Notes-full-set-notes.pdf

Dr Paul Johnson School of Mathematics The University of Manchester

MATH20912: Introduction to Financial Mathematics Semester 2 2018

Course Outline MATH20912 Introduction to Financial Mathematics

Dr Paul Johnson Lecturer in Financial Mathematics 2.107 Alan Turing Building School of Mathematics Office hours: Tuesday 10am-12 email: [email protected] website: http://www.maths.manchester.ac.uk/~pjohnson/pages/math20912.html

Getting in Contact • If you need to see me in person come along during my office hour (Tuesday 10-12) or catch me in one of the examples classes (there are 4 per week).

• Preferred method of answering queries is to use the Blackboard forum (so that everyone

can see the answer and I don’t get asked same questions again and again). I am automatically notified every time someone posts and will try answer queries here within one working day.

• When answering emails I will normally post the query and response on the blackboard forum (anonymised) or point you towards a similar question on the forum.

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MATH20912 Textbooks • J. Hull, Options, Futures and Other Derivatives, 7th Edition, Prentice-Hall, 2008. • P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, 1995

• Blackwells have informed us they will match (or beat) the Amazon price on these titles. The Hull book is extensive and covers all the material in the course and beyond (would be of use in third/fourth year modules) and the latest edition is priced accordingly. The Wilmott book is more of an introductory text that gives a mathematicians view on the material.

Assessment • Coursework test in week 6, 12pm Friday 9th March 2018: 20% • 2 hours examination: 80% • The exam will be the same format as previous years

Support • Lectures 2hr pw (total 21hrs): – Monday 9am-10am Stopford Building Theatre 1

– Friday 12pm-1pm Stopford Building Theatre 1 • Lecture slides complement the notes and provide no extra information (apart from a few pictures). Please don’t waste paper printing them off.

• Notes will be provided in advance of lectures. There are gaps in the notes that we will fill

in on the visualiser in the lecture. The idea is to print the notes off before the lecture but consistent page/example numbering should help you use written notes if you wish.

• Podcasts will be available online following all lectures. They will include a video stream of the visualiser along with audio recording.

• It is up to you to fill in the gaps in the notes by attending lectures (preferable!) or catching up on podcasts.

• Check your emails for course announcements.

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• Tutorials 1hr pw starting week 2: – 50% time working through additional examples or detailed explanation on board, – 50% time to ask questions about examples sheets.

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Lecture 1 Introduction To The Course 1.1

Background Economics

Financial Markets are where financial contracts are bought and sold. Markets act as the third party in all transactions. They do not set the prices of things they merely allow the buyers and sellers to meet or advertise products. They also perform the function of recording all transactions and making that knowledge public, so that all participants in the market can see the price at which things are sold.

Financial Contract is a written agreement between two parties to exchange payments according to some specified criteria. The two parties are normally called the holder and seller.

Contract Holder is normally the buyer of a contract, who pays money at the beginning in exchange for receiving some payments at a later date. Because payments may not always be positive a contract could be free to enter at the start or even one where you pay to hold it.

Contract Seller holds the opposite position to the holder, which normally mean they receive money at the beginning in exchange for giving out some payments at a later date. Example 1.1. Draw up a contract to sell a phone for a fixed price at some future date. Solution 1.1. Fill in figure 1.1.

Stock Market or stock exchange are where stocks (shares) are bought and sold. 1

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Lecture 1

Figure 1.1: An example financial contract. 2

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Lecture 1

Share or stock (can be used interchangeably) refers to the financial contract in which a company sells a percentage of the ownership of the company to the holder, which entitles the holder to a percentage share of all future profits. Stocks can be used to refer to contracts with several different companies, whilst shares normally refers to a set of contracts with just one company. This is why it is called the stock exchange (not the share exchange). We will just use the term share from now on, as we are primarily concerned with models of just one company’s share price.

Bond Market is where bonds and other debt securities are bought and sold. Bond is a debt guarantee in which the holder pays now to receive a guaranteed (in as much as anything can ever be guaranteed) fixed payment in the future.

Asset implies ownership or property, so sometimes stocks and shares may be called assets because they confer ownership of the company.

Price is the amount required as payment to buy or sell a contract. You should note that price is not necessarily the same value, as the second term tends to incorporate more than just numbers.

Market Price is the amount quoted on the market to buy or sell a contract. What is the value of anything? Value in almost any context is extremely subjective, and the idea of something being subjective does not sit well with mathematics. You might note that the economics department in Manchester is considered a Social Science rather than a Physical Science, and there are some who believe that we should not mix the two. What we learn in this course are the techniques that have enabled academics and the finance industry to remove the subjectivity (or risk, we will talk later about the value of risk) and develop a framework under which complex financial contracts can be valued. Because certain economic assumptions (which we will not go into here) apply to contracts sold in markets, it follows that the value under that framework must be the same as the market price.

1.2

Money

Money is the circulating medium of exchange as secured by the government (and hence its citizens). The most important thing we need to know about money is that it is engineered to be inflationary, which means that the government strives to ensure that the value of the money in the future will be worth less than it is now. This is because a government wishes to encourage its

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citizens to spend, invest and take part in the economy, and this inflationary property is delivered through fiscal policy (which is something we won’t touch on in this course). One consequence of the fact that money is inflationary is that banks will pay customers to deposit money with them, and that payment is called the interest rate. This means that as an agent in the economy I know that £10 now > a promise of £10 in the future because £10 in the future plus interest > a promise of £10 in the future . Again, because of the subjectivity that might arise when different people value receiving sums of money in the future (even if it is guaranteed) we must compare eggs with eggs. So if we want to value a payment from a contract in the future, we can only compare it to money in the future, so we need to know how money changes with time.

Interest rate is a payment made in exchange for being in debt to another party. Once you deposit money with a bank they owe you the money back, so they are in debt to you and will pay you interest in return. The rate part comes in the fact that it is expressed as a % payment per annum.

Time value of money we define the time value of money using the interest rate. There are several ways in which interest rate may be paid. • Simple Interest Rate If there is a single payment made at time t, then we call it a simple interest rate. The value of an investment V (t) at time t is V (t) = P (1 + rt) where V (0) = P is the initial investment, r is the interest rate and t is the current time.

• Compound Interest Rate If there is are multiple payments made at fixed intervals during over the life of the investment then we call it a compound interest rate. The value of an investment V (t) at (integer) time t after mt payments are made is  r mt V (t) = P 1 + m where P is the initial investment, r is the interest rate, m is the number of payments per year and t is the current time (integer years).

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• Continuously Compounded Interest Rate For many reasons (mainly practical) it

is much easier to deal with the continuously compounded interest rate rather than a discrete set of payments. In the continuously compounded interest rate we let the number
z of payments per year increase (m → ∞) and since e = limz→∞ 1 + z1 we obtain the formula

V (t) = P ert where P is the initial investment, r is the interest rate and t is the current time. In fact in real life payments are always made discretely at fixed intervals, but the interest rates can easily be adjusted so that they match up with real investments. Example 1.2. Write down the value of an initial investment of £100 at t = 2 years with: • a simple interest rate with r = 0.05 paid at t = 2 • a compound interest rate with r = 0.05 in two payments per year • a continuously compounded interest rate with r = 0.05 Solution 1.2.

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Why buy shares? In a capitalist society ownership of shares is encouraged through tax breaks and other incentives. The general idea is that ownership of companies can be spread out through the population and the best performing companies will be able to get investment to expand their operations. Most of you will end up owning shares probably through investments made on your behalf by pension companies. Example 1.3. Consider an investor has £1000 to spend. They see Apple shares are trading £100 and they think it will go up to £125 by the end of the year, and Google shares are trading at £50 and they think it will go up to £65 by the end of the year. Which shares should they invest in? Solution 1.3.

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Return on investment or return for short will always be defined as the relative return which can be calculated as

1.3

change in value . initial investment

Modelling Shares

We let St denote the market price of a share at time t. There is an abundance of data available for the historical prices of shares. Obviously the models that have been developed over the last hundred years or so have done so because they match in some way the properties of the data. In a market there are many things which drive the changing of price, but they can be broadly split into two parts.

The Deterministic Part contains all known or expected trends. These long term trends that can be estimated either by analysing market data or by using fundamental economic knowledge and insight.

The Stochastic Part this contains all the uncertain parts. Uncertainty about what may happen in the future plays a part, as even events such as extreme weather can have an effect. But also the inherent complexity of the interactions between thousands and thousands of participants. For example, the holder of some shares might lose his job and need to sell them off quickly, which will cause a short dip in the price, the next day somebody gets his job and buys them back! It is this second effect (market interactions) that drives the price over the shortest time scales, and is what we really try to capture in this course.

A Stochastic Model for Shares So we model the return on a share as dS = µdt + σdW. S

(1.1)

Here µdt is a measure of the deterministic expected rate of growth of the stock price. In general, µ = µ(S, t). In simple models µ is taken to be constant (µ = 0.1 yr−1 = 10 %yr−1 ). The last term σdW describes the stochastic change in the stock price, where dW stands for ∆W = W (t + ∆t) − W (t) as ∆t → 0. We will describe what W (t) is in the next lecture, it is a special random process 1

1

called the Wiener process. We call σ the volatility (σ = 0.2 yr− 2 = 20 %yr− 2 ).

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Volatility is a statistical measure of the standard deviation of returns for a share price. It 1

is normally given as a percentage per year− 2 since it is relative to the current price of the share. Example 1.4. What does this model mean in words? What should be the properties of the Weiner process to make it consistent with the real data? Solution 1.4.

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Lecture 2 Properties of SDEs Background Material Throughout this course we adopt the following notations • ∆t is a small but finite change in time • ∆S, ∆V etc is a change in value of something that depends on t over a small finite change in time ∆t

• dt represents an infinitesimally small change in time that can be used to build integrals or differential equations

• dS, dV etc. will be the corresponding change in value over an infinitesimally small change in time

So when you see ∆ in front of the variable we are talking about discrete approximations, whereas a d will mean we are talking about the continuous limit.

Normal Distributions A normal distribution is a continuous probability distribution with the probability density: f (x; µ, σ) =

(µ−x)2 1 √ e− 2σ2 σ 2π

(2.1)

where µ is the mean and σ 2 is the variance of the distribution. If the random variable Y is drawn from a normal distribution with mean µ and variance σ 2 then we can say Y ∼ N (µ, σ 2 )

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Adding normal distributions together If we have Y1 and Y2 are independent random variables that are normally distributed then if Y1 ∼ N (µY1 , σY2 2 ) Y2 ∼ N (µY2 , σY2 2 ) Z = Y1 + Y2 we have Z ∼ N (µY1 + µY2 , σY2 1 + σY2 2 ). Then next result we might need in this course to know is what happens when we construct a new random variable by multiplying and adding constants. Suppose that Z = a + bX where X ∼ N (0, 1) is a standard normal distribution with zero mean and unit variance. Then it follows that Z ∼ N (a, b2 ).

