Advanced Derivatives Course Chapter 1

Week 1: Introduction. Financial derivatives: Forwards, Futures, Swaps, Options Lecture I.1 Why derivatives? There is not a single investment bank which does not have a derivatives desk. Moreover, now even some non-financial institutions have their own derivatives analysts. For example oil companies spend quite a lot of money on derivatives research which may seem as an odd activity unrelated to the industry’s main business. Why then derivatives are so popular among so many? It turns out that different businesses love derivatives for different reasons. Banks use derivatives as a powerful instrument to generate profits and hedge their risks. Businesses use derivatives as sources of additional investments and also as risk management instruments. The derivatives users base is extremely large. It even includes pensioners who can now buy options on places at retirement houses. First of all we have to define what are financial derivatives. Generally speaking, a derivative is a financial instrument whose value is derived from the price of a more basic asset called the underlying. The underlying may not necessarily be a tradable product. Examples of underlyings are shares, commodities, currencies, credits, stock market indices, weather temperatures, sunshine, results of sport matches, wind speed and so on. Basically, anything which may have to a certain degree an unpredictable effect on any business activity can be considered as an underlying of a certain derivative. All derivatives can be divided into two big classes: q q

Linear Non-linear

Linear are derivatives whose values depend linearly on the underlying’s value. This includes q q

Forwards and Futures Swaps

Non-linear are derivatives whose value is a non-linear function of the underlying. This includes q q q q

Options Convertibles Equity Linked Bonds Reinsurances

One can add some other instruments to both of the two classes. For example, bonds can be viewed as non-linear derivatives with the interest rate being a non-tradable underlying. During the course we will talk about each of the listed above types of derivative products. Although the main goal of my course is not to teach how to use various derivatives, but rather

how to price them, I will try to explain the most common applications of some of the derivatives.

Forwards and Futures Forward is a contract between two parties agreeing that at certain time in the future one party will deliver a pre-agreed quantity of some underlying asset (or its cash equivalent in the case of non-tradable underlyings) and the other party will pay a pre-agreed amount of money for it. This amount of money is called the forward price. Once the contract is signed, the two parties are legally bound by its conditions: the time of delivery, the quantity of the underlying and the forward price. While the delivery time and the delivery quantity of the underlying asset can be fixed without any problem, the question is how the parties can agree on the future price of the underlying when the latter can change randomly due to market price fluctuations. It turns that in the case of forward contracts there exists what is called the fair future price of the asset. This can be found as follows. Suppose we sell a one year forward contract meaning that we take a responsibility to deliver in one year a certain quantity, say n, of the underlying asset whose current market price is S. In order to avoid any expose to the market risk, we can borrow from a friendly bank the amount n×S and buy the necessary quantity of the underlying. In other words, we sell a covered forward. At the end of the year, we will deliver the asset to the buyer of the forward contract who will pay us the forward price n×F. From this amount we have to repay the bank our loan which obviously grew to (1 + r)×S, where r is the one-year interest rate quoted by our bank. Thus, at the end of the year our cash flow is F − (1 + r ) × S . Since we started with no money, we have to end with no money. Otherwise, by selling or buying forward contracts we would be able to make unlimited profit without taking any risk. This is not possible in practice. Therefore, we have to impose the following constrain F − (1 + r ) × S = 0 . This gives us the forward price F = (1 + r) S . If any other price is written in the forward contract, one of the parties will be able to make a risk-less profit by selling or buying the contracts in large (theoretically unlimited) quantities. The considered above example shows us how all specifications of a forward contract can be fixed by the parties to their mutual satisfaction. The argument which helped us to discover the forward price was that there must be no trading strategy allowing for a risk-free profit. This is called the no-arbitrage principle. We will talk more about this principle in future lectures.

There is another requirement which has to be fulfilled when we look for the forward price. This is that the cash flow must remain equal to zero at any time to expiration of the forward contract. We only showed that our cash flow is zero at the starting and ending points of the contract. However, we did not consider what is happening to the cash flow at intermediate moments. In order to investigate this problem, we need to know the model describing the behaviour of the underlying asset. We will return to this question when we will talk about random motion of share prices. For the time being we will assume that in the example considered the cash flow does vanish at all times before the maturity. We will prove this assumption later on. Another assumption is that the interest rate r does not change in time. This might be true for short periods of time. However, for one year it will be most certainly a wrong conjecture. However, the effects of changing interest rates are quite negligible and can be ignored in many cases. Let us consider a specific example based on real data taken from the Financial Times Example: The underlying =British Airways The spot price=£334.25 (31 August 2000) The time to maturity=Six months (0.5 a year) The six-month risk -free interest rate =6.84%

