the claim X. The following proposition ensures that the value process of replicating strategies coincide, thus proving uniqueness of the value process. Proposition 10 Suppose the market is arbitrage-free. Then any attainable contingent claim X is uniquely replicated: for all ϕ, ψ ∈ Φ such that VTϕ = VTψ = X we have that for all 0 ≤ i ≤ T Viϕ = Viψ This uniqueness property allows us now to define the important concept of an arbitrage price process. Definition 11 Suppose the market is arbitrage free. Let X be any attainable contingent claim with time T to maturity. Then the arbitrage price process π iX , 71

0 ≤ i ≤ T or simply the arbitrage price of X is given by the value process of any replicating strategy ϕ for X. The construction of hedging strategies that replicate the outcome of a contingent claim is an important problem in both practical and theoretical applications. Hedging is central to the theory of option pricing. The classical arbitrage valuation models, such as the Binomial tree models and the BlackScholes Model (see the next Chapters), depend on the idea that an option can be perfectly hedged using the underlying risky asset and a risk-free asset. Analysing the arbitrage-pricing approach we observe that the derivation of the price of a contingent claim doesn’t require any specific preferences of the agents other than that they prefer more to less, which rules out arbitrage. So, the pricing formula for any attainable contingent claim must be independent of all preferences that do not admit arbitrage. In particular, an economy of risk-neutral investors must price a contigent claim in the same manner. This fundamental insight simplifies the pricing formula enormously. In its general form the price of an attainable contingent claim is just the expected value of the discounted payoff with respect to an equivalent martingale measure. Proposition 12 The arbitrage price process of any attainable contingent claim X is given by the risk-neutral valuation formula πiX =

Bi EP ∗ [X|Fi ], BT

i = 0, 1, . . . , T

where EP ∗ is the expectation operator with respect to an equivalent martingale measure P ∗ . 72

Proof: Since we assume that the market is arbitrage-free there exists (at least) an equivalent martingale measure P ∗ for the discounted price process S˜i . Furthermore because the claim is attainable there exists (at least) one self-financing replicating strategy ϕ. First we prove that the discounted value process V˜iϕ = Bi−1 Viϕ is a P ∗ -martingale: Indeed, by the self-financing property of ϕ = (βi , ζi ) ϕ EP ∗ [V˜iϕ |Fi−1 ] − V˜i−1 ϕ = EP ∗ [V˜iϕ − V˜i−1 |Fi−1 ] ϕ −1 = EP ∗ [Bi−1 Viϕ − Bi−1 Vi−1 |Fi−1 ] −1 = EP ∗ [Bi−1 (βi Bi + ζi Si ) − Bi−1 (βi−1 Bi−1 + ζi−1 Si−1 )|Fi−1 ] −1 = EP ∗ [Bi−1 (βi Bi + ζi Si ) − Bi−1 (βi Bi−1 + ζi Si−1 )|Fi−1 ] −1 = EP ∗ [ζi (Bi−1 Si − Bi−1 Si−1 )|Fi−1 ]

= ζi EP ∗ [S˜i − S˜i−1 |Fi−1 ] = 0.

73

The last equality follows because S˜i = Bi−1 Si is P ∗ -martingale. So we have for each i = 0, 1, . . . , T πiX

= Viϕ = Bi V˜iϕ = Bi EP ∗ [V˜Tϕ |Fi ] = Bi EP ∗ [BT−1 VTϕ |Fi ] = (Bi /BT )EP ∗ [VTϕ |Fi ] = (Bi /BT )EP ∗ [X|Fi ]

3.4

Complete Markets

The last section made clear that attainable contingent claims can be priced using an equivalent martingale measure. In this section we will discuss the question of the circumstances under which all contingent claims are attainable. This would be a very desirable property of the market, because we would then have solved the pricing question (at least for contingent claims) completely under the assumption that the market is arbitrage free. Since contingent claims are merely non-negative FT -measurable random variables in our setting, it should be no suprise that we can give a criterion in terms of probability measures. We start with: Definition 13 A market is complete if every contingent claim is attainable, i.e. for every non-negative FT -measurable random variables X ∈ X there exists a 74

replicating strategy ϕ ∈ Φ such that VTϕ = X. In the case of an arbitrage-free discrete market, one can insist on replicating contingent claims by an admissible strategy ϕ ∈ Φa . Based on the no-arbitrage assumption one can prove: Theorem 14 (Completeness Theorem) An arbitrage free market is complete if and only if there exists a unique probability measure P ∗ equivalent to P under which the discounted price process of the stock S˜i = Bi−1 Si is a martin˜ = {P ∗ }. gale, i.e. P(S)

3.5

The Fundamental Theorem of Asset Pricing

We summarise what we have achieved so far. We call a measure P ∗ under which the discounted price S˜ is a P ∗ -martingale a martingale measure. Such a P ∗ equivalent to the actual probability measure P is called an equivalent martingale measure. Then: • No-Arbitrage Theorem: A market is arbitrage free if and only if at least one equivalent martingale measure exists. • Completeness Theorem: An arbitrage-free market is complete (all contingent claims can be replicated) if and only if there exists a unique equivalent martingale measure. So

75

Theorem 15 (Fundamental Theorem of Asset Pricing) In an arbitragefree complete market, there exists a unique equivalent martingale measure P ∗ . The above theorem establishes the equivalence of an economic modelling condition such as no-arbitrage and completeness to the existence of the mathematical modelling condition, viz. the existence and uniqueness of equivalent martingale measures. Assume now that the market is arbitrage-free and complete and let X ∈ X be any contingent claim, ϕ a replicting strategy (which exists by completeness), then: VTϕ = X Furthemore, we have seen that πiX = Viϕ =

Bi EP ∗ [X|Fi ], BT

i = 0, 1, . . . , T

and call πiX = Viϕ the the arbitrage price of the contingent claim X at time i. For, if an investor sells the claim X at time i for πiX , he can follow strategy ϕ to replicate X at time T and clear the claim; an investor selling this value is perfectly hedged. To sell the claim for any other amount would provide an arbitrage opportunity. We note that, to calculate prices as above, we need to know only: • Ω the set of all possible states, • the filtration

F,

• P ∗. 76

We do not need to know the underlying probability measure P (only its null sets, to know what ’equivalent to P ’ means and actually in our finite model there are no non-empty null-sets, so we do not need to know even this). Now pricing of contingent claims is our central task, and for pricing purposes P ∗ is vital and P itself irrelevant. We thus may – and shall – focus attention on P ∗ , which is called the risk-neutral probability measure. To summarize, we have: Theorem 16 (Risk-Neutral Pricing Formula) In an arbitrage-free complete market, arbitrage prices of contingent claims are their discounted expected values under the risk neutral (unique equivalent martingale measure) P ∗ .

3.5.1

Examples

The One-step Binomial Model We return to model given in Figure 2.2. There exists only two possible outcomes. There is an upperstate u if price after one time step equals S1 = uS0 and a down-state d if the stock price changes to S1 = dS0 , Ω = {u, d}. In both cases the riskfree asset goes from 1 to a price b say (b is typically equal to er or 1 + r0 ). A probability measure on Ω is completely determined by the number 0 < P ({u}) < 1; we then have P ({d}) = 1 − P ({u}). In order that the discounted price process is a martingale with respect to a (P -equivalent) probability measure P ∗ , with say 0 < P ∗ ({u}) = p∗ < 1, on Ω, it has to satisfy only one equation: EP ∗ [b−1 S1 |F0 ] = S0 77

or equivalently b−1 uS0 p∗ + b−1 dS0 (1 − p∗ ) = S0 .

(3.1)

Rewriting (3.1) gives p∗ =

b−d u−d

(3.2)

In order that this gives rise to a probability measure, we should have 0 < p∗ < 1, which is equivalently with u > b > d ≥ 0.