2.1

Properties of SDEs

In the first lecture of the course, we proposed a continuous model for the share price of the form dS = µdt + σdW S

(2.2)

where the first term captures the average growth rate of the share price or the trend and the second part captures the unpredictable nature of share price. As a result of the many observations on stock prices over the course of the past 100 plus years it has been found that one of the bast ways to model the dW term is to use a random walk model. It is easiest to explain a random walk model in discrete steps and we will do so next. In fact the actual model we use is the continuous version, also called a Wiener process, which has special properties that make it easier to work with.

2.1.1

Random Walk

In general to create a random walk we take successive draws from a random distribution and add them together. In our example the random distribution will be defined as ∆W , and we will

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define Wk as the value of the random walk at the kth step. To create our special random walk we take successive draws from a normal distribution with mean zero and variance ∆t and add it to our previous result. If we are talking in discrete time, and we observe the process Wk at the kth step, then Wk+1 = Wk + ∆W

(2.3)

where ∆W ∼ N (0, ∆t) Example 2.1. Sketch this random walk (2.3) with ∆t = 1. You can use https://www.random.org to generate random numbers. Solution 2.1. Fill in figure 2.1.

Now if we write the discrete random walk (2.3) as Wk =

X

∆W

k

it is trivial to show that Wk ∼ N (0, k∆t). Obviously if we write t = k∆t we have W (t) ∼ N (0, t). These results also hold for the continuous limit but proving it is beyond the scope of this course. Therefore in the limit ∆t → 0 we can write it as the integral Z t dW W (t) =

(2.4)

0

where dW ∼ N (0, dt).

2.2

Properties of the Wiener Process

Wiener Process One way to describe the Wiener process W (t) is as the continuous limit of a random walk process, or given the random walk as defined above we take the limit ∆t → 0. It has the following properties

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Figure 2.1: Sketching a random walk. 14

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• W (0) = 0 • W (t) has independent increments: if u ≤ v ≤ s ≤ t, then W (t) − W (s) and W (v) − W (u) are independent

• W (s + t) − W (s) ∼ N (0, t) • W (t) has continuous paths Example 2.2. Show that the following results hold: E[W ] = 0;

(2.5)

E[W 2 ] = t;

(2.6)

E[∆W ] = 0;

(2.7)

E[(∆W )2 ] = ∆t;

(2.8)

1 2

∆W = X (∆t) ,

where

Solution 2.2.

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X ∼ N (0, 1) .

(2.9)

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Probability Density Function for the Wiener Process The probability density function for a Wiener process is given by f (y; µ, σ) = √

1 − y2 e 2t 2πt

(2.10)

so the probability that Wt is contained within some interval [a, b] at a future time t is given by Z b P(a ≤ Wt ≤ b) = f (y; µ, σ)dy. (2.11) a

Example 2.3. Draw the probability distribution for the Weiner process. How does this distribution change over time? Solution 2.3. Fill in figure 2.2.

2.3

An Approximation to the Model for the share Price

Remember our continuous model for share price (1.1) is written dS = µSdt + σSdW. Over a small instant of time we can write ∆S ≈ µS∆t + σS∆W and using the result (2.9) we obtain 1

∆S ≈ µS∆t + σS(∆t) 2 X.

(2.12)

Now given the standard result for a normal distribution we know that ∆S ∼ N (µS∆t, σ 2 S 2 ∆t)

(2.13)

S(t + ∆t) ∼ N (S + µS∆t, σ 2 S 2 ∆t).

(2.14)

and

Note that although S(t + ∆t) is normal, we can’t simply add another ∆S and find that the result S(t + 2∆t) is still normal. Although the increments ∆W are all independent random draws the ∆S at the next step depends on the value at S(t + ∆t), which is itself a distribution. The only way we can get round this is by using some clever results from stochastic calculus that we introduce in lecture 3.

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Figure 2.2: The probability distribution for a Wiener Process. 18

Example 2.4. Consider a share that has volatility 30% and provides expected return of 15% p.a. Find the increase in share price for one week if the initial share price is 100. Solution 2.4.

Example 2.5. Given the result from Example 2.4, what is the distribution of the share price in that case after one week? Solution 2.5.

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Example 2.6. Show that the return

∆S S

is normally distributed with mean µ∆t and variance

2

σ ∆t. Solution 2.6.

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Lecture 3 Log Normal Distribution 3.1

Itˆ o’s Lemma for Continuous Stochastic Variables

Mathematical Finance is about pricing (or valuing) financial contracts, and in particular those contracts which depend of the value of another contract. Now we are going to write down a result which tells us how a function that depends on a random variable changes in time. We assume that the function f (S, t) is a smooth function of S and t. Then if S satisfies the SDE dS = µSdt + σSdW Itˆ o’s Lemma states that the distribution of the change in f is given by   ∂f ∂f 1 2 2 ∂2f ∂f df = + µS + σ S dt + σS dW ∂t ∂S 2 ∂S 2 ∂S

(3.1)

where df = f (S + dS, t + dt) − f (S, t). There are two things to note here. Firstly, the resulting df is a distribution characterised by the term in blue containing the dW term. Because the function f depends on something that is random and unpredictable, the value of the function itself will also be random and unpredictable in the future. Secondly, we have this extra red term appearing from nowhere, so where does it come from? Well, we are not going to spend time going through this but if we think about a Taylor series approximation normally the dS 2 which this term is related to is much much smaller than all the other terms as dS 2 << dS. However, in stochastic calculus we find that dS 2 = O(dt)

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which comes from the earlier result that E[W 2 ] = t. Expanding the above we have dS 2 = (µSdt + σSdW )(µSdt + σSdW ) = σ 2 S 2 dW 2 + o(dt) ≈ σ 2 S 2 dt and we can see where the red term has come from. You are not required to know how to derive this result for the purpose of this course only how to apply Itˆo’s Lemma in practical situations. Later on in the course we will need to apply it in order to derive the value of contracts. Example 3.1. Find the SDE satisfied by f = S 2 . Solution 3.1.

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3.2

Log Normal Distribution

Example 3.2. Show that the stochastic differential equation (SDE) for f = ln S is:   σ2 ln(S(t)) = ln(S0 ) + µ − t + σW (t) 2

(3.2)

Hint: remember that adding together normal distributions results in another normal distribution. This means that constant coefficient SDE’s can be integrated using the result Z t dW = W (t). (3.3) 0

Solution 3.2.

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From (3.2), we can deduce that the logarithm of the share price ln(S(t)) is normally dis-

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 tributed with mean ln(S0 ) + µ −

σ2 2



t and variance σ 2 t. It follows then that

ln(S(t)) ∼ N





σ2 ln(S0 ) + µ − 2



2

t, σ t



(3.4)

Example 3.3. Consider a share with an initial price of 40, an expected return of 16% and a volatility of 20%. Find the probability distribution of ln S in six months. Solution 3.3.

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3.3

Lecture 3

Geometric Brownian Motion

Definition A stochastic process S(t) is said to follow a Geometric Brownian Motion if dS = µSdt + σSdW where W is a Wiener process, and µ and σ are constants. Given that our share price model is a GBM, we have some properties and results that can be useful: 1. We have an exact formula for the share price at time t given by S(t) = S(0)e(µ−σ

2

/2)t+σW (t)

2. There is a closed form expression for the probability density function at time t given by
2 ! ln s − ln(S0 ) − (µ − σ 2 /2)t 1 exp − . fSt (s; µ, σ, t) = √ 2σ 2 t sσ 2πt The two results stated here are particularly important when it comes to evaluating financial contracts depending on S using numerical methods. The first result is particularly useful when generating random samples for S at some future time t as we only need a single random number to be generated for each path. The second result is used in more complex numerical methods that take advantage of the fact that expectations can be evaluated as some integral involving the probability distribution. Example 3.4. Write down a formula for S(t) in terms of the standard normal distribution. Solution 3.4.

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Example 3.5. Try to draw a sketch of the log-normal distribution for S(t). Solution 3.5. Fill in Figure 3.1.

3.3.1

Is It A Good Model?

How good is GBM as a model of the share price? Well, it certainly has a lot of the properties that we would like to see in a good model of the share price such as independent increments and it works very well for the most part but there are two main flaws. They are: • the observed volatility σ in real markets is not constant in time, • there are often large non normal jumps in the stock market, when external events cause a large number of participants to want to either buy or sell at the same time.

Trying to come up with models that can overcome these flaws has been the focus of much research over the years, however in the light of recent market failures academics and practitioners are beginning to question whether such a simple model can ever really capture everything about the market.

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Figure 3.1: Drawing a log-normal distribution. 29

Lecture 4 Derivatives 4.1

Simple Derivatives

Financial contracts have been written and exchanged between parties for thousands of years, even back to ancient Greece. Before shares were traded in a market the most common type of financial contract would be one involving a commodity, such as food crops or fuel (wood/coal/oil), in which the holder would agree a price now for which they would buy the commodity at some future time. The trading of such contracts has over time become more and more popular not just because there is money to be made but also because they can have some real positive social benefits. For example, imagine the farmer who is able to lock in the price of crops now before they start growing them, this means they can be more certain about the future, perhaps encouraging them to invest in new equipment to increase yields. In order for the farmer to be more certain about the price he receives, somebody has to be on the other side of the contract, and they will be less certain. Take for instance the following scenario, a Spanish farmer is able to forward sell 100 Tons of oranges to Tescos that must be delivered in June 2017 for e600 per Ton. Now the current price of oranges is e657 per Ton, so is this a good price or not? Well let us consider two extreme cases: • The weather is good, crop yields are high, and as a result prices of oranges in June are

much lower than e600 per Ton. In this scenario the farmer is happy, and Tescos are

sad, they would have been better off buying from the market in June. • The weather is bad, crop yields are low, and as a result prices of oranges in June are much higher than e600 per Ton. In this scenario the farmer is sad, he could have sold his oranges on the market and made more profits so Tescos are happy.