The Forward Price=(1+0.0684)×334.25=£357.11 Futures are standardised forwards which are traded on exchanges. All futures positions are marked to the market at the end of every working day. To illustrate this procedure let us suppose that we bought a three months futures contract on crude oil for €30 per barrel. The next day the futures closing price for the same delivery date is €31 per barrel. This means that our contract has gained one Euro, because at the maturity we still have to pay only €30. In this case, the seller of the futures contract immediately pays €1 into our account. Suppose that one day after, the futures closing price dropped to €29. Now we have to pay two Euros to the seller’s account. By this time our contract lost one Euro. This process continues to the maturity date. Because of the specific mechanism adopted by futures exchanges, contracts are settled in cash and only in some special cases the seller has to physically deliver the asset (especially, in commodities markets). We can draw the following table:

Time

Futures Price

Buyer’s account

Seller’s account

0

€30

0

0

1

€31

+1

-1

2

€29

-2

+2

3

€28

-1

+1

-2

+2

Balance

At the end of the third day, the seller of the futures contract is better off by €2. Forwards and futures are designed to reduce risks related to the uncertainty of future market prices for both sellers and buyers of underlying assets. By entering this type of contracts, both sides achieve complete certainty about their future positions, which may help them to have a better control over their financial resources. However, many traders take futures positions for purely speculative reasons. For instance, if we sell an uncovered futures contract (i.e., when we do not have a long position in the underlying asset), then when the asset price goes down, our futures position will gain a profit and vice versa. In what follows we will be using the following definitions. A long position in an asset is a position that benefits from price increases in that security (an investor who buys a share has a long position, but an equivalent long position can also be established with derivatives). A short position benefits from price decreases in the security. A short position is often established through a short-sale. To sell a security short, one borrows the security and sells it. When one unwinds the short-sale, one has to buy the security back in the market to return it to the lender. One then benefits from the short-sale if the asset’s price is lower when one buys it back than it was when one sold it.

Swaps Swaps, as the name suggests, are instruments which allow a swap holder to receive a floating interest rate from and pay a fixed interest rate to a swap seller for a certain period of time. The interest rates are paid on the same fixed notional principal. Swaps can be arranged in various ways. For example, there are swaps between different currencies, in which case the parties swap a domestic and a foreign rates. A swap can be priced as a combination of bonds.

Bonds

Bonds are securities which pay a certain fixed amount on a certain fixed date in the future. Since we know how much we will get on some future date, we can find the present value of the notional by discounting this amount to the present time with respect to a certain interest rate. If the rate was known in advance, the price of the bond would be very easy to calculate. For example, if the rate is fixed and equal to 5% per annum, then the one-year bond with the notional value €1000 should now cost 1000 B= = 666 .67 1 + 0 .5 However, in reality interest rates are not known in advance, at least, not for very long periods of time. Therefore, the pricing of bonds represents a challenging problem, which involves various assumptions about the behaviour of interest rates. Swaps can be priced in terms of bonds. Let us consider a swap with N payments at times Ti. The outgoing fixed-rate part of the swap is given by N

Out = r fixed ∑ B (t,T i ). i =1

The incoming floating-rate part can be presented as follows N

In = ∑ ( B(t , Ti −1 ) − B(t , Ti )). i =1

Indeed, the bond B(t,Ti) can be expressed in terms of the bond B(t,Ti-1) as follows B(t , Ti ) = B (t , Ti −1 ) − δ i ,i −1 , where δi,i-1 is the interest earned from the time Ti-1 to Ti discounted to the present time. Obviously, δi,i-1 coincides with the incoming floating-rate contribution into the swap. Taking into account that B(t,T0) = 1, we find In = 1 − B(t , T N ). All in all, the value of the swap is given by N

Swap = [1 − B (t , T N ) − r fixed ∑ B (t ,Ti )] × Notional. i =1

We can now see that a swap from the floating-rate receiver side can be presented as a combination of short positions in bonds with different maturities. Normally, the fixed rate is chosen so that the swap present value is equal to zero. Swaps are extremely liquid instruments and, therefore, their market values can be used to price bonds.

Lecture I.2 Options Options are the most flexible of all derivatives because they multiple choice at various moments during the life time of the an option seller does not have such a flexibility and always holder’s requests. For this reason, the option buyer has to pay seller.

give an option holder a option contract. However, has to fulfil the option a premium to the option

There are three main categories of options: European, American and Bermudan. European options can be exercised only at expiration time. American options can be exercised at any moment prior to maturity. Bermudan options can be exercised prior to maturity but on certain pre-determined days. Put options give the right to sell the underlying asset. Calls give the right to buy the underlying assets. There exist also chooser options, when the option holder has the right to chose between call and put payoffs. There are hundreds of different types of options which differ in their payoff structures, path-dependence, payoff trigger and termination conditions. Pricing some of these options represent a complex mathematical problem.