(3.3)

In conclusion a martingale measure P ∗ ∈ P for the discounted stock price exists if and only if (3.3) is satisfied. If (3.3) holds true, then there is a unique such measure in P characterised by (3.2). So in conclusion, if (3.3) is satisfied the one-step binomial model is arbitrage free and complete. Note that (3.3) means that by investing in a stock one can have a bigger return than the risk-free return (u > b), but also can have a greater loss (b > d). Note also that one can easily show that the multi-period model of Section 2.3 is complete if and only if the underlying single-period model is complete. If we now have a contingent claim with payoff fu in the upstate and fd in the down state, the initial price of this claim is equal to f = b−1 p∗ fu + b−1 (1 − p∗ )fd

78

Figure 3.1: The One-step Trinomial Model In order to hedge or replicated this claim one has to solve the equations ξuS0 + ηb = fu ξdS0 + ηb = fd

Note that this system of equations has a unique solution if and only if    uS0 det   dS0

b   6= 0,  b

which is equivalent with S0 6= 0, b 6= 0, and u 6= d (all which are ruled out). The One-step Trinomial Model Suppose now the following one-step trinomial model: In one time step there exists three possible outcomes as shown in picture 3.1. There is an upperstate u if the stock price changes to S1 = uS0 , a middle state m if the stock price after one step is S1 = mS0 , and a down-state d if the stock price changes to S1 = dS0 , 0 ≤ d < m < u: Ω = {u, m, d}. Again, in all cases the riskfree asset changes in a deterministic way from 1 to a price b say. A probability measure on Ω is now

79

completely determined by two numbers 0 < P ({u}) < 1 and 0 < P ({m}) < 1; we then have P ({d}) = 1 − P ({u}) − P ({m}). In order that the discounted price process is a martingale with respect to a probability measure P ∗ , with say 0 < P ∗ ({u}) = p∗ < 1 and 0 < P ∗ ({m}) = q ∗ < 1, on Ω, it has to satisfy again only one equation: EP ∗ [b−1 S1 |F0 ] = S0 or equivalently b−1 uS0 p∗ + b−1 mS0 q ∗ + b−1 dS0 (1 − p∗ − q ∗ ) = S0 . Unfortunately this equation has more than one solution as can be easily been seen after a simple rewriting: p∗ =

(b − d) − (m − d)q ∗ u−d

For every 0 < q ∗ < 1 there is a corresponding p∗ . If we then take also into account that the values of p∗ and q ∗ must give rise to a probability distribution, i.e. 0 < p∗ , q ∗ < 1 and p∗ + q ∗ < 1, there still are infinitely many solutions. In conclusion there exist more then one martingale measure for the discounted stock price. So the one-step trinomial model is arbitrage free, but is not complete. If we have a contingent claim with payoff fu in the upstate, fm in the middle state and fd in the down state it can only be replicated if there exists a solution

80

to the equations ξuS0 + ηb = fu ξmS0 + ηb = fm ξdS0 + ηb = fd

This is only the case if 

 uS0   det   mS0   dS0



b fu    b fm   = 0.   b fd

Because we assume that S0 6= 0 and b 6= 0, this is equivalent with    u 1 fu   det   m 1 fm   d 1 fd

    = 0.   

So only contingent claims which payoff function satisfies the above condition are attainable and can be replicated and priced in an arbitrage-free way.

81

Chapter 4

Exotic Options Derivatives with more complicated payoffs than the standard European or American calls and puts are referred to as exotics options. Most exotics options are traded in the OTC market and have been designed to meet particular needs of investors. In this chapter we describe different types of exotic options and discuss their valuation. Option of an European nature can typically be price by Monte-Carlo simulation. The main problem with using Monte-Carlo simulation to value path-dependent derivatives is that the computation time necessary to achieve the required level of accuracy can be unaccaptable high. Moreover Americantype option can not be handled.

82

4.1

Monte Carlo Pricing

When the payoff depends on the path followed by the underlying variable S in theory one has to consider every possible path. When using 30 time steps in the Binomial tree model, there are about a billion different paths and one has to relay on (Monte Carlo) simulations, which are computationally very time consuming. The expected payoff in a risk-neutral world is calculated using a sampling procedure. It is then discounted at the risk-free interest rate: 1. Sample a random path for S in a risk-neutral world. 2. Calculate the payoff from the derivative. 3. Repeat steps one and two to get many sample values of the payoff from the derivative in a risk neutral world. 4. Calculate the mean of the sample payoff to get an estimate of the expected payoff in a risk-neutral world. 5. Discount the estimated expected payoff at the risk-free rate to get an estimate of the value of the derivative.

4.2

Lookback Options

In everything we have encountered so far, uncertainty has unfolded with time, and our task has been to make optimal use of the information available to date. For options, at expiry T the investor is in posssesion of the history of the price evolution over time interval [0, T ] of the option’s life, and it may well be that 83

one could have been doing better. It is only natural to look back with regret. If only one could buy at the low, and sell at the high ... In order to provide some investor the right to do that lookback options were created. We write S for the stock price process and consider a time interval [0, T ]. Let us denote the maximum and minimum process, resp., of a process X = {Xt , 0 ≤ t ≤ T } as MtX = sup{Xu ; 0 ≤ u ≤ t} and mX t = inf{Xu ; 0 ≤ u ≤ t},

0 ≤ t ≤ T.

The two basic types of continuously montiored lookback options are the lookback call, with payoff LC cont (T ) = ST − mST , giving one the right to buy at the low over [0, T ], and the lookback put with payoff LP cont (T ) = MTS − ST , giving one the right to sell at the high over [0, T ]. One can approximate their value by their discretely monitored counterparts. Consider an a partition of [0, T ] into n equal time intervals with size ∆t = T /n. Write Si for the stock price value at time i∆t and MiS , mSi for its maximum and minimum over [0, i∆t]. The discretely monitored versions payout LC discr (T ) = Sn − mSn , and LP discr (T ) = MnS − Sn , 84

respectively. We illustrate first how such a discrtely monitored European-type lookback put can be priced using binomial trees. Later, we will comment on the American version. When exercised, this provides a payoff equal to the excess of the current maximum stock price over the current stock price. Set Yi =

MiS Si

and produce a binomial tree for the stock price (using the Cox-Ross-Rubenstein setting). See the left tree in Figure 4.1. From this tree produce a corresponding tree for Y . Initially Y0 = 1, because M0S = S0 . If there is an up-move in S during the first step, both the maximum and the stock price increase by a proportional amount u and remains Y = 1. If there is a down movement in S during the √ first step, the maximum stays at S0 , so that Y = 1/d = 1/ exp(−σ ∆t) = u. Continuing with these types of arguments, we produce the tree shown in Figure 4.1 (σ = 0.40, r = 0.10). The rules defining the geometry of the tree are 1 When Yi = 1, then Yi+1 is either u or 1. 2 When Yi = um , then Yi+1 is either um+1 or um−1 . An up-movement in Y corresponds to a down-movement in S and vice versa. The probability of an up movement in Y is, therefore, always 1 − p, with p the probability of a up-movement in the stock. Note thus that p is also the probability of a down-movement of Y . We will use the Y -tree to value the lookback option in units of the stock

85

price (rather then in euros). In euros, the payoff from the option is S n Yn − S n . In stock price units, the payoff from the option is Yn − 1. We roll back through the tree in the usual way, valuing a derivative that provides this payoff except that we adjust for the differences in the stock prices (i.e. the units of measurement) at the nodes. If fi,j is the value of the lookback at the jth node at time iδt, and Yi,j is the value of Y at this node: Yi,j = uj , the rollback procedure gives eur = exp(−r∆t)((1 − p)fi+1,j+1 d + pfi+1,j−1 u), fi,j

when j ≥ 1. Note that fi+1,j+1 is multiplied by d and fi+1,j−1 is multiplied by u in this equation. This takes into account that the stock price at node (i, j) is the unit of measurement. The stock price at node (i + 1, j + 1), which was the unit of measurement for fi+1,j+1 is d times the stock price at node (i, j). Similarly, the stock price at node (i + 1, j − 1), which was the unit of measurement for fi+1,j−1 is u times the stock price at node (i, j). When j = 0, the roll back procedure gives eur fi,0 = exp(−r∆t)((1 − p)fi+1,1 d + pfi+1,0 u),

The tree is initialized at the final nodes with the boundary conditions eur fn,j

= Yn,j − 1

eur fn,0

= 0. 86

Figure 4.1: Lookback tree example The tree (with 5 time steps) in the Figure 4.1 estimates the value of the option at time zero (in stock price units) as 0.230 for the European version. This means that the value of the option is 0.230 × S0 = 11.50 euros. In case of an American type option, these two equations can be adjusted by comparing the european price with the early exercise price (Yi,j − 1) and taking the maximum of both: amer fi,j

= max {Yi,j − 1, exp(−r∆t)((1 − p)fi+1,j+1 d + pfi+1,j−1 u)} ,

amer fi,0

= exp(−r∆t)((1 − p)fi+1,1 d + pfi+1,0 u),

87

j≥1

the boundary conditions remain the same: amer fn,j

= Yn,j − 1

amer fn,0

= 0.