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So it looks as though there is always one side that loses - so why would either side agree to this? Well there are benefits to both parties from the certainty about the price to be paid and therefore cash flows in the future. And to answer if they can agree on this, well, you might think that it depends on the probability that the price goes above or below e600 per Ton. This type of contract is called a Derivative and it is a financial instrument whose value depends on the values of other underlying variables. Other names are financial derivative, derivative security, derivative product. A share option, for example, is a derivative whose value is dependent on the share price. Examples of these contract are forward contracts, futures, options, swaps, CDS, etc. In the example of Tescos and the farmer they trade a forward contract and the underlying is oranges. Example 4.1. Draw a cartoon to illustrate the buying and selling of a contract outlined in Lecture 1 - Paul wishes to sell his phone for £300 on 5th May 2018. Solution 4.1. Fill in Figure 4.1.

Example 4.2. Now on 5th May 2018, the p10 phone is trading on the market at either £250 or £350. Who is happy in each case? Solution 4.2. Fill in Figure 4.2.

Example 4.3. Now the holder wishes to include an option to cancel the purchase. What happens when this new contract is agreed in the two scenarios above? Solution 4.3. Fill in Figure 4.3.

4.2

Options

Options have been written into financial contracts almost as long as they have been traded. In our example of the farmer and Tescos, the farmer might think he is well placed to mitigate bad weather and would like to include an option to cancel the contract with Tescos if he thinks he

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Figure 4.1: A cartoon to illustrate the buying and selling of a contract. 32

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Lecture 4

Figure 4.2: A cartoon to illustrate different scenarios at maturity. 33

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Lecture 4

Figure 4.3: A cartoon to illustrate different scenarios at maturity when a cancellation option is included. 34

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Lecture 4

can get a better price on the market. If such an option is included in the contract, and it is the farmer that makes the decision on whether to sell or not, he will be the one buying the option from Tescos even though he is selling the oranges. In order to get Tescos to agree to the cancellation option, they will most likely want the farmer to pay them some (non-refundable) upfront free – the option price. Options are very attractive to investors, both for speculation and for hedging.

Option Holder The option holder (buyer) is the one that is able to make a decision, to exercise or not, to buy or not, to sell or not, and they pay the option seller a fee for the privilege.

Option Writer The option writer (seller) has a decision forced upon them at some future time, so they receive some compensation in return.

Option Value is Positive By definition, because the option holder (buyer) is the one that is able to make a decision, and as that usually includes doing nothing, the value of an option is always at least zero.

4.2.1

Options In Stock Markets

All of the options we describe in this course will be written on share prices. Because the value of the option contracts will be derived from the price of the underlying share price, the contract is called a derivatives contract. Any financial contract whose price is derived from the value of another fits into this category. The most commonly traded options are put and call options.

European Call Option The holder of a call option has the right to buy (but not the obligation) the underlying share (sometimes we call this the asset) at a pre-agreed exercise price E on the specified expiry date T of the contract. Mathematically we will usually denote the price of a call option by C(S, t). Sometimes we may include some of the parameters, such as an exercise price of E = 10 and expiry date T = 1 like this C(S, t; E = 10, T = 1). The formula to derive the value of the contract on the expiration date is C(S, T ) = max(S − E, 0)

European Put Option The holder of a put has the option to sell or not at a pre-agreed exercise price on the specified expiry date of the contract. Mathematically we will usually denote the price of a put option by

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P (S, t). The formula to derive the value of the contract on the expiration date is P (S, T ) = max(E − S, 0) Example 4.4. Consider a three-month European call option on a BP share with exercise price E = 15 (T = 0.25). If you enter into this contract you have the right but not the obligation to buy one share for E = 15 in a three months time. What happens at expiry if: 1. The share price is £25? 2. The share price is £5? Solution 4.4.

Exercise Price This can be used interchangeably with the strike price and is denoted by E. The pre-agreed strike price or exercise price E is written into the contract and therefore the value of the option will depend on its chosen value.

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Underlying Share/Asset Price The share price S may also be called the underlying share price or underlying asset price (the second is more generic), both just mean S. The underlying signifies the thing which underlies the contract price.

Expiry Date/Maturity Date The final termination date as specified in the contract at which the option ends or must be enacted. Unfortunately I have a habit for using these to words interchangeably but essentially in finance they mean the same thing. People tend to use expiry when talking about options, and maturity when talking about bonds but they can both be used.

4.2.2

What Actually Happens At Expiry?

If you currently hold an option, and the option is going to make a profit there are two possible ways you could go • wait until the option expiry date and then exercise the option, so if you have a call option

you will need to pay E to receive the share in exchange. If you have a put option you will

receive E in exchange for the share, if you don’t currently hold it you will need to go out and buy it on the market. • close out your position by taking the opposite position to the one you hold, i.e. a call

option holder will sell an identical call option and receive the profit on sale of the option.

The second option might be more attractive for a couple of reasons, for instance you might not actually want the share or have the money E to pay to buy it. By closing out your position you can trade in options without ever owning a share! This is very popular and you will see a large amount of trading in the last few days of an option’s lifetime as people close out their positions.

Position The term position is used here to denote the amount of financial contracts held, so closing out a position means return that number to zero.

4.3

How To Price An Option - Payoff Diagrams

We need to look at the payoff to calculate the price. The price or value of something does NOT depend on how much you paid for it. The price is what someone else is willing to pay for it, and they will only be interested in what the future profits might be. In our example above, the farmer will buy a European put option from Tescos. This means that they have an option to sell oranges at the preagreed exercise price of e600 per Ton in June 2016. If for example, the price of oranges in markets has gone down to e400 per Ton in June 2016, the farmer will agree to sell (exercising his option) and make a e600 − e400 = e200 per

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Lecture 4

Ton profit above the price he would have made selling on the market. If however, the price of oranges in markets has gone up to e800 per Ton in June 2016, he walks away from the agreement (neither making nor receiving any payments from Tescos) and is free to sell his oranges on the market. To know what price Tescos will charge the farmer for the cancellation option, we must look at the profit/loss for the holder/writer of the option under different scenarios.

Payoff Diagrams The payoff diagram of a financial contract spells out the cash flows under all possible scenarios on the expiry date, i.e. over all possible share prices on that date. Example 4.5. Draw the payoff diagrams for a call option and a put option. Solution 4.5. Fill in Figure 4.4 and Figure 4.5.

Profits This is not to be confused with the payoff which only considers what happens at expiry, profits must take account of the price paid (or the amount received if we are selling) on the day the option is bought or sold. To compare cash flows at different times we must compare like with like, so if we pay money for an option now and receive cash in the future, we must compare it with investing the money in a bank over the same period of time. The formula for profit is given by (using a call option as an example) profit = −C0 erT + C(S, T ). For the holder of European put option, the profit at time T is profit = P (S, T ) − P0 erT = max (E − ST , 0) − P0 erT .

38

(4.1)

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Lecture 4

Figure 4.4: Payoff diagram for the call option. 39

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Lecture 4

Figure 4.5: Payoff diagram for the put option. 40

Example 4.6. Find the share price on the expiry date in three months, for a European call option with an exercise price of £10 to give a gain (profit) of £14 if the option is bought for £2.25, financed by a loan with continuously compounded interest rate of 5% Solution 4.6.

41

Lecture 5 Trading With Portfolios How Can I Sell Something I Don’t Own? Often market participants will wish to take negative positions in the stock price, that is to say they will look to profit when the stock goes down. This practice is called short selling. But how can you sell something if you don’t own it? Well, traders are able to do this by exchanging an “IOU” contract with a stock holder to gain ownership of the stock for a period of time. They sell the stock now and invest the cash, waiting for the stock price to drop so they are able to buy it back for a lower price. Once the trader has bought it back they are able to hand the stock back to the holder cancelling out their IOU. There are laws surrounding short selling as it can be used to artificially raise supply (the number of stocks for sale) above the actual number of holders who want to sell, and since an increase in supply normally results in a drop in price there is obviously an incentive to do this. It has been blamed for several market crashes and is normally restricted or banned under certain circumstances.

5.1

Portfolio

A portfolio is range of investments held by an individual. We assume that portfolios may contain both positive and negative positions in stocks, bonds, call and put options. Mathematically, we tend to denote portfolios as Π, and your position in each asset will be positive if you are holding it (long) and negative if you are selling it (short). Example 5.1. Starting from zero, create a portfolio that is long or short one share by borrowing or investing money.

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Lecture 5

Solution 5.1.

43

Therefore the following portfolio Π = S + 2C − P describes a situation where the investor is long a stock, long two call options and short one put option. Example 5.2. Draw the payoff diagrams for going both long and short on both a put and call. Solution 5.2. See presentation slides for Lecture 5.

5.2

Trading With Portfolios

There are a variety of reason why investor may choose to buy a combination of options, stocks and bonds rather than just one or the other. We call the combination that an investor holds the portfolio, and in this lecture we discuss some of the common combinations that might be held by investors. We look at the payoff of those portfolios, and how to construct the payoff diagrams for the resulting portfolio. It is common for students to ask why an investor would take a certain position, particularly if looks as though the payoff at the end could be negative. To understand this you need to remember that the price of taking a certain position will be related to the likely payoffs – there is no easy way to make money on the stock market. If there is a large probability that there will be a negative payoff at the end of the contract (you pay out money rather than receiving it) then it is likely that the portfolio will be very cheap (or even negative in price, so you receive money at the start) to set up. The investor has to balance up the risk of winning and losing against their view on the market. Setting up different portfolios will allow them to maximise returns given what they think will happen in the future.

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5.2.1

Lecture 5

Straddle

One of the most common portfolios is the straddle. This involves buying both a call option and a put option with the same strike price and maturity date at the same time. The value of this portfolio is given by Π(S, t) = C(S, t; E) + P (S, t; E) and therefore the payoff at maturity (t = T ) is ( E − S if S < E Π(S, T ) = S − E if S ≥ E

(you exercise the put) (you exercise the call)

(5.1)

.

(5.2)

So this has large positive winnings if the stock price has a large downward movement or if the stock has a large upward movement. Given that the investor wins both ways, it follows that the cost of setting up such a portfolio will be relatively high. Example 5.3. Draw the payoff diagram for going long on a straddle and short on a straddle. Solution 5.3. Fill in Figure 5.1 and Figure 5.2.

The opposite position of a short straddle is what you get if you sell those same two options. We write the value of this portfolio as Π(S, t) = −C(S, t; E) − P (S, t; E).

(5.3)

At maturity we have Π(S, T ) =

(

−(E − S) −(S − E)

if S < E

(the buyer exercises the put)

if S ≥ E

(the buyer exercises the call)

.

(5.4)

Now this contract has negative payoff in all scenarios, so why would you do it? Well if the payoff is always negative then the person you sell it to you will have to give you a lot of money at the start. This means that you make money by hoping that there isn’t a large movement up or down. As long as movements in stock price are small you are able to charge enough at the start to still make a profit after you have paid out any winnings at the end. This can be risky as you tend to win small amounts very often but there is always a small chance you might lose big and potentially go bankrupt – making this a very risky strategy!

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Lecture 5

Figure 5.1: Payoff diagram for a long straddle. 46

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Lecture 5

Figure 5.2: Payoff diagram for a short straddle. 47

Example 5.4. An example of large profits: S0 = 40, E = 40, C0 = 2, P0 = 2. Can you find the expected return if the stock price at T is given by the following tree? 60   S0 = 40  HH H 20 p= 1 4 3 p= 4

Solution 5.4.

5.2.2

Bull Spread

A bull spread is a slightly cheaper way to bet on an upward movement in the stock price than just buying the call option. We create it by buying a call then selling one with a slightly higher exercise price, this gives Π(S, t) = C(S, t; E1 ) − C(S, t; E2 )

(5.5)

where E2 > E1 . So why would you invest like this? Well if you expect only a small movement in price this is a cheaper way to exploit this than buying the call option.

48

Example 5.5. Sketch the payoff diagram for a bull spread. Solution 5.5.

Fill in Figure 5.3.

You may hear the term bull to describe market conditions, as a bull market is one in which confidence is high and stocks are growing. The term bear indicates the opposite, and it refers to a situation where stocks are losing value, so a bear spread is used exploit down movements in

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Lecture 5

Figure 5.3: Payoff diagram for bull spread. 50

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Lecture 5

price. We create a bear spread by buying and selling a put option, or Π(S, t) = P (S, t; E1 ) − P (S, t; E2 )

(5.6)

where E1 > E2 . Example 5.6. Sketch the payoff diagram for a bear spread. Solution 5.6. Left as an exercise.

5.3

Risk Free Investments

When dealing with contracts that make payments in the future there is always a risk that you won’t get paid, even banks sometimes run out of money! In economics, we like to imagine that there does exist an investment in the economy that is completely risk-free, and we can then use this investment to compare against our options, portfolios etc. Even before we begin applying our model to the stock price there are things we can say something about the price of financial contract, for instance the upper and lower bounds, that must be true under any situation. The risk free investment is so important because the true market price (the price everyone agrees on) is known, since the only thing that makes a value subjective is risk. We talk about this more in the next lecture, but the main focus of this course is describing a strategy in which you can take risky financial derivatives and make them completely risk free. This ability to effectively remove risk is what transformed the world of investment banking.

5.3.1

Bond

A Bond is a contract that yields a known amount F , called the face value, on a known time T , called the maturity date. The authorised issuer (for example, government) owes the holder a debt and is obliged to repay the face value at maturity and may also make interest payments (the coupon). We will assume in this course that the only bonds traded are those issued by governments that can be assumed to be risk free. The term B(t) will denote a risk free investment in government bonds.

Face Value/Principle The final payment amount as described in the contract, usually denoted F . The holder of the bond can collect the payment of F on the maturity date.

Maturity The maturity date is the date written in the contract on which the holder with receive the face value payment.

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Interest Rate This is the growth rate of the bond, or how much value is added for the holder over a period of time. If I deposit money into a restricted access savings account in a bank then the bank is effectively selling me a bond. Imagine I pay them say £1000 now to receive £1100 in five years time. I have bought a bond with face value F = 1100 and maturity date T = 5, the interest rate could be worked out from er×5 = 1.1 . The bank now owes me money and I am hoping that they will have the money in five years time to pay me!

Coupon These are payments from the seller of the bond to the buyer. Usually if the bond has a very long maturity date (they can be up to 50 years or more!) the value of the bond would be very low without these payments. Coupons enable the seller of the bond to receive more money from the initial sale.

Zero Coupon Bond Simply a bond that doesn’t pay coupons. Example 5.7. Write down the return on a risk free bond if the interest rates are constant, and calculate the value of the bond if B(t = T ) = F and r is constant. Solution 5.7.

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No-Arbitrage Principle In the next lecture we introduce the no-arbitrage principle which is the bedrock of mathematical finance. It allows us to draw equivalence between different types of financial contracts and construct arguments on how to price financial contracts.

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Lecture 6 Risk and No Arbitrage 6.1

Risk

Risk A risky investment is one where the investment’s actual return will be different than what is expected. Risky investments may go up as well as down and they include the possibility of losing. If a financial contract that is trading in a market is risky, economic theory tells us that different individuals will value that contract differently according to their own current situation. If for instance one investor is extremely rich, and another is extremely poor, they will have a different view on an investment according to what they feel as though they are able to lose. The rich investor may feel they are able to gamble for a big win since small losses won’t hurt them whereas the poor investor cannot take risks in case they end up without enough money to pay the bills. Dealing with risk in a market is extremely difficult because of this non-uniformity and before we show how to remove risk from valuations it is important to note that valuing with risk is possible but very subjective! Example 6.1. Two products are being sold on the market and they offer a payoff according to some future events at time T . Everyone on the market knows (and agrees on) the probabilities associated the events. Product A is described by • Pays £500 with 50% probability at time T • Pays £1500 with 50% probability at time T Product B is described by • Pays £0 with 99% probability at time T

54

• Pays £100000 with 1% probability at time T Imagine both products are available to buy at £750, which offers the better investment? Solution 6.1.

Example 6.2. Imagine there are two similar products being sold on the market, where Product A is described by • Pays £1000 with 100% probability at time T Product B is described by • Pays £250 with 100% probability at time T Imagine the products are available to buy at £800 and £200, which offers the better investment?

55

Solution 6.2.

Example 6.3. Imagine there are two products being sold on the market, where Product A is described by • Pays £1000 with 100% probability at time T Product B is described by • Pays £1000 with 99% probability at time T • Pays £2000 with 1% probability at time T What can we say about the price of these two products? Solution 6.3.

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6.2

No Arbitrage

Arbitrage Opportunity An arbitrage opportunity is a way to make a risk-free profit. Mathematically we define such an opportunity as one which yields a positive payoff (in all circumstances) with a zero investment. This can be written Π0 = 0 ⇒ ΠT > 0. Note that it must be strictly greater than zero to represent arbitrage. Note that in this course all circumstances means all possible values of ST at maturity, so profit must be non zero everywhere in ST ∈ [0, ∞).

No-Arbitrage In a complete market there are never opportunities to make risk-free profit. Therefore, any price of a contract which admits arbitrage can never exist in a market. Example 6.4. To see why this must hold true let us consider the opposite, that there is an opportunity. Assume that there are two risk free products on the market offering to pay the holder £10 on the same date T in the future. Imagine that one of those products, Product A is selling for £5 and the other Product B for £7. What happens? Well, if everyone can make a free choice to buy either product then everyone with choose to buy A, as it offers best value. So in a market, can you say what happens to the demand (and hence price) of each product?

57

Solution 6.4.

Results of No-Arbitrage Equivalent Contracts So if two products are identical in every way they must have ˆ are two portfolios with the same payoff the same price in a market. Assume that Π and Π ˆT ΠT = Π ˆ t for all t. at maturity t = T . If no-arbitrage holds, then Πt = Π

Examples If we were to construct a risk-free portfolio Π, then the return on that must be the same as that of a risk-free investment, or dΠ dB = = rdt, (6.1) Π B where B is a risk free bond and r is the risk-free interest rate. This result must hold for any risk-free investment and for all time t.

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Finally, if the portfolio we hold has no profit or loss at maturity, i.e. ΠT = 0, then there is no risk in holding the portfolio, so it must be true that ⇒ Πt = 0

ΠT = 0

∀t.

(6.2)

ˆ are two portfolios with Comparing Contracts Now consider the situation when Π and Π payoff that satisfy the following condition under all circumstances ˆT. ΠT ≥ Π ˆ t for all t. at maturity t = T . If no-arbitrage holds, then Πt ≥ Π

Examples In the simplest case, we can say that a risky contract with positive payout ΠT ≥ 0 must at least as valuable that holding nothing. Mathematically we can write this as ΠT ≥ 0

⇒ Πt ≥ 0

∀t.

(6.3)

Now consider a portfolio Π such that its payout at T is bounded below by some fixed amount E, so that ΠT ≥ E. Then consider the risk-free bond contract B that pays E with maturity T , we have ΠT ≥ B T . We know then for certain that the value of the contract Π must be greater than or equal to the value of the bond B for all time, or Πt ≥ B t

6.3

or

Πt ≥ Ee−r(T −t) .

(6.4)

Put Call Parity

This result shows a relationship between the price of put and calls in a market that must hold at all times. In fact, often only the price of a call is calculated and the corresponding put values are derived from this relationship.

59

Example 6.5. Consider a portfolio of the form Π = S + P − C.

(6.5)

Calculate the payoff at maturity and by using no arbitrage theory the so-called put-call parity: St + Pt − Ct = Ee−r(T −t) .

(6.6)

Solution 6.5.

It shows that the value of European call option can be found from the value of European

60

put option with the same strike price and maturity: C0 = P0 + S0 − Ee−rT . Example 6.6. Use this formula for a call option along with the No-Artbitrage Theorem, derive the lower bound for the call option: Ct ≥ St − Ee−r(T −t) . Solution 6.6.

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(6.7)

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Lecture 6

Example 6.7. (a) Find a lower bound for a six month European call option with the strike price £35 when the initial stock price is £40 and the risk-free interest rate is 5% p.a. (b) Consider the situation where the European call option is £4. Show that there exists an arbitrage opportunity. Solution 6.7.

62

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Lecture 7 Arbitrage 7.1

No Arbitrage Theorem

“Arbitrage cannot exist in a market” No arbitrage is a strong assumption about the prices that can exist in a real market. If we make an assumption about the price of something, and that assumption allows for arbitrage to exist, then our assumption is clearly wrong. For example, to prove bounds on our option price, say Ct ≥ 0, we can assume that the opposite is true and exists in the market, namely Ct < 0.

(7.1)

If we can show that assuming (7.1) to be true leads to an arbitrage opportunity, this contradicts our no arbitrage assumption. Then it can only be that our assumption about the call price was false and in fact Ct ≥ 0 must hold true.