Increasing the number of time-steps n, will give a more precise estimate of a continuously montinored lookback options. It is quite well known that the treevalues converge slowly to this value. This is due because, one is actually missing all the situations where a maximum/minimum has been attained in between two discrete montoring points and where the stock price has fallen/risen back before the end of that interval.

4.3

Barrier Options

The payoff of a barrier option depends on whether the price of the underlying asset crosses a given threshold (the barrier) before maturity. The simplest barrier options are “knock in” options which come into existence when the price of the underlying asset touches the barrier and “knock-out” options which come out of existence in that case. For example, an up-and-out call has the same payoff as a regular plain vanilla call if the price of the underlying asset remains below the barrier over the life of the option but becomes worthless as soon as the price of the underlying asset crosses the barrier. Let us denote with 1(A) the indicator function, which has a value 1 if A is true and zero otherwise. For single barrier options, we will focus on the following types of call options: 88

• The down-and-out barrier call is worthless unless its minimum remains above some low barrier H, in which case it retains the structure of a European call with strike K. Its initial price is given by: DOBC

= exp(−rT )EQ [(ST − K)+ 1(mST > H)].

• The down-and-in barrier call is a standard European call with strike K, if its minimum went below some low barrier H. If this barrier was never reached during the life-time of the option, the option is worthless. Its initial price is given by: DIBC

= exp(−rT )EQ [(ST − K)+ 1(mST ≤ H)].

• The up-and-in barrier call is worthless unless its maximum crossed some high barrier H, in which case it retains the structure of a European call with strike K. Its price is given by: U IBC

= exp(−rT )EQ [(ST − K)+ 1(MTS ≥ H)].

• The up-and-out barrier call is worthless unless its maximum remains below some high barrier H, in which case it retains the structure of a European call with strike K. Its price is given by: U OBC

= exp(−rT )EQ [(ST − K)+ 1(MTS < H)].

The put-counterparts, replacing (ST − K)+ with (K − ST )+ , can be defined along the same lines.

89

We note that the value, DIBC, of the down-and-in barrier call option with barrier H and strike K plus the value, DOBC, of the down-and-out barrier option with same barrier H and same strike K, is equal to the value C of the vanilla call with strike K. The same is true for the up-and-out together with the up-and-in: DIBC + DOBC = C = U IBC + U OBC.

(4.1)

The above options are so-called continuously monitored. Their value can be approximated by the discretely monitored counterparts, like in the lookback case. These discretely monitored barrier options (of european and american type) can again be priced using the binomial tree setup. For example an American down-and-out put can be valued as in the same way as an regular American option except that, when we encounter a node below the barrier, we set the value at that note equal to zero. With the usual notation we have for 0 ≤ i < n fi,j fi,j

 = max K − S0 uj di−j , exp(−r∆t)((1 − p)fi+1,j+1 + pfi+1,j ) if S0 uj di−j ≥ H

= 0 if S0 uj di−j < H

and for i = n fn,j

= K − S0 uj dn−j if Sn ≥ H

fn,j

= 0 if Sn < H

Similar schemes can be easily deduced for the other combinations. Unfortunately, convergence of the price of the discretely monitored option to the price 90

of the continuouss is also here very slow when this approach is used. A large number of time steps is required to obtain a reasonably accurate result. The reason for this is that the barrier being assumed by the tree is different fom the true barrier. Define the inner barrier as the barrier formed by nodes on the side of the true barrier (i.e., closer to the center of the tree) and the outer barrier as the barrier formed by nodes just outside the true barrier (i.e., farther away from the center of the tree). Figure 4.2 shows the inner and outer barrier for a trinomial tree on the assumption the true barrier is horizontal. Figure 4.3 does the same for a binomial tree. The usual tree calculations implicitly assume that the outer barrier is the true barrier because the barrier conditions are first used at nodes on this barrier. For coping with this barrier-problem, one alternative is to calculate when rolling back through the tree, two values of the derivative (for the nodes on the inner barrier). The first one is obtained by assuming the inner barrier is correct; the second one is obtained by assuming the outer barrier is correct. A final estimate for the value of the derivative for the true barrier is then obtained by interpolating between these two values. For example, suppose that at time i∆t, the true value barrier is 0.2 above the inner barrier and 0.6 below the outer barrier and suppose further that the value of the derivative on the inner barrier is zero if the inner barrier is assumed to be correct and 1.6 if the outer barrier is assumed to be correct. The interpolated value (for the inner barrier node) is then 0.4. Once we have adjusted the value at the inner barrier node, we can roll back through the tree to obtain the initial

91

Figure 4.2: Trinomial tree: inner and outer barrier

92

Figure 4.3: Trinomial tree: inner and outer barrier

93

value of the derivative in the usual way.

4.4

Asian Options

In this section we consider the pricing of a European-style arithmetic average call option with strike price K, maturity T and n averaging days 0 ≤ t1 < . . . < tn ≤ T . Its payoff is given by AAC =

 Pn

k=1

S tk

n

−K

+

.

The american versions allows early exercise and in that case pays out the surplus over the strike price K of the running average. For the put version just switch the sum and the strike price : AAP =



K−

Pn

k=1

n

S tk

+

.

Average price options are typically less expensive than regular options and are arguably more appropriate than regular options for meeting some of investors needs. Asian options are widely used in pratice - for instance, for oil and foreign currencies. The averaging complicates the mathematics, but e.g., protects the holder against speculative attemps to manipulate the asset price near expiry. Assume for simplicity that t = 0 and that the averaging has not yet started. First note, that for any K1 , . . . , Kn ≥ 0 with K = n X

k=1

Stk − nK

!+

Pn

k=1

Kk , we have

n  + X + = (St1 − nK1 ) + · · · + (Stn − nKn ) ≤ (Stk − nKk ) . k=1

94

Hence the intial price AAC0 (K, T ) AAC0 (K, T ) =

  !+ n X exp(−rT )  EP ∗  Stk − nK F0 n k=1

≤

exp(−rT ) n

=

exp(−rT ) n

n X

k=1 n X

i h + EP ∗ (Stk − nKk ) F0 exp(rtk )EC0 (κk , tk ),

(4.2)

k=1

where EC0 (κk , tk ) denotes the price of a European call option at time 0 with strike κk = nKk and maturity tk .

In terms of hedging, this means that we have the following static superhedging strategy: for each averaging day tk , buy exp(−r(T − tk ))/n European call options at time t = 0 with strike κk and maturity tk and hold these until their expiry. Then put their payoff on the bank account. Since the upper bound (4.2) holds for all combinations of κk ≥ 0 that satisfy Pn

k=1

κk = nK, one still has the freedom to choose strike values.

Note that, if 0 ≤ r, the choice κk = K (k = 1, . . . , n) immediately implies,

since EC0 (K, t) ≤ EC0 (K, s), for t ≤ s, that AAC0 (K, T ) ≤ EC0 (K, T ). However, one naturally look for that combination of κk ’s which minimizes the right-hand side of (4.2). This can be done using comonotonic theory, but will lead us to far in this course. Next, we discuss the pricing of the European and American AAC using binomial tree models. However, the procedure is not as simple as in the barrier 95

case and this because at each node we do not know the running average when we reached that node. Typically, all different paths to reach the node lead to different average prices, and the number of paths grow exponentially. Luckily, the tree-approach can be extended to cope with this under certain circumstance. We illustrate the nature of the calculation by condidering the case of an European Asian arthimetic option. The payoff of this options depends on a single function of the path followed, namely the average stock price. We call this average function is the determining path function. The trick is tp carry out, at each node, the calculations for a small number of representative values of the path function. When the value of the derivative is required for other values of the path function, we calculate it from the known values using interpolation. Suppose the initial stock price is 50, the strike price is 50, r = 0.10 and the volatility is 0.40, and the time to maturity is one year. We use a tree with 20 time steps. The parameters describing the binomial tree parameters are ∆t = 0.05, u = 1.0936, d = 0.9144, p = 0.5056. Figure 4.4 shows the calculations that are carried out in one small part of the tree. Node X is the central node at time 0.2 year (at the end of the fourth time step). Nodes Y and Z are the two nodes at time 0.25 years that are reachable from node X. The stock price is X is 50. Forward induction shows that the maximum average stock price achievable in reaching node X is 53.83. The minimum is 46.65. (We include both the initial and final stock prices when calculating the average, i.e. t1 = 0 and tn = T .) From node X, we branch to