7.2

Bounds on a Put and Call Option

Example 7.1. Sketch the upper and lower bounds for a call St − Ee−r(T −t) ≤ Ct ≤ St

(7.2)

Ee−r(T −t) − St ≤ Pt ≤ Ee−r(T −t)

(7.3)

and a put option

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Figure 7.1: A sketch of the bounds on options prices. 65

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Lecture 7

Solution 7.1. Fill in Figure 7.1.

7.3

Put Call Parity by No Arbitrage

We have already shown that the put call parity will hold given our assumption on the return of a risk free portfolio. Now we try to use a proof by contradiction argument to show that any deviation from this rule will lead to an arbitrage opportunity. Let us assume that the price of the put option P0 is too high relative to the call option price C0 . If this is the case then we can write P0 > C0 − S0 + Ee−rT .

(7.4)

We should note here that neither P0 nor C0 need to be outside the no-arbitrage bounds for their individual prices, it is only the combination that admits arbitrage.

66

Example 7.2. Show that the correct portfolio for arbitrage will be Π=C −P −S+B Solution 7.2.

67

(7.5)

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Lecture 7

Example 7.3. Find arbitrage and show that since our initial assumption on the put price was false the following must hold true Pt ≤ Ct − St + Ee−r(T −t) Solution 7.3.

68

(7.6)

It is relatively trivial to show that the opposite case will also admit arbitrage, namely if the put option is priced too. So let us now assume that the put option satisfies the following condition Pt < Ct − St + Ee−r(T −t) .

(7.7)

Example 7.4. Show that the correct portfolio for arbitrage will be Π=P −C +S−B

(7.8)

Solution 7.4.

Example 7.5. Find arbitrage and show that since our initial assumption on the put price was false the following must hold true Pt ≥ Ct − St + Ee−r(T −t)

69

(7.9)

Solution 7.5.

Given the two conditions (7.6) and (7.9) we see that Pt = Ct − St + Ee−r(T −t) must hold.

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(7.10)

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Lecture 7

Example 7.6. Three month European call and put options with the exercise price £12 are trading at £3 and £6 respectively. The stock price is £8 and interest rate is 5%. Show that there exists arbitrage opportunity. Solution 7.6.

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Lecture 8 Binomial Trees Midterm Test Don’t forget the midterm test is on Friday 9th March at 12pm! Check your emails for guidance on which room your test will be in (unless you have special requirements and have been assigned another room by the admin office).

8.1

One Step Binomial Model

Assume that we have a binomial model for the stock price. Initially the stock price is S0 , but at the future time t = T the stock price can either move up from S0 to S0 u or down from S0 to S0 d ( u > 1; d < 1) with probability q and 1 − q respectively. Now for the option price we have

C0 = C(S0 , t = 0) is the value of the option at the initial stock price and the initial time. If the

stock price moves up to S0 u, we say that the option price under this scenario is Cu = C(ST = S0 u, t = T ). Similarly, if the stock price moves down we get Cd = C(ST = S0 d, t = T ). Example 8.1. Sketch the binomial trees for the stock and option price. Solution 8.1.

We draw the trees in figure 8.1 and figure 8.2.

From (8.1) we can calculate the mean and variance of the stock price at t = T . For instance E[ST ] = qS0 u + (1 − q)S0 d.

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(8.1)

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Lecture 8

Figure 8.1: One step stock price tree

Figure 8.2: One step option price tree

Given that the value of the option is known at time t = T as it is just given by the payoff formula it implies that we should be able to say something about the price of the option C0 at t = 0 given the expected value at T is E[CT ] = qCu + (1 − q)Cd . Example 8.2. Can we use (8.2) to price our option? Solution 8.2.

74

(8.2)

8.2

Risk Free Portfolio

The trick here is to sell the call and buy the stock at the same time. We can trade of any likely increase/decrease in one (say selling the call) by cancelling it with a decrease/increase in the other (holding the stock) in order to get the same payoff in both cases. From earlier lecture we know how to deal with guaranteed payoffs (by no-arbitrage they must have a risk-free return) so we will be able to come up with the true price of the option. Consider the portfolio Π = ∆S − C

(8.3)

where ∆ is some number that we can choose. Note here that if ∆ is chosen to be a fraction then we can’t really trade fractions of a share. However, if we are talking about a large institution such as a bank they may be selling 10000 call options, so if ∆ = 0.6251 then they buy 6251 shares. Example 8.3. Using this portfolio show that this value for ∆ ∆=

Cu − Cd . S0 (u − d)

gives a constant payoff ΠT = K at maturity. Solution 8.3.

75

(8.4)

Can we price this now? With no risk in the payoff the answer is YES! If ΠT = K we have Π0 = Ke−rT since the payoff is fixed we just discount at the risk-free rate. Example 8.4. Using this value for Π0 and ∆, sub back into earlier equations and rearrange to obtain the following formula for the call option price C0 = e−rT [pCu + (1 − p)Cd ] . Solution 8.4.

76

This final simplified formula (8.4) is called the “Risk Neutral Valuation”. We can in fact interpret p as a probability, but it is not the same as q. The probability q does not appear in the formula because the payoff K is the same in both cases so q does not matter. The formula states that the price is equal to the expected value under the risk neutral probabilities discounted back to t = 0, or C0 = e−rT Ep [CT ]

(8.5)

So in order to interpret what the probability p means, you need to think about it in reverse. The value of p is not the probability of the stock moving up or down, it is the implied probability we would need to put in the formula to get the price we observe on the market. We assume that agents on the market that set the price (Banks, large institutions) are always hedging away their risk. Therefore they set the price using the no-arbitrage hedging strategy, but a single observer will see that price on the market and may ask the question what is the probability that will produce that price - and the answer will be p not q.

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Example 8.5. A stock price is currently $40. At the end of three months it will be either $44 or $36. The risk-free interest rate is 12%. What is the value of three-month European call option with a strike price of $42? Use noarbitrage arguments and risk-neutral valuation. Solution 8.5.

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Lecture 9 Two Step Binomial Trees 9.1

Risk Neutral Valuation

In the last lecture we derived the value of the Call option price as C0 = e−rT (pCu + (1 − p)Cd ) , where

erT − d u−d This formula is known as a risk-neutral valuation. p=

Example 9.1. Show that in order for the No-Arbitrage to hold in a one step binomial tree model we require d < erT < u

(9.1)

0 ≤ p ≤ 1.

(9.2)

and hence

Solution 9.1.

79

80

Given that (9.1) holds, we can interpret the variable 0 ≤ p ≤ 1 as a probability of an up

movement in the stock price. The probability of up q or down movement 1 − q in the stock price plays no role whatsoever! Why???

Let us find the expected stock price under the risk neutral probabilities at t = T : erT − d erT − d S0 u + (1 − )S0 d = S0 erT . u−d u−d This shows that stock price grows on average at the risk-free interest rate r. Since the Ep [ST ] = pS0 u + (1 − p)S0 d =

expected return is r, this is a risk-neutral world.

In the Real World: E [ST ] = S0 e

In a Risk-Neutral World:

µT

Ep [ST ] = S0 erT

Example 9.2. What is a Risk-Neutral World? Solution 9.2.

81

We have our Risk-Neutral valuation formula, C0 = e−rT Ep [CT ]

(9.3)

This formula can is the expected payoff in a risk-neutral world (at maturity), discounted at risk-free rate r. It is taking the expectations under the risk-neutral measure which allows us to ignore risk and discount at the risk-free rate to derive a single price for the option.

9.2

Two Step Tree

Now the stock price changes twice, each time by either a factor of u > 1 or d < 1. We assume that the length of the time step is ∆t such that T = 2∆t. After two time steps the stock price will be S0 u2 , S0 ud or S0 d2 . Example 9.3. Draw the two step tree for stock prices. Solution 9.3.

82

The call option expires after two time steps producing payoffs Cuu , Cud and Cdd respectively. Example 9.4. Write down formula for Cuu , Cud and Cdd in terms of S0 , E, u and d. Solution 9.4.

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Lecture 9

Example 9.5. Draw and annotate the two step tree for option prices. Solution 9.5.

The purpose is to calculate the option price C0 at the initial node of the tree. Before we can find C0 we must apply the risk-neutral valuation backwards in time by first finding Cu and Cd . We simply use the one step formula to price the option at these nodes to obtain Cu = e−r∆t (pCuu + (1 − p)Cud ) and Cd = e−r∆t (pCud + (1 − p)Cdd ) . Then the final step is to use a one step tree for the current option price C0 = e−r∆t (pCu + (1 − p)Cd ) .

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Lecture 9

Example 9.6. Substitution gives
C0 = e−2r∆t p2 Cuu + 2p(1 − p)Cud + (1 − p)2 Cdd ,

(9.4)

for the option price. What do p2 , 2p(1 − p) and (1 − p)2 represent? Solution 9.6.

Finally, the current call option price is C0 = e−rT Ep [CT ] ,

T = 2∆t.

The current put option price can be found in the same way: P0 = e−2r∆t p2 Puu + 2p(1 − p)Pud + (1 − p)2 Pdd or P0 = e−rT Ep [PT ] .

85




(9.5)

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Lecture 9

Example 9.7. Consider six months European put with a strike price of £32 on a stock with current price £40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. Solution 9.7.

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Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test • Friday 9th March - 12pm • Examples Sheet 1 – 4 (not qu 3 or qu 5 on sheet 4) • Lectures 1-9

10.1

A Discrete Model for Stock Price

Reminder: The continuous model of Stock Price Earlier in Lecture 2 we proposed the continuous random model for the stock price: dS = µSdt + σSdW Example 10.1. Given current stock price S0 at t = 0, what possible values of the stock price at time t in this model? Solution 10.1.

88

The discrete model of Stock Price The binomial model for the stock price we have described in the last 2 lectures is a discrete time model: • The stock price S changes only at discrete times ∆t, 2∆t, 3∆t, ... • The price either moves up S → Su or down S → Sd with d < er∆t < u. • The probability of up movement is q. Example 10.2. If there are n steps in the tree, how many possible stock prices can we observe (at all times)? Solution 10.2.

89

10.2

Binomial Stock Price Tree

So let us build up a tree of possible stock prices. The tree is called a binomial tree, because the stock price will either move up or down at the end of each time period. Each node represents a possible future stock price. We divide the time to expiration T into several time steps of duration ∆t = T /N , where N is the number of time steps in the tree. What we want to do is have the ability to increase N to a large enough number so that the binomial tree approximates the continuous model. Example 10.3. Sketch the binomial tree for a stock price with N = 4. Solution 10.3. See figure 10.1.

We introduce the following notations: • Snm is the n-th possible value of stock price at time-step m∆t. • Snm = un dm−n S00 ,

where n = 0, 1, 2, ..., m.

• S00 is the stock price at the time t = 0. Note that u and d are the same at every node in the tree.

For example, at the third time-step 3∆t, there are four possible stock prices:

S03 = d3 S00 ,

S13 = ud2 S00 , S23 = u2 dS00 and S33 = u3 S00 . At the final time-step N ∆t, there are N + 1 possible values of stock price.

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Figure 10.1: A sketch of the 4 step stock binomial tree 91

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10.3

Lecture 10

Binomial Call Option Tree

Example 10.4. Sketch the binomial tree for a call option price with N = 4. Solution 10.4. See figure 10.2.

We denote by Cnm the n-th possible value of call option at time-step m∆t. In order to calculate the price of the call option at S = S0 and t = 0, we must solve recursively just as we did with the two step tree. At each substep in the tree we apply a Risk Neutral Valuation according to the formula
m+1 Cnm = e−r∆t pCn+1 + (1 − p)Cnm+1 .

Here 0 ≤ n ≤ m, 0 ≤ m < N and p =

er∆t −d u−d .

Now of course, before we can move recursively through the tree, we need a final condition to

apply at t = T . For a call option we have
CnN = max SnN − E, 0 ,

where n = 0, 1, 2, ..., N and E is the strike price.

The current option price C00 is again the expected payoff in a risk-neutral world, discounted at risk-free rate r: C00 = e−rT Ep [CT ] . Example 10.5. Why not use arbitrage arguments? Solution 10.5.

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Figure 10.2: A sketch of the 4 step call option binomial tree 93

10.4

Approximating the Continuous Model

If we wish to come up with a binomial model that approximates the continuous model, we need to choose the parameters of the binomial model u, d and p so that they can match the properties of the continuous model, in particular the mean and variance of the model. We assume that the stock price starts at the value S0 and the time step is ∆t. Let us find   the expected stock price, E [S] , and the variance of the return, var ∆S , for continuous and S discrete models.

Expected stock price For the continuous model we have E [S] = S0 eµ∆t . On the binomial tree: E [S] = qS0 u + (1 − q)S0 d. Example 10.6. Combine these two results for the first equation needed to match the models. Solution 10.6.

94

Variance of the stock price For the Continuous model we have: 

 ∆S var = σ 2 ∆t S Example 10.7. Derive the variance for the binomial tree and hence the second equation. Solution 10.7.

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What is Left? This gives us two equation for three unknowns, so what to do? In fact we have a free choice, one of the most popular models is the CRR model which imposes u = d−1 so that an up movement followed by a down movement takes you back to where you started.

How to solve these equations? The solution to these equation is rather tedious to derive but you can have a go at it in examples sheet 4!

96

Lecture 11 American Options and Replicating Portfolios American Option is an option that may be exercised at any time prior to expire (t = T ). The decision about when to exercise is up to the holder of the option. If we are going to price the option we will need to determine when it is in the holders best interests to exercise the option. In fact we can show that this is not subjective! It can be determined in a systematic way! Example 11.1. Take a look at the historical observation of a stock price St in figure 11.1. Can you decide when it would have been best for the holder to exercise the option? Solution 11.1.

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10

Stock Price S(t)

8

6

t0

t1

4

2

0 0

0.25

0.5

0.75

1

Time t

10

Exercise value S(t) − E

8

t0

t1

6

4

2

0 0

0.25

0.5

0.75

1

Time t

Figure 11.1: A simulated plot of a stock price S(t) as a function of time. Underneath we have the corresponding payoff at exercise E − S(t) where E = 10

98

We are able to price the American put option value by using the simple condition that P (S, t) ≥ E − S

(11.1)

for all S and t. This simply says that the value must be greater than or equal to the payoff function. Example 11.2. If P < max(E − S, 0), show there is an arbitrage opportunity. Solution 11.2.

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This condition looks simple enough but it results in a non-linear problem with no analytic solution. However, we can derive a rather simple solution using the binomial tree – which is one of the reasons why binomial trees are popular with quantitative analysts.

11.1

American Option Tree

We denote by Pnm the n-th possible value of put option at time-step m∆t. Now the formula we had for the European Put Option is
m+1 Pnm = e−r∆t pPn+1 + (1 − p)Pnm+1 .

Here 0 ≤ n ≤ m and the risk-neutral probability p =

er∆t −d u−d .

In order to make this an American Put Option we simply include the condition (11.1) into

our calculation by taking the maximum of the value of continuing to the next step as before, and the value of exercising now. This looks like:

Pnm



= max e

−r∆t

m+1 pPn+1

+ (1 −

p)Pnm+1




, max(E −

Snm , 0)



,

(11.2)

where Snm is the n-th possible value of stock price at time-step m∆t. The final condition is the same as before


PnN = max E − SnN , 0 ,

where n = 0, 1, 2, ..., N , and E is the strike price.

Example 11.3. Can we still use the simplified formulas (9.4) and (9.5) to price an American option? Solution 11.3.

100

Example 11.4. Evaluating an American Put Option on a Two-Step Tree.

• We assume that over each of the next two years the stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%.

• Find the value of a 2-year American put with a strike price of $52 on a stock whose current price is $50.

Solution 11.4.

101

102

11.2

Replicating Portfolios

This next part of the course asks whether we replicate an options contract with a trading strategy (just by buying and selling stocks). Being able to do this is extremely important for the big players in the stock market (investment banks) as it allows them to lock in the profit of a contract at the point of sale, rather than having to wait until the contracts are settled. The term strategy here indicates that we will look to change the number of stocks we hold according to some objective. Our aim is now to establish a portfolio with a trading strategy using stocks and bonds so that the payoff of a call option is completely replicated, i.e. ΠT = CT = max (S − E, 0) Then to prevent a risk-free arbitrage opportunity, if the terminal values are the same the current values should also be identical. We can then say that the portfolio replicates the option, and by “The Law of One Price” we have Πt = Ct . Consider the replicating portfolio of ∆ shares held long and N bonds held short. The value of the portfolio is: Π = ∆S − N B. A pair (∆, N ) is called the trading strategy. The values of ∆ and N may be different for every value of S and t. The portfolio is called self-financing if any changes in ∆ (buying/selling shares) are offset by a change in N (selling/buying bonds).

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Lecture 11

Replicating the One-Step Binomial Tree

Example 11.5. For the one-step binomial tree model, can we find (∆, N ) such that ΠT = CT and Π0 = C0 ? Hence show that C0 = e−rT (pCu + (1 − p)Cd ) , where p=

erT − d . u−d

Solution 11.5.

104

(11.3)

105

Lecture 12 The Black-Scholes Model

The Black-Scholes model for option pricing was developed by Fischer Black, Myron Scholes in the early 1970’s. This model is the most important result in financial mathematics. Up until that point the traders and banks were using hedging techniques to minimise their risk but did not do so in a way that was consistent with the market and pricing framework. Black and Scholes were the first to show how risk could be eliminated from the their framework – this was the most important step forward. The Black-Scholes model can be used to calculate an options contract using a small set of parameters: contract specific parameters (eg strike price E, time to expiration T ), the stock price S0 , the volatility σ and the risk free interest rate r. The most important parameter intentionally missing in that list is µ, the growth rate of the stock. A second benefit of the model was to not only remove risk but also remove the need to estimate the growth rate of the stock.

106

12.1

Model Assumptions

Example 12.1. Can this model be used to price any financial contract? Solution 12.1.

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12.2

Lecture 12

Deriving the Black-Scholes Equation

In the following we denote by V (S, t) the value of an option. We use the notations C(S, t) and P (S, t) for the value of a call and a put when the distinction is important. The first step is to set up a portfolio consisting of a long position in one option and a short position in ∆ shares. The value of the portfolio is Π = V − ∆S.

(12.1)

The next step is to find the number of shares that makes this portfolio risk free. To do this we need to return to Itˆ o’s lemma to find out how the value of this portfolio changes over time. Example 12.2. Use Itˆ o’s lemma to obtain     1 2 2 ∂2V ∂V ∂V ∂V + σ S + µS − ∆µS dt + σS − ∆σS dW dΠ = ∂t 2 ∂S 2 ∂S ∂S for the change in portfolio value. Solution 12.2.

108

(12.2)

Can we eliminate risk? Yes, we can choose ∆ = Example 12.3. Put ∆ =

∂V ∂S

into (12.2).

Solution 12.3.

109

∂V ∂S

to remove the dW terms.

No-Arbitrage Principle The return on a risk-free portfolio must be rdt. So we may write

dΠ = rdt Π

(12.3)

for the return on the portfolio. Example 12.4. Using the no-arbitrage principle (12.3) and (12.2) derive the Black Scholes equation ∂V ∂V 1 ∂2V + σ 2 S 2 2 + rS − rV = 0 ∂t 2 ∂S ∂S Solution 12.4.

110

(12.4)

Scholes received the 1997 Nobel Prize in Economics. It was not awarded to Black in 1997, because he died in 1995. Black received a Ph.D. in applied mathematics from Harvard University. Example 12.5. How do we solve for the price of a financial contract? Solution 12.5.

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E

C(S, t = 0) C(S, t = T ) 0

E

2E

ST

Figure 12.1: Plot of a European Call Option value against stock price.

For a call option, the final condition is given by C(S, t = T ) = max(S − E, 0). and an example solution is plotted in figure 12.1.

112

Lecture 13 Exact Solution to the Black-Scholes Equation 13.1

Boundary Conditions

Boundary condition are important when solving PDE problems such as the Black Scholes equation (12.4), especially if you need to solve them numerically. In fluid dynamics there are theorems and a variety of techniques to describe how a solution behaves near a boundary. There are different types of boundaries but most in finance are quite passive – that is to say that essentially nothing happens near the boundaries.

Call Option Example 13.1. Consider a call option, if the stock price is close to or equal to zero, what is the likelihood that the option will be exercised? Hence show that the value of the option at S = 0 is simply C(S = 0, t) = 0 Solution 13.1.

113

(13.1)

Example 13.2. Consider a call option, if the stock price is extremely large, what is the likelihood that the option will be exercised? Hence show that the value of the option as S → ∞ is simply C(S, t) → S as S → ∞ Solution 13.2.

114

(13.2)

Put Option Example 13.3. In a similar way, show that the boundary conditions for a put are P (0, t) = Ee−r(T −t)

(13.3)

and P (S, t) → 0

as

Solution 13.3.