96

Figure 4.4: Part of tree for Asian option one of the two nodes Y and Z. At node Y , the stock price is 54.68 and the bounds for the average are 47.99 and 57.39. At node Z, the stock price is 45.72 and the bounds for the average stock price are 43.88 and 52.48. Suppose that we have chosen the representative values of the average to be four equally spaced values at each node. This means that at node X, we consider the averages 46.65, 49.04, 51.44, and 57.83. At node Y , we consider the averages 47.99, 51.12, 54.26, and 57.39. At node Z, we consider the averages 43.88, 46.75, 49.61, and 52.48. We assume backward induction has already been used to calculate the value of the option for each of the alternative valuesof the average at node Y and Z. The values are shown in Figure 4.4. For exemple, at

97

node Y when the average is 51.12, the value of the option is 8.101. Consider the calculation at node X for the case where the average is 51.44. If the stock price moves up to node Y , the new average will be 5×51.44+54.68)/6 = 51.98. The value of the derivative at node Y for this average can be found by interpolating between the values when the average is 51.12 and when it is 54.26. It is (51.98 − 51.12) × 8.635 + (54.26 − 51.98) × 8.101 = 8.247. 54.26 − 51.12 Similarly, if the stock price moves down to node Z, the new average will be 5 × 51.44 + 45.72)/6 = 50.49 and by interpolation the value of the derivative is 4.182. The value of the derivative at node X when the average is 51.44 is, therefore, exp(−0.1 × 0.05)(0.5056 × 8.247 + (1 − 0.5056) × 4.182) = 6.206. The other values at node X are calculated similarly. Once the values at all nodes at time 0.2 year have been calculated, we can move on to the nodes at time 0.15 year. The value given by the full tree for the option at time zero is 7.17. As the number of time steps and the number of averages considered at each node is increased, the value of the option converges to the correct answer. With 60 time steps and 100 averages at each node, the value of the option is 5.58. The true value of the option is around 5.62. A key advantage of the method here is that it can handle American options. The calculations are as we have described them except that we test for early

98

exercise at each node for each of the alternative values of the path function at the node. The approach just described can be used in a wide range of different situations if the following conditions are satisfied: • the payoff from the derivative must depend on a single function, the path function, of the path followed by the underlying asset; • it must be possible to calculate the value of the path function at time t+∆t from the value of this function at time t and the value of underlying asset at time t + ∆t. Efficiency is improved somewhat if quadratic rather than linear interpolation is used at each node.

99

Chapter 5

The Black-Scholes Option Price Model In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made a major breakthrough in the pricing of stock options by developing what has become known as the Black-Scholes model. The model has had huge influence on the way that traders price and hedge options. In 1997, the importance of the model was recognized when Myron Scholes and Robert Merton were awarded the Nobel prize for economics. Sadly, Fischer Black died in 1995, otherwise he also would undoubtedly have been one of the recipients of this prize. This chapter shows how the Black-Scholes model for valuing European call and put options on a non-dividend-paying stock is derived.

100

5.1

Continuous-Time Stochastic Processes

This section develops a continuous-time, continuous-variable stochastic process for stock prices. An understanding of this process is the first step to understanding the pricing of options and other more complicated derivatives. It should be noted that, in practice, we do not observe stock prices following continuousvariable, continuous-time processes. Stock prices are restricted to discrete values (often multiples of 0.01 euro) and changes can be observed only when the exchange is open. Nevertheless, the continuous-variable, continuous-time process proves to be a useful model for many purposes.

5.1.1

Information and Filtration

The underlying set-up is as in the discrete time case. We assume a fixed finite planning horizon T . We need a complete probability space (Ω, FT , P ), equipped with a filtration, i.e. a nondecreasing family

F = (Ft )0≤t≤T of sub-σ-fields of

FT : Fs ⊂ Ft ⊂ FT for 0 < s < t ≤ T ; here Ft represents the information available at time t, and the filtration

F = (Ft ) represents the information flow

evolving with time. We assume that the filtered probability space (Ω, FT , P, F) satisfies the ’usual conditions’: a) F0 contains all P -null sets of F. This means intuitively that we know which events are possible and which not, and b) (Ft ) is right-continuous, i.e. Ft = ∩s>t Fs ; a technical condition. A stochastic process X = (Xt )0≤t≤T is a family of random variables defined on (Ω, FT , P, F). We say X is

F-adapted if Xt ∈ Ft (i.e. Xt is Ft -measurable) 101

for each t: thus Xt is known at time t.

5.1.2

Martingales

A stochastic process X = (Xt )t≥0 is a martingale relative to (P, F) if • X is

F-adapted

• E[|Xt |] < ∞ for all t ≥ 0 • E[Xt |Fs ] = Xs , P -a.s., (0 ≤ s ≤ t), A martingale is ’constant on average’, and models a fair game. This can be seen from the third condition: the best forecast of the unobserved future value Xt based on information at time s, Fs , is the at time s known value Xs .

5.2

Brownian Motion

The Scottish botanist Robert Brown observed pollen particles in suspension under a microscope in 1828 and 1829, and observed that they were in constant irregular motion. In 1900 L. Bachelier considered Brownian motion as a possible model for stock-market prices. In 1905 Albert Einstein considered Brownian motion as a model of particles in suspension and used it to estimate Avogadro’s number. In 1923 Norbert Wiener defined and constructed Brownian motion rigorously for the first time. The resulting stochastic process is often called the Wiener process in his honour.

102

Definition 17 A stochastic process X = {Xt , t ≥ 0} is a standard Brownian motion on some probability space (Ω, F, P ), if 1. X0 = 0 a.s. 2. X has independent increments. 3. X has stationary increments. 4. Xt+s −Xt is normally distributed with mean 0 and variance s: Xt+s −Xt ∼ N(0, s). 5. X has continuous sample paths. We shall henceforth denote standard Brownian motion by W = {Wt , t ≥ 0} (W for Wiener).

Construction No construction of Brownian motion is easy: one needs both some work and some knowledge of measure theory. We take the existence of Brownian motion for granted. To gain some intuition on its behaviour, it is good to compare Brownian motion with a simple symmetric random walk on the integers. More precisely, let X = {Xi , i = 1, 2, . . . } be a series of independent and identically distributed random variables with Pr(Xi = 1) = Pr(Xi = −1) = 1/2. Define the simple symmetric random walk Z = {Zn , n = 0, 1, 2, . . . } as Z0 = 0 and Zn =

Pn

i=1

Xi , n = 1, 2, . . . . Rescale this random walk as √ Yk (t) = Zbktc / k, 103

where bxc is the integer part of x. Then from the Central Limit Theorem, Yk (t) → Wt as k → ∞, with convergence in distribution (or weak convergence). In Figure 5.1, one sees a realization of the standard Brownian motion. In Standard Brownian Motion 1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.1: A sample path of a standard Brownian motion

Figure 5.2, one sees the random-walk approximation of the standard Brownian motion. The process Yk = {Yk (t), t ≥ 0} is shown for k = 1 (i.e. the symmetric random walk), k = 3, k = 10 and k = 50. Clearly, one sees the Yk (t) → Wt . The universal nature of Brownian motion as a stochastic process is simply the dynamic counterpart – where we work with evolution in time – of the universal nature of its static counterpart, the normal distribution – in probability, statistics, science, economy etc. Both arise from the same source, the central limit theorem. This says that when we average large numbers of independent and comparable objects, we obtain the normal distribution in a static context, 104

k=1

k=3

10

10

5

5

0

0

−5

−5

−10

0

10

20

30

40

−10

50

0

10

20

k=10 10

5

5

0

0

−5

−5

0

10

20

40

50

30

40

50

k=50

10

−10

30

30

40

−10

50

0

10

20

Figure 5.2: Random walk approxiamtion for standard Brownian motion

or Brownian motion in a dynamic context. What the central limit theory really says is that, when what we observe is the result of a very large number of individually very small influences, the normal distribution or Brownian motion will inevitably and automatically emerge. Next, we look at some of the classical properties of Brownian motion.

Martingale Property Brownian motion is one of the most simple examples of a martingale. We have for all 0 ≤ s ≤ t, E[Wt |Fs ] = E[Wt |Ws ] = Ws .

105

We also mention that one has: E[Wt Ws ] = min{t, s}.

Path Properties One can proof that Brownian motion has continuous paths, i.e. Wt is a continuous function of t. However the paths of Brownian motion are very erratic. They are for example nowhere differentiable. Moreover, one can prove also that the paths of Brownian motion are of infinite variation, i.e. their variation is infinite on every interval. Another property is that for a Brownian motion W = {Wt , t ≥ 0}, we have that Pr(sup Wt = +∞ and inf Wt = −∞) = 1. t≥0

t≥0

This result tells us that the Brownian path will keep oscillating between positive and negative values.

Scaling Property There are a well-known set of transformations of Brownian motion which produce another Brownian motion. One of this is the scaling property which says that if W = {Wt , t ≥ 0} is a Brownian motion, then also for every c 6= 0, ˜ = {W ˜ t = cWt/c2 , t ≥ 0} W is a Brownian motion.

106

(5.1)

5.3 5.3.1

Itˆ o’s Calculus Stochastic Integrals

Stochastic integration was introduced by K. Itˆ o in 1941, hence its name Itˆ o calculus. It gives meaning to Z

t

Xu dYu 0

for suitable stochastic processes X = {Xu , u ≥ 0} and Y = {Yu , u ≥ 0}, the integrand and the integrator. We shall confine our attention here to the basic case with integrator Brownian motion: Y = W . Because Brownian motion is of infinite (unbounded) variation on every interval, the first thing to note is that stochastic integrals with respect to Brownian motion, if they exist, must be quite different from the classical deterministic integrals. We take for granted Itˆ o’s fundamental insight that stochastic integrals can be defined for a suitable class of integrands. We only show how these integrals can be defined for some simple integrands X.

Indicators If Xt = 1[a,b] (t), i.e. it equals 1 between a and b and is zero elsewhere, we define R

XdW :     0    Z t  It (X) = Xs dWs = Wt − W a  0       Wb − W a 107

if t ≤ a if a ≤ t ≤ b if t ≥ b

Simple deterministic functions We can extend the above definition by linearity: if X is a linear combination of indicators, Xt =

Pn

i=1 ci 1[ai ,bi ] (t),

It (X) =

Z

we define

t

Xs dWs = 0

n X i=1

R

ci

Z

XdW : t

1[ai ,bi ] (s)dWs 0

Simple stochastic processes X is called a simple stochastic process if there is a partition 0 = t0 < t1 < · · · < tn = T < ∞ and uniformly bounded Ftk -measurable random variables ξk (|ξk | ≤ C for all k = 0, . . . , n for some C) and if Xt can be written in the form Xt = ξ0 10 (t) +

n−1 X

ξi 1(ti ,ti+1 ] (t),

i=0

0 ≤ t ≤ T.

Then if tk ≤ t ≤ tk+1 , k = 0, . . . , n − 1, It (X) =

Z

t

Xs dWs = 0

k−1 X i=0

ξi (Wti+1 − Wti ) + ξk (Wt − Wtk )

It is not so hard to prove some simple properties of the stochastic integrals defined so far: • It (aX + bY ) = aIt (X) + bIt (Y ). • It (X) is a martingale. • Itˆ o isometry: E[(It (X))2 ] =

Rt 0

The Itˆ o isometry above suggests that with

Rt 0

E[(Xu )2 ]du. Rt 0

XdW should be defined only for processes

E[(Xu )2 ]du < ∞ for all t and this is indeed the case. Each such X may

be approximated by a sequence of simple stochastic processes and the stochastic integral may be defined as the limit of this approximation. Furthermore 108

the three above properties remain true. We will not include the technical and detailed proofs of this procedure in this book. Note that one also can construct a closely analogous theory for stochastic integrals with the Brownian integrator W above replaced by a (semi-)martingale integrator M .

5.3.2

Itˆ o’s Lemma

The price of a stock option is a function of the underlying stock’s price and time. More generally, we can say that the price of any derivative is a function of the stochastic variables underlying the derivative and time. Therefore, we must acquire some understanding of the behavior of functions of stochastic variables. An important result in this area was discovered by K. Itˆ o, in 1951. It is known as Itˆ o’s lemma. Suppose that F : R2 → R is a function, which is continuously differentiable once in its first argument (which will denote time), and twice in its second argument: F ∈ C 1,2 . Denote the partial derivatives Ft (t, x)

=

Fx (t, x)

=

Fxx (t, x)

=

∂F (t, x) ∂t ∂F (t, x) ∂x ∂2F (t, x) ∂x2

Theorem 18 (Itˆ o’s lemma) Let W = {Wt , t ≥ 0} be Standard Brownian motion and let F (t, x) ∈ C 1,2 , then F (t, Wt )−F (s, Ws ) =

Z

t

Fx (u, Wu )dWu + s

Z

s

109

t

Ft (u, Wu )du+

1 2

Z

t

Fxx (u, Wu )du. s

or 1 dF = Fx dWt + Ft dt + Fxx dt. 2 for short. As an application of Itˆ o’s lemma we compute x2 . Then Wt2 = W02 +

Z

t

2Wu dWu + 0

1 2

Z

Rt 0

Wu dWu by using F (t, x) =

t

2du = 2 0

Z

t

Wu dWu + t. 0

So that Z

t

Wu dWu = 0

t Wt2 − 2 2

Note the contrast with ordinary calculus ! Itˆ o calculus requires the second term on the right – the Itˆ o correction term.

5.4

Stochastic Differential Equations

Like with any ordinary and partial differential equations in a deterministic setting (ODEs and PDEs), the two most basic questions are those of existence and uniqueness of solutions. To obtain existence and uniqueness results, one has to impose reasonable regularity conditions on the coefficients occuring in the differential equation. Naturally, stochastic differential equations (SDEs) contain all the complications of their non-stochastic counterparts, and more besides. Consider the stochastic differential equation dXt = b(t, Xt )dt + σ(t, Xt )dWt ,

110

Xs = x,

(5.2)

where the coefficients b and σ satisfy the following Lipschitz and growth conditions |b(t, x) − b(t, y)| + |σ(t, x) − σ(t, y)| ≤ K|x − y| |b(t, x)|2 + |σ(t, x)|2 ≤ K 2 (1 + |x|2 ) for all t ≥ 0, x, y ∈ R, for some constant K > 0. To see that the SDE (5.2) has a solution, we first define recursively (0)

Xt

(n+1)

= x, Xt

=x+ (n)

One can then prove that Xt

Z

t s

b(u, Xu(n) )du +

Z

t s

σ(u, Xu(n) )dWu .

converges (in some sense), to Xt say; Xt is the

unique (strong) solution to (5.2), i.e. X0 = x, Xt = x +

Z

t

b(u, Xu )du + s

Z

t

σ(u, Xu )dWu . s

The next result, which is an example for the rich interplay between probability theory and analysis, links SDEs with PDEs. Suppose we consider a stochastic differential equation (satisfying the above Lipschitz and growth conditions), dXu = µ(u, Xu )du + σ(u, Xu )dWu ,

Xs = x,

s≤u≤T

Consider a function F (t, x) ∈ C 1,2 of it. Then we have the following extension of Itˆ o’s lemma: Theorem 19 (Itˆ o’s lemma for SDE’s) Let F (t, x) ∈ C 1,2 , then Z

t

F (t, Xt ) − F (s, Xs ) = σ(u, Xu )Fx (u, Xu )dWu + (5.3) s  Z t σ(u, Xu )2 Ft (u, Xu ) + µ(u, Xu )Fx (u, Xu ) + Fxx (u, Xu ) du. 2 s 111

Now suppose that F satisfies the PDE (σ(t, x))2 Fxx (t, x) = 0, 2

Ft (t, x) + µ(t, x)Fx (t, x) + with boundary condition

F (T, x) = h(x). Then the above expression for (5.3) gives F (s, Xs ) = F (t, Xt ) −

Z

t

σ(u, Xu )Fx (u, Xu )dWu s

The stochastic integral on the right is a martingale, so has constant expectation, which must be 0 as it starts at 0. So F (s, x) = E[F (t, Xt )|Xs = x] which leads for t = T to the Feynman-Kac Formula F (s, x) = E[h(XT )|Xs = x]. The Feynman-Kac formula gives a stochastic representation to solutions of PDEs. We shall return to the Feynman-Kac formula below in connection with the Black-Scholes partial differential equation.

5.5

Geometric Brownian Motion

Now that we have both Brownian motion W and Itˆ o’s Lemma to hand, we can introduce the most important stochastic process for us, a relative of Brownian motion – geometric Brownian motion.