115

S→∞

(13.4)

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13.2

Lecture 13

Solution for a Call Option

The Black-Scholes equation ∂V ∂V 1 ∂2V + σ 2 S 2 2 + rS − rV = 0 ∂t 2 ∂S ∂S with the final conditions for a call option C(S, T ) = max(S − E, 0) and boundary conditions, (13.1) and (13.2), has the explicit solution: C(S, t) = SN (d1 ) − Ee−r(T −t) N (d2 ), where

1 N (x) = √ 2π

and d1 =

Z

x

e−

y2 2

dy

(13.5)

(cumulative normal distribution)

−∞

ln(S/E) + (r + σ 2 /2)(T − t) √ , σ T −t

√ d2 = d1 − σ T − t.

We simply state this solution in the course – it will not be derived! See figure 13.1 for a plot of the solution of the call option. It demonstrates several features, including the shape of the solution along with an upper and lower bound on the value. The cumulative normal distribution has the following properties: N (−x) = 1 − N (x) N (−∞) = 0 1 2 N (∞) = 1 N (0) =

(13.6) (13.7) (13.8) (13.9)

These properties are important if we wish to investigate how the solution behaves when parameters in the problem are varied.

116

Example 13.4. Why does the cumulative normal distribution appear in the solution? Solution 13.4.

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E

Lecture 13 Maximum Value

Actual Value

Speculative Value    Intrinsic Value

0

0

E

  

ST

Figure 13.1: A plot of the call option solution showing some features. Example 13.5. Calculate the price of a three-month European call option on a stock with a strike price of £25 when the current stock price is £21.6. The volatility is 35% and the risk-free interest rate is 1% p.a. Solution 13.5.

118

Example 13.6. Find the limit lim C(S, t).

σ→∞

Solution 13.6.

119

120

Lecture 14 The Greeks

In practice, options are traded in a market and if you want to know the price of the option we don’t use the Black-Scholes model to find it – you just look on the market to see how much someone is selling it for! So why do we need the Black Scholes model? Remember how the Black-Scholes model is derived, the first step is to set up portfolio with Π = V − ∆S and choose ∆ to make the portfolio risk free. So how do we choose ∆ in the real world? We work out what it should be using the Black-Scholes model. Example 14.1. How might a trader use the Black Scholes model in a real stock market? Solution 14.1.

121

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14.1

Lecture 14

Calculating ∆

So if we are going to trade on the market with options, we will need to know how to calculate the value of ∆. So let us show that the value of ∆ can be calculated from the simple formula ∆=

∂C = N (d1 ). ∂S

(14.1)

The method we are going to use is to differentiate the explicit solution (13.5) from Lecture 13 with respect to S. Example 14.2. Differentiate (13.5) with respect to S to obtain   ∂d 1 ∆ = N (d1 ) + SN 0 (d1 ) − Ee−r(T −t) N 0 (d2 ) ∂S for the first stage. Solution 14.2.

123

(14.2)

Example 14.3. Find the value of ∆ for a 6-month European call option on a stock with a strike price equal to the current stock price (t = 0). The interest rate is 6% p.a. and the volatility σ = 0.16. Solution 14.3.

124

Example 14.4. Find the Delta for a European put option by using the put-call parity: S + P − C = Ee−r(T −t) . Solution 14.4.

Example 14.5. Sketch the value of the call option and its corresponding ∆. Solution 14.5. Fill in figure 14.1.

Example 14.6. Sketch the value of the put option and its corresponding ∆. Solution 14.6. Fill in figure 14.2.

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Figure 14.1: A plot of the call option and its delta. 126

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Lecture 14

Figure 14.2: A plot of the put option and its delta. 127

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14.2

Lecture 14

Greeks

We may write the option value under the Black-Scholes model as V = V ( S, t ; σ, r, T ). |{z} | {z } variables parameters

Any of those variables or parameters may change over time, so how to traders manage this risk? In fact, in exactly the same way we could eliminate the risk in S by hedging with

∂V ∂S

,

we can do the same to other types risk using the differential with respect to that quantity. As a result, the differential of the value function with respect to different parameters has become extremely important to traders. Because of the mathematical notation associated with each of the quantities, these became known as the Greeks. They represent the sensitivities of options to a change in an underlying variable or parameter on which the value of an option is dependent. • Delta: ∆ =

∂V ∂S

measures the rate of change of option value with respect to changes in the

underlying stock price

• Gamma: Γ =

∂2V ∂S 2

=

∂∆ ∂S

measures the rate of change in ∆ with respect to changes in the

underlying stock price. See Examples Sheet 7 where we show that Γ =

• Vega: ∂V ∂σ

measures the sensitivity to volatility σ.It can be shown that

• Rho:

∂V ∂r measures −r(T −t) t)e N (d2 ).

ρ =

∂V ∂σ

N 0 (d1 ) √ . Sσ T −t

√ = S T − tN 0 (d1 )

the sensitivity to interest rate r. One can show that ρ = E(T −

Example 14.7. Try sketching the other Greeks for a put and call. Solution 14.7. Left as an exercise.

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Lecture 15 More on Replicating Portfolios 15.1

Replicating Portfolios

If we can replicate the payoff of a contract with a combination of products that are already sold on the market, we can find the value of that contract. But more importantly, it allows banks to lock in risk-free profit at the point of sale rather than having to wait until maturity. We will go through an example of this later on in the lecture, but first let us show how to setup the replicating portfolio strategy. Example 15.1. What does it mean to replicate a financial contract? Solution 15.1.

129

Our aim is to find a replicating strategy with stocks and bonds that can produce the BlackScholes equation. Given some initial investment, we can divide it between ∆ shares held long and N bonds held short. Recall that a pair (∆, N ) is called a trading strategy. The value of the portfolio is: Π = ∆S − N B. Now we introduce the familiar models for stocks and bonds • SDE for a stock price S(t): • Equation for a bond price B(t):

dS = µSdt + σSdW . dB = rBdt.

Example 15.2. Consider a financial contract Vt written on the underlying asset S, use the portfolio Π under a self-financing constraint to replicate the contract and derive the Black-Scholes equation ∂V 1 ∂2V ∂V + σ 2 S 2 2 + rS − rV = 0. ∂t 2 ∂S ∂S Solution 15.2.

130

(15.1)

131

Example 15.3. Assume that the risk-neutral valuation of a call option is C0 = £10, but a trader is able to sell the option for £11. How does a trader use replication to lock in a guaranteed profit now as opposed to at maturity? Solution 15.3.

132

15.2

Static Hedging

Let us remember the Put-Call Parity. We set up the portfolio consisting of a long position in one stock, long position in one put and a short position in one call both with the same maturity and strike price. The portfolio is Π = S + P − C. Example 15.4. (a) What is the payoff of this portfolio at maturity? (b) What is the risk when holding this portfolio? Hint: The risk of a portfolio is the variance of the return. Solution 15.4.

The Put-Call Parity is an example of complete risk elimination where we carry out only one transaction in call/put options and underlying security.

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Lecture 15

Dynamic Hedging

Let us consider the dynamic risk elimination procedure. We could set up a portfolio consisting of a long position in one call option and a short position in ∆ shares Π = C − ∆S We know that we can eliminate the random component in Π by choosing ∆=

∂C . ∂S

This is a ∆-hedging strategy! It required a continuous rebalancing of the number of shares in the portfolio Π. Example 15.5. ∆ is a function of S and t, so how does this work in the real world? Solution 15.5.

134

135

Lecture 16 Extensions to the Black Scholes Model 16.1

Dividends

Dividend is a sum of money paid regularly (typically annually) by a company to its shareholders out of its profits (or reserves). In this lecture we are interested in how to model dividends in a stock model, and then how to include them into the Black-Scholes model. We present the most simple possible model for dividends, and that is to assume that in a time dt the underlying stock pays out a cash sum proportional to the stock price S equal to D0 Sdt where D0 is the constant dividend yield. Example 16.1. What happens to the stock price when a dividend is paid out? Solution 16.1.

136

In reality, dividends are paid out discretely on a regular basis, but we choose to model them as a continuous, proportional payment to make things easier. If we include the dividend payment D0 Sdt in our stock model, we get −D0 Sdt | {z }

dS = µSdt + σSdW

cash paid to each shareholder

137

which we can rewrite as dS = (µ − D0 )Sdt + σSdW. Example 16.2. How do these dividend payments work in a portfolio? Solution 16.2.

138

16.2

Options on Dividend Paying Shares

Now, we set up a portfolio consisting of a long position in one call option and a short position in ∆ shares. The value is Π = C − ∆S. Example 16.3. Show that the change in value of this portfolio in the time interval dt is dΠ = dC − ∆dS − ∆D0 Sdt.

(16.1)

Solution 16.3.

Example 16.4. Using Itˆ o’s Lemma written like this:   ∂C 1 ∂2C ∂C dC = + σ 2 S 2 2 dt + dS, ∂t 2 ∂S ∂S

(16.2)

derive the modified Black-Scholes PDE: ∂2C ∂C 1 ∂C + σ 2 S 2 2 + (r − D0 )S − rC = 0. ∂t 2 ∂S ∂S

139

(16.3)

Solution 16.4.

140

How can we find a solution to this modified Black-Scholes equation? The equation itself looks almost identical to (12.4), except we have an extra term multiplying the first derivative of S. Luckily there are a range of tricks we can use to solve a problem like this, by first guessing at the form of a solution and then seeing if the resulting equation is simplified in any way. To proceed, guess at a solution of the form C(S, t) = e−D0 (T −t) C1 (S, t). After this has been substituted into (16.3), we should find that C1 satisfies the Black-Scholes equation except that the r is replaced by r − D0 . Since we have already stated a solution to the Black-Scholes equation, we can then construct a solution to (16.3).

In Examples Sheet 8 you should substitute C(S, t) = e−D0 (T −t) C1 (S, t) into (16.3) to show that the modified Black-Scholes equation has the explicit solution for the European call C(S, t) = Se−D0 (T −t) N (d10 ) − Ee−r(T −t) N (d20 ), where d10 = and

ln(S/E) + (r − D0 + σ 2 /2)(T − t) √ σ T −t √ d20 = d10 − σ T − t.

141

(16.4)

16.3

Early Exercise

Recall that an American Option is one that may be exercised at any time prior to expiry (t = T ). We have already discussed how this works in a discrete time binomial model, so what happens when we move to continuous time? Example 16.5. How do we price an American option in continuous time? Solution 16.5.

In fact in the continuous limit, the resulting problem can be formulated and solved in several different ways • Optimal Stopping Problem: this is popular with probability academics. The problem can be stated as the optimal time at which to exercise (and hence stop holding the option).