112

Suppose we wish to model the time evolution of a stock price St . Consider how S will change in some small time interval from the present time t to a time t + ∆t in the near future. Writing ∆St for the change St+∆t − St , the return on S in this interval is ∆St /St . It is economically reasonable to expect this return to decompose into two components, a systematic part and a random part. The systematic part could plausibly be modeled by µ∆t, where µ is some parameter representing the mean rate of the return of the stock. The random part could plausibly be modeled by σ∆Wt , where ∆Wt represent the noise term driving the stock price dynamics, and σ is a second parameter describing how much effect the noise has – how much the stock price fluctuates. Thus σ governs how volatile the price is, and is called the volatility of the stock. The role of the driving noise term is to represent the random buffeting effect of the multiplicity of factors at work in the economic environment in which the stock price is determined by supply and demand. Putting this together, we have the following SDEs ∆St = St (µ∆t + σ∆Wt ),

S0 > 0.

In the limit as ∆t → 0, we have the stochastic differential equation dSt = St (µdt + σdWt ),

S0 > 0.

The differential equation above has the unique solution St = S0 exp



σ2 µ− 2





t + σWt .

For, writing f (t, x) = exp



µ−

113

σ2 2



t + σx



Itˆ o’s lemma gives δf δf 1 δ2 f dt + dWt + dt δt δx 2 δx2   1 σ2 f dt + σf dWt + σ 2 f dt = µ− 2 2

df (t, Wt ) =

= f (µdt + σdWt ) so f (t, Wt ) is a solution of the stochastic differential equation. This means that σ2 log St = log S0 + µ − 2 



t + σWt

has a normal distribution. Thus St itself has a lognormal distribution. This geometric Brownian motion model, and the log-normal distribution which it entails, are the basis for the Black-Scholes model for stock-price dynamics in continuous time. In Figure 5.3 one sees the realization of the geometric Brownian motion based on the sample path of the standard Brownian motion of Figure ??.

5.6

The Market Model

We consider a frictionless security market in which two assets are traded continously. Investors are allowed to trade continuously up to some fixed finite planning horizon T , where all economic activity stops. The first asset is one without risk (the bank account). Its price process is given by Bt = ert , 0 ≤ t ≤ T . The second asset is a risky asset, usually refered to as stock. The price process of this stock, St , 0 ≤ t ≤ T , is modelled by the

114

Figure 5.3: Sample path of a geometric Brownian motion (S0 = 100, µ = 0.05, σ = 0.40) linear stochastic differential equation dSt = St (µdt + σdWt ),

S0 = x > 0,

where Wt is standard Brownian motion, defined on a filtered probability space (Ω, F, P, F). This means that under P , Wt has a Normal(0, t) distribution. Furthermore, in the previous chapter we derived that St follows a geometric Brownian motion: St = S0 exp



σ2 µ− 2





t + σWt .

µ is reflecting the drift and σ models the volatility and are assumed to be constant. We assume as underlying filtration, the Brownian filtration

F =

(Ft ), basically Ft = σ(Ws , 0 ≤ s ≤ t), slightly enlarged to satisfy the usual conditions. Consequently, the stock price process St follows a strictly positive adapted process. We call this market model the Black-Scholes model. 115

Our principle task will be the pricing and hedging of contingent claims, which we model as non-negative FT -measurable random variables. This implies that the contingent claims specify a stochastic cash-flow at time T and that they may depend on the whole path of the underlying in [0, T ] – because FT contains all information. We will often have to impose further (integrability) conditions on the contingent claims under consideration. As before, the fundamental concept in (arbitrage) pricing and hedging contingent claims is the interplay of self-financing replicating portfolios and risk-neutral probabilities. Although the current (timecontinuous) setting is on a much higher level of sophistication, the key ideas remain the same. We call a two-dimensional adapted (predictable), locally bounded process ϕ = {ϕt = (βt , ξt ),

t ∈ [0, T ]}

a trading strategy or dynamic portfolio process. The conditions ensure that the stochastic integral

Rt 0

ξt dWt exists. Here βt denotes the money invested in the

riskless asset and ξt denotes the number of stocks held in the portfolio at time t. Remark: In a more general setting the trading strategy has to be predictable in stead of adapted. Predictability of these processes imply that (βt , ξt ) has to be determined on the basis of information available strictly before time t, Ft− : the investor selects his time t portfolio just before the observation of the price St . Because our Brownian filtration is continuous we have Ft− = Ft and predictablity and adaptedness are the same. • 116

The components of ϕt may assume negative as well as positive values, reflecting the fact that we allow short sales and assume that the assets are perfectly divisible. Definition 20

(i) The value of the portfolio ϕ at time t is given by Vt = Vtϕ = βt Bt + ξt St = βt ert + ξt St

The process Vtϕ is called the value process, or wealth process, of the trading strategy ϕ. (ii) The gains process Gϕ t is defined by Gt = G ϕ t =

Z

t

βu dBu + 0

Z

t

ξu dSu 0

(iii) A trading strategy ϕ is called self-financing if the wealth process V tϕ satisfies Vtϕ = V0ϕ + Gϕ t for all t ∈ [0, T ]. The financial implications of the above equations are that all changes in the wealth of the portfolio are due to market changes, as opposed to withdrawals of cash or injections of new funds.

5.7

Equivalent Martingale Measures and RiskNeutral Pricing

Next, we develop a pricing theory for contingent claims. Again the underlying concept is the link between the no-arbitrage condition and certain probability 117

measures. We begin with Definition 21 A trading strategy ϕ is called tame, if the associated wealth process is always positive: Vtϕ ≥ 0, for all t ∈ [0, T ]. Similarly as in the discrete case (admissible strategies), tame strategies prevent the broker from unbounded short sales. Using tame strategies the investor’s wealth may never go negative at a time, even if he is able to cover his debt at the final date. If we would later on allow non-tame strategies, one can show that it is possible to construct doubling strategies that can attain arbitrarily large values of wealth starting with zero initial capital. Such strategies are examples of arbitrage opportunities, which we define in general as: Definition 22 A self-financing trading strategy ϕ is called an arbitrage opportunity if the wealth process V ϕ satisfies the following set of conditions: V0ϕ = 0,

P (VTϕ ≥ 0) = 1 and P (VTϕ > 0) > 0.

Arbitrage opportunities represent the limitless creation of wealth through riskfree profit and thus should not be present in a well-functioning market. We say that our market is arbitrage-free if there are no tame self-financing arbitrage opportunities. The main tool in investigating arbitrage opportunities is the concept of equivalent martingale measures: Definition 23 We say that a probability measure P ∗ defined on (Ω, FT ) is an equivalent martingale measure if: 118

(i) P ∗ is equivalent to P (ii) the discounted price process S˜t = e−rt St is a P ∗ -martingale. We denote the set of equivalent martingale measures by P. As in the discrete case, one can prove that one can preclude arbitrage opportunities if an equivalent martingale measure exists. Furthermore, in the more general continuous-time setting, we have the following partial analogue of the completeness theorem in the discrete setting: If P = {P ∗ }, then the market is complete, in the restricted sense that for every contingent claim X satisfying EP ∗ [X 2 ] < ∞ there exists at least an admissible self-financing trading ϕ strategy such that VTϕ = X. Remark: Having seen the above results, a natural question is to ask whether converse statements are also true. One has to put some further requirements on portfolios to establish such converse results. These requirements should of course be economically meaningful. A lot of effort has been put into solving this question, and several alternatives have been proposed, but the details will lead us to far. • By the risk-neutral valuation principle the price Vt at time t, of a contingent claim with payoff function G({Su , 0 ≤ u ≤ T }) is given by Vt = exp(−(T − t)r)EP ∗ [G({Su , 0 ≤ u ≤ T })|Ft ],

t ∈ [0, T ],

(5.4)

where P ∗ is an equivalent martingale measure. In a general setting their is not a unique martingale measure ( incomplete market models). Roughly speaking incompleteness means that a general contingent claim can not be perfectly hedged. 119

Most models are not complete, and most practitioners believe the actual market is not complete. we have to choose an equivalent martingale measure in some way and this is not always clear. Actually, the market is choosing the martingale measure for us. In the Black-Scholes world however, one can prove (Girsanov Theorem) that there is a unique equivalent martingale measure and we do not have to deal with coosing an appropriate one. It is not hard to see that under P ∗ , the stock price is following a Geometric Brownian motion again. This risk-neutral stock price process has the same volatility parameter σ, but the drift parameter µ is changed to the continuously compounded risk-free rate r: St = S0 exp



σ2 r− 2





t + σWt .

Equivalent, we can say that under P ∗ our stock price process S = {St , 0 ≤ t ≤ T } is satisfying the SDE: dSt = St (rdt + σdWt ),

S0 > 0.

This SDE tells us that in a risk-neutral world the total return from the stock must be r. Next, we will calculate European call option prices under this model.

5.7.1

The Pricing of Options under the Black-Scholes Model

If the payoff function is only depending on the time T value of the stock, i.e. G({Su , 0 ≤ u ≤ T }) = G(ST ), then the above formula can be rewritten as (we

120

set for simplicity t = 0): V0

= exp(−T r)EP ∗ [G(ST )] = exp(−T r)EP ∗ [G(S0 exp((r − q − σ 2 /2)T + σWT ))] Z +∞ = exp(−T r) G(S0 exp((r − q − σ 2 /2)T + σx))fN ormal (x; 0, T )dx. −∞

Explicit Formula for European Call and Put Options In some cases it is possible to evaluate explicitly the above expected value in the risk-neutral pricing formula (5.4). Take for example an European call on the stock (with price process S) with strike K and maturity T (so G(ST ) = (ST − K)+ ). The Black-Scholes formulas for the price C(K, T ) at time zero of this European call option on the stock (with dividend yield q) is given by C(K, T ) = C = S0 N(d1 ) − K exp(−rT )N(d2 ), where d1 d2

=

log(S0 /K) + (r + √ σ T

σ2 2 )T

=

log(S0 /K) + (r − √ σ T

σ2 2 )T

,

(5.5) √ = d1 − σ T ,

(5.6)

and N(x) is the cumulative probability distribution function for a variable that is standard normally distributed (Normal(0, 1)). From this, one can also easily (via the put-call parity) obtain the price P (K, T ) of the European put option on the same stock with same strike K and

121

same maturity T : P (K, T ) = −S0 N(−d1 ) + K exp(−rT )N(−d2 ). For the call, the probability (under P ∗ ) of finishing in the money corresponds with N(d2 ). Similarly, the delta (i.e. the change in the value of the option compared with the change in the value of the underlying asset) of the option corresponds with N(d1 ).

Black-Scholes PDE If moreover G(ST ) is a sufficiently integrable function, then the price at time t is only a function of t and St : Vt = F (t, St ). We show that F solves the Black-Scholes partial differential equation ∂ ∂ 1 ∂2 F (t, s) + (r − q)s F (t, s) + σ 2 s2 2 F (t, s) − rF (t, s) ∂t ∂s 2 ∂s

= 0,

(5.7)

F (T, s) = G(s) This will basically follow from the Feynman-Kac representation for Brownian motion. Indeed, let H(t, s) be a solution of 1 Ht (t, s) + rsHs (t, s) + σ 2 s2 Hss (t, s) = 0, 2 H(T, s) = e−rT G(s). Then we know from the Feynman-Kac representation that H has the representation H(t, St ) = e−rT EP ∗ [G(ST )|Ft ]; 122

Note that by the risk-neutral valuation principle Vt = F (t, St ) = exp(−r(T − t))EP ∗ [G(ST )|Ft ] = exp(rt)H(t, St ). By computing the partial derivatives of F we obtain the PDE (5.7).

5.7.2

Hedging

For a European call option on a non-dividend-paying stock, it can be shown from the Black-Scholes formulas that ξt = ∆call = N (d1 ) = N

ln(S0 /K) + (r + √ σ T

σ2 2 )T

!

For a European put option on a non-dividend-paying stock, it can be shown from the Black-Scholes formulas that delta is given by ξt = ∆put = N (d1 ) − 1 = N

ln(S0 /K) + (r + √ σ T

σ2 2 )T

!

− 1 = ∆call − 1

∆ is the rate of change of the option price with respect to the price of the underlying asset. Suppose that the delta of a call option on as stock is 0.6. This means that when the stock price changes by a small amount, the option price changes by about 60 percent of that amount. Suppose further that the stock price is 100 euro and the option price is 10 euro. Imagine an investor who has sold 2000 option contracts – that is, options to buy 2000 shares. The investor’s position could be hedged by buying 0.6 × 2000 = 1200 shares. The gain (loss) on the option position would then tend to be offset by the loss (gain) on the stock position. For example, if the stock goes up by 1 euro (producing a gain of 1200 euro on the shares purchased), the option price will tend to go up by 123

0.6 × 1 = 0.60 euro (producing a loss of 2000 × 0.6 = 1200 euro on the options written); if the stock price goes down by 1 euro (producing a loss of 1200 euro on the stock position), the option price will tend to go down by 0.60 (producing a gain of 1200 euro on the option position). It is important to realize that, because delta changes (with time and stock price movements), the investor’s position remains delta-hedged (or delta neutral) for only a relatively short period of time. In order to have a perfect hedge, the positions have to be adjusted continuously. In practice however one can only adjust periodically. This is known as rebalancing. For example, suppose that an increase in the stock leads to an increase in delta, say from 0.60 to 0.65. An extra of 0.05 × 2000 = 100 shares would then have to be purchased to maintain the hedge. Tables 5.1 and 5.2 provide two simulations of the operation of periodical delta-hedging. The hedge is assumed to be rebalanced weekly. Assume we have to hedge a position of 100000 written call options on a non-dividend paying stock with strike price K, with S0 = 49, K = 50, r = 0.05 (compound interest rate per year), σ = 0.2 (per year) and T = 20 weeks = 0.3846 years From this we can easily compute the initial value of the call: C = 2.40047; and the delta which equals ∆ = 0.52160. This means that as soon as the option is written we have to buy 0.52160 × 100000 = 52160 shares at a price of 49 euro, for the total amount of 49 × 52160 = 2555840 euro. So we must borrow this 124

amount of 2555840 euro to buy 52160 shares. Because the interest rate is 0.05, the interest cost totaling 2555840(exp(0.05/52) − 1) = 2459 euro are incurred in the first week. In Table 5.1, the stock price falls to 48.125 euro by the end of the first week. ∆ is recomputed at the end of the first week using S0 = 48.125, K = 50, r = 0.05 (compound interest rate per year), σ = 0.2 (per year) and T = 19 weeks = 0.3654 years and is equal to ∆ = 0.45835. A total of 52160 − (0.45835 × 100000) = 6325 shares must be sold to maintain the hedge. This realizes 6325×48.125 = 304391 cash and the cumulative borrowings at the end of week one are reduced to 2555840 − 304391 + 2459 = 2253908 euro. During the second week the stock price reduces to 47.375 euro and the delta declines again; and so on. Towards the end of the life of the option it becomes apparent that the option will be exercised and the delta approaches 1. By week 20, therefore, the hedger has a fully covered position. The hedger receives 5000000 euro for the stock held, so that the total cost of writing the option and hedging it is 5000000 − 5263157 = 263157 euro. Table 5.2 illustrates an alternative sequence of events such that the option closes out of the money. As it becomes clearer that the option will not be exercised, delta approaches zero. By week 20, the hedger has a naked position and has incurred costs totaling 256558 euro. In Table 5.1 and 5.2, the costs of hedging the option, when discounted to the beginning of the period, i.e. 258145 and 251672 are close to but not exactly

125

Table 5.1: Hedging simulation; call option closes in the money

Table 5.2: Hedging simulation; call option closes out of the money

126

the same as the Black-Scholes price of 240047. If the hedging scheme worked perfectly, the cost of hedging would, after discounting, be exactly equal to the theoretical price of the option on every simulation. The reason that there is a variation in the cost of delta hedging is that the hedge is rebalanced only once a week. As rebalancing takes place more frequently, the uncertainty in the cost of hedging is reduced. Of course the simulations above are idealized in that they assume that the volatility and interest rate are constant and that there are no transaction costs. In Figure 5.4, one can see the underlying Standard Brownian Motion, the related Geometric Brownian Motion, the option prices of a European call option and the associated hedge over the one year life-time of the option (S0 = 100, K = 105, r = 0.03, µ = 0.09, σ = 0.4). Note how fast ∆ is near maturuity going to 1 (the option ends in the money).

5.8

The Greeks

The Black-Scholes option values depend on the (current) stock price S, the volatility σ, the time to maturity T , the interest rate r, and the strike price K. The sensitivities of the option price with respect to the first four parameters are called the Greeks and are widely used for hedging purposes. Recall the Black-Scholes formula for a European call: C = C(S, T, K, r, σ) = SN

ln(S0 /K) + (r + √ σ T −Ke

−rT

127

N

σ2 2 )T

!

ln(S0 /K) + (r − √ σ T

σ2 2 )T

!