Some results can be derived but to get an actual value one of the next two methods must be used. • Variational Inequalities: this formulation is the most robust way to formulate the problem, and there are many numerical techniques (no analytic ones though) available to solve problems of this type. • Free Boundary: this formulation is popular since it can means that analytic solutions can be derived in some cases. Unfortunately it is not very robust since assumptions have to be made about the existence of the barrier and numerical solutions are difficult to code although they are very accurate. As such, formulating and then valuing a contract such as this can be very difficult as there are often no explicit analytic solutions.

142

Example 16.6. Write the American put option problem as a free boundary problem (commonly found in fluid mechanics). Solution 16.6.

143

Lecture 17 Bonds and Interest Rates 17.1

Time Varying Interest Rates

Bonds are contracts that yields a known amount (nominal, principal or face value) on the maturity date, t = T . The bond may pay a coupon (interest payment) at fixed times. If there is no coupon payment, the bond is known as a zero-coupon bond. Let us introduce the following notation: • V (t) is the value of the bond, • r(t) is the interest rate which is now a function of time, • and K(t) is the coupon payment rate. Example 17.1. Show that a bond with a time varying interest rate r(t) paying a coupon at the rate K(t) and face value F should satisfy the following equation dV = r(t)V − K(t) dt with the final condition V (T ) = F. Solution 17.1.

144

(17.1)

Example 17.2. Sketch the value of a bond as a function of time in the following cases 1. r(t) > 0 and K(t) = 0; 2. r(t) = r0 > 0 and K(t) = K0 > r0 F where r0 and K0 are constants; 3. r(t) = r0 > 0 and K(t) is single one-off payment of K0 at time t = T /2. Solution 17.2.

145

146

17.2

Zero Coupon Bonds

Let us now consider the case when the coupon payment rate is K(t) = 0 and the interest rate r(t) is a function of time. We have that the bond price must satisfy dV = r(t)V dt

(17.2)

Example 17.3. Show that the solution of (17.2) with V (T ) = F can be written as ! Z T

V (t) = F exp −

Solution 17.3.

147

r(s)ds .

t

(17.3)

Example 17.4. A zero-coupon bond, V , issued at time t = 0, is worth V (t = 1) = 1 at maturity T = 1. Find the bond price V (t) at time t < 1 and V (0), when the continuous interest rate is r(t) = t2 . Solution 17.4.

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17.3

Lecture 17

Coupon Bonds

Let us consider the case when the continuous coupon payment rate K(t) 6= 0. We can solve (17.1) using the integrating factor. The integrating factor we need is I(t) = e−

Rt

r(s)ds

,

(17.4)

note that we use the dummy variable s inside the integral as t appears in the limits of integration. Example 17.5. Using (17.4), solve (17.1) with final condition V (T ) = F to obtain the solution for the coupon bond V (t) = F e−

RT t

r(s)ds

+

Z

t

Solution 17.5.

149

T

e−

Rs t

r(u)du

K(s)ds.

(17.5)

We can also solve (17.1) by guessing that the solution can be written as ! Z T

V (t) = F exp −

r(s)ds

+ V1 (t),

t

and V1 (t) = C(t) exp −

Z

t

T

!

r(s)ds .

Substituting this into (17.1) results in a simple equation for C(t) that can be solved by separation of variables. See Examples Sheet 9 for more details.

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Lecture 18 Term Structure of Interest Rates 18.1

Yield

Bonds as debt contracts may involve different maturities and coupon payments, and also any number of other options may also be included. Take for example a set of risk free zero coupon bonds are on sale with different maturities and they are priced as follows: Current Time t

Maturity Date T

Face Value

Current Sale Price V (t, T )

0

1

£100

98.24

0

2

£250

244.47

0

3

£100

96.43

0

5

£1000

939.00

0

10

£500

448.95

Here we use the notation V (t, T ) for the bond price at time t maturing at time T . As we show in the table, bond prices are quoted at time t for different values of T . If we look at the same table in six months time all the t’s change to 0.5 and the prices of the bonds will all change but the maturity dates T will stay fixed. Investors would like an easy way to compare bonds, something that tells us about the return on investment. Since it is not the actual price of the bond that is important, but the expected return on investment, investors will compare what is called the yield.

Yield to Maturity for zero coupon bonds is defined as Y (t, T ) = −

ln(V (t, T )) − ln(V (T, T )) , T −t

(18.1)

and it is a measure of the future values of interest rate, where V (t, T ) can be taken from financial data.

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There is a more complicated problem to be solved when calculating the yield to maturity for coupon bonds, but we will not go through it in this course. Example 18.1. Calculate the yields Y (0, 2), Y (0, 3) and Y (0, 5) from the table above. Solution 18.1.

Given our formula for a zero coupon bond (17.3) we can write   R  T ln F exp − t r(s)ds − ln F Y (t, T ) = − T −t so that Y (t, T ) =

1 T −t

Z

T

r(s)ds

(18.2)

t

Clearly can see that Y (t, T ) is the average value of the interest rate r(t) in the time interval [t, T ]. Therefore the bond price can be written as V (t, T ) = F e−Y (t,T )(T −t)

152

(18.3)

MATH20912

Lecture 18

Example 18.2. Using arbitrage arguments, what is the value of a zero coupon bond at time t = 3 maturing at T = 5 with F = 100, denoted V (3, 5), and hence the future yield Y (3, 5)? Solution 18.2.

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In the example we have that V (0, 5) = 93.90 and V (0, 10) = 89.79, so what can this tell us about the interest rates in the future and what bond prices will be in 5 years time? Well from the above we know that the future yield can be calculated. But what about the future interest rates r(T )? Imagine we have a quoted bond price for all future maturity dates so that V (t, T ) is available for T ∈ (t, ∞), what is the relationship between these quoted prices and the interest rate in the future?

Example 18.3. By differentiating (17.3) with respect to T , show that the forward interest rate is given by r(T ) = −

∂V (t, T ) 1 . V (t, T ) ∂T

Solution 18.3.

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(18.4)

MATH20912

18.2

Lecture 18

Term Structure of Interest Rates

We define the term structure of interest rates (yield curve): Z ln(V (0, T )) − ln(V (T, T )) 1 T Y (0, T ) = − r(s)ds = T T 0 as a plot of the average value of interest rate in the future. The usual examples that are tested in this course is to assume that we have already modelled the interest rate with some function r(t) = f (t), so that we can calculate the bond price, yield and term structure of interest rates. However in the real world it works the other way round, we are given a set of prices V (t, T ), and we must work out the best function r(t) which matches that set of prices. The term structure of interest rates you see quoted on market websites is calculated from market prices not from some r(t), you are asked to do this for the table in this lecture in example 18.5. Example 18.4. Assume that the instantaneous interest rate r(t) is r(t) = r0 + at, where r0 and a are positive constants. Calculate the bond price formula V (t, T ) and then sketch the term structure of interest rates. Solution 18.4.

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Example 18.5. Fill in all the yields for the table on page 151 and plot the term structure of interest rate for the data. Solution 18.5. Left as an example

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18.3

Default Risk

Example 18.6. In what ways can we extend our model for interest rates? Solution 18.6.

If there exists a risk of a default bond, V (t, T ), when the principal is not paid to lender as promised by the borrower. Given there is a chance we don’t get the money back, we should expect to pay less for the bond, therefore the yield should go up. Consider a 1 year bond, V (0, 1), that has probability p of defaulting on repayments, we can show that the increase s in yield Y (0, 1) = r + s on the bond is related to probability that the payment is not made. Bond Tree:  p   

u  V (0, 1) H H HH 1−p

Price: u0

V (0, 1) = e−r (F (1 − p) + 0.p) and therefore the yield is Y (0, 1) = − ln(e−r F (1 − p)) + ln F

HHu F

and Y (0, 1) = r + s

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Example 18.7. Show that the yield spread, s, is approximately p. Solution 18.7.

In fact, if we model default as a Poisson process with intensity λ(t) we find the yield spread is s(T ) =

1 T

Z

0

158

T

λ(s)ds

Lecture 19 Exotic Options Example 19.1. Given this is a course on finance, is there anything I have learned that will help me get rich? Solution 19.1.

Options that are not traded in the market are usually called exotic options. They might

159

include special clauses in the contract, or have payoffs that make them difficult to calculate. As opposed to call and put options trading in efficient markets, they are normally traded over the counter (OTC) which means that two participants must agree on the price for a trade to take place. The price they come up with they must calculate themselves and they are not helped out by seeing a market where other people state what price they would put on that contract. Example 19.2. Why might exotic options present an opportunity to make money? Solution 19.2.

Some examples of exotic options are: • Asian • Bermudan • Parisian • ParAsian (cross between Asian and Parisian!) • Barrier • Compound

160

• Lookback Often these options can be combined or included in other financial contracts. Example 19.3. How do we come up with a price for these options? Solution 19.3.

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19.1

Asian Options

Asian Option is a contract giving the holder the right to buy/sell an underlying asset for its average price over some prescribed period. Example 19.4. What is the average of the stock price over the period [0, t]? Solution 19.4.

The floating strike Asian put option has the final condition: ! Z 1 T S(t)dt, 0 = max(S − A, 0). V (S, T ) = max S − T 0 A floating strike option uses A as the underlying and S as the exercise price. A fixed strike option uses A as the underlying and E as the exercise price.

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Example 19.5. What is the payoff for a fixed strike Asian call option? Solution 19.5.

19.2

Black Scholes for Asian Options

To proceed we must first introduce a new variable: Z t I(t) = S(t)dt or 0

dI = S(t). dt

Example 19.6. Write down the payoff for a floating strike Asian put option in terms of I. Solution 19.6.

Now we have that V = V (S, I, t) is a function of three variables. Example 19.7. Using the adjusted Itˆo’s lemma:   ∂V 1 2 2 ∂2V ∂V ∂V dV = + σ S dt + dS + dI, ∂t 2 ∂S 2 ∂S ∂I

163

(19.1)

set up a hedging portfolio to eliminate risk and obtain the modified Black-Scholes equation for the Asian option ∂V ∂V 1 ∂V ∂2V + σ 2 S 2 2 + rS − rV + S = 0. ∂t 2 ∂S ∂S ∂I

(19.2)

Solution 19.7.

Even though this PDE looks very similar to the BS PDE it is actually much harder to solve. The value of an Asian option must be calculated numerically (no analytic solution!). If you are solving PDEs like this in finance it does not just mean pricing options, there are actually many other applications in economics.

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