.

Figure 5.4: Wt , St , Ct and ∆t , t ∈ [0, 1] (S0 = 100, K = 105, r = 0.03, µ = 0.09, σ = 0.4)

128

We therefore get ln(S0 /K) + (r + √ σ T

σ2 2 )T

!

δC ∆= δS

= N

δC V= δσ

√ = S Tn

ln(S0 /K) + (r + √ σ T

σ2 2 )T

!

>0

δC Θ= δT

Sσ √ n 2 T

ln(S0 /K) + (r + √ σ T

σ2 2 )T

!

+

=

Kre−rT N δC ρ= δr δ2 C Γ= δS 2

= T Ke n =



−rT

N

>0

ln(S0 /K) + (r − √ σ T

ln(S0 /K) + (r − √ σ T  2

ln(S0 /K)+(r+ σ2 )T √ σ T

√ Sσ T

σ2 2 )T

σ2 2 )T

!

!

>0

>0

> 0,

where as usual N is the cumulative normal distribution function and n is its density. As discussed before ∆ measures the change in the value of the option compared with the change in the value of the underlying asset. Furthermore, ∆ gives the number of shares in the replication portfolio for a call option. Vega, V, measures the change of the option price compared with the change in the volatility of the underlying, and similar statements hold for theta Θ, rho ρ. Gamma Γ measures the sensitivity of our replicating portfolio to the change in the stock price.

5.9

Drawbacks of the Black-Scholes Model

Over the last decades the Black-Scholes model turned out to be very popular. One should bear in mind however, that this elegant theory hinges on several

129

Figure 5.5: Simulated Normally and Nasdaq Composite log-returns crucial assumptions. We assumed that there were no market frictions, like taxes and transaction costs or constraints on the stockholding, etc. Moreover, most empirical evidence suggests that the classical Black-Scholes model does not describe the statistical properties of financial time series very well. Real markets exhibit from time to time very large discontinuous price movements. Moreover, according to the Black-Scholes model, the log-returns, i.e. differences of the form log St+h −log St , are independent and identically normally distributed. Figure 5.5 shows daily log-returns of the American NasdaqComposite Index over the period 1-1-1990 until 31-12-2000 and simulated i.i.d. normal variates with variance equal to the sample variance of the NasdaqComposite log-returns.

130

This picture makes two stylized facts immediately apparent, which are typical for most financial time series. • We see that large asset prize movements occur more frequently than in a model with normal distributed increments. This feature is often refered to as excess kurtosis or fat tails; it is the main reason for considering asset price processes with jumps. • There is evidence for volatility clusters, i.e. there seems to be a succession of periods with high return variance and with low return variance. This observation motivates the introduction of models for asset process where volatility is itself stochastic. Typically we enter the realm of incomplete markets whenever we want to use models for asset price dynamics which are more ’realistic’ than the BlackScholes model. For example markets are incomplete if we consider asset price processes with random volatility or with jumps of varying size.

131

Chapter 6

Miscellaneous 6.1

Decomposing Options into Vanilla Position

Consider an option with a payoff function that only depends on the terminal stock price value, i.e. assume that the payoff function is of the form f (ST ). Assume furthermore (for technical reasons) that the function f is twice differentiable. The fundamental theorem of calculus implies that for any fixed κ: f (x) = f (κ) + 1(x > κ)

Z

= f (κ) + 1(x > κ)

Z

−1(x < κ)

Z

x κ x κ κ

x

f 0 (L)dL − 1(x < κ) " Z L

f 0 (κ) + 0

f (κ) −

Z

κ

κ x

f 0 (L)dL #

f 00 (K)dK dL

κ



Z

00



f (K)dK dL. L

Noting that f 0 (κ) does not depend on L and interchanging the order of integra-

132

tion (Fubini’s theorem) yields: f (x)

= f (κ) + f 0 (κ)(x − κ) + 1(x > κ) +1(x < κ)

Z

κ

Z

x

K

Z

x κ

Z

x

f 00 (K)dLdK

K

f 00 (K)dLdK.

x

Performing the integral over L yields: Z x f (x) = f (κ) + f 0 (κ)(x − κ) + 1(x > κ) f 00 (K)(x − K)dK κ Z κ 00 +1(x < κ) f (K)(K − x)dK x Z ∞ Z κ− = f (κ) + f 0 (κ)(x − κ) + f 00 (K)(x − K)+ dK + f 00 (K)(K − x)+ dK. κ

0

Thus, the payoff decomposes into bonds, forward contracts with delivery price κ, calls struck above κ, and puts struck below κ. Letting V0f denote the initial value of the contract with payoff f (ST ) at T , then the absence of arbitrage implies: V0f

= BT−1 EQ [f (ST )] =

f (κ)BT−1 +

Z

0

+ f (κ)(S0 − κ

κBT−1 )

+

Z

∞

f 00 (K)EC0 (K, T )dK

κ

f 00 (K)EP0 (K, T )dK,

0

where EC0 (K, T ) and EP0 (K, T ) denotes the initial value of resp. an European call and put option with strike K and time to maturity T . Note that if we choose κ = BT S0 , i.e. the forward price of the stock, the second term cancels out.

133

6.2

Variance Swap

Consider a finite set of discrete times {t0 = 0, t1 , . . . , tn = T } at which the path of the underlying is monitored. We denote the price of the underlying at these points, i.e. Sti , by Si for simplicity. Typically, {t0 = 0, t1 , . . . , tn = T } corresponds to daily closing times and Si is the closing price at day i. Note that then log(Si ) − log(Si−1 ),

i = 1, . . . , n,

correspond to the daily log-returns. The so-called realized variance (or better, 2nd moment) is then calculated using the estimator given by n X

1 n

i=1

(log(Si ) − log(Si−1 ))

2

!

.

A contract with as payoff "

1 VS =N × n

n X i=1

(log(Si ) − log(Si−1 ))

2

!

#

−K ,

where N denotes the notational amount, is called a variance swap. The value of K is typically chosen such that the contract has a zero value when it is initiated (just like in the case of futures and forwards). Basically this contract swaps fixed (annualized) second moment, K, (variance) by the realized second moment (variance) and as such provides protection against unexpected or unfavorable changes in second moment (variance) or the related volatility. Next, we will show how in a general setting this contract can be hedged using more basic contracts. 134

We start with the following (Taylor-like) expansion of the 2nd power of the logarithmic function
2 (log(x)) = 2 x − 1 − log(x) + O((x − 1)3 ) . Substituting x by Si /Si−1 leads to 2

(log(Si /Si−1 )) = 2



 ∆Si − log(Si /Si−1 ) + O((∆Si /Si−1 )3 ) , Si−1

where ∆Si = Si − Si−1 . Summing over i gives the following decomposition: n X

(log(Si /Si−1 ))

2

(6.1)

i=1

= 2

n  X ∆Si i=1

Si−1

− log(Si /Si−1 ) + O((∆Si /Si−1 )3 )



n n X X ∆Si = −2(log(ST ) − log(S0 )) + 2 + O( (∆Si /Si−1 )3 ) S i−1 i=1 i=1

(6.2)

due to telescoping. Thus up to 3rd-order terms the sum of the squared logreturns decomposes into the payout from a log-contract (−2(log(ST ) − log(S0 ))) and a dynamic strategy (2

Pn

∆Si i=1 Si−1 ).

The log-contract can be hedge by a dynamic trading strategy in combination with a static position in bonds, European vanilla call and put options maturing at time T. More precisely, first note that for any L > 0 log(ST ) − log(S0 ) =

1 (ST − S0 ) − u(ST ) + u(S0 ), L

for u(x) =



 x−L − log(x) + log(L) . L 135

(6.3)

Moreover with the technique explained in Section 6.1, one can show that u(ST ) = Since ST − S0 = n X i=1

Z

L 0

Pn

(log(Si /Si−1 ))

i=1

2

1 (K − ST )+ dK + K2

Z

+∞ L

1 (ST − K)+ dK. K2

(6.4)

∆Si , substituting (6.4) in (6.3) and (6.2) implies ≈

 n  X 1 2 − ∆Si − 2u(S0 ) Si−1 L i=1 Z L Z +∞ 2 2 + + (K − S ) dK + (ST − K)+ dK T 2 2 K K 0 L

136