2. Derivatives

Classical financial mathematics deals first of all with basic financial instruments like stocks, foreign currencies and bonds. A derivative (derivative security or contingent claim) is a financial instrument whose value depends on the value of others, more basic underlying variables. In this chapter we consider forward contracts, futures contracts and options as well as some combinations.

Simple derivatives have been known on European stock exchanges since the turn of the $ 19$th century. While they lost popularity between World War I and II, they revived in the seventies accompanied by the work of Black, Scholes and Merton, who developed a theoretical foundation to price such instruments. Their entrepreneurial approach, which is not only applied to price derivatives but everywhere in finance where the risk of complex financial instruments is measured and controlled, was awarded the Nobel price in economics in 1997. At the same time, it triggered the development of modern financial mathematics whose basics we describe in the first part of this book. Since we concentrate only on the mathematical modelling ideas, we introduce the required financial terminology as we pass by. We leave out numerous details which are of practical importance but which are of no interest for the mathematical modelling, and refer to, for example, Hull (2000), Welcker et al. (1992).

Particularly simple derivative securities are forward and future contracts. Both contracts are agreements involving two parties and calling for future delivery of an asset at an agreed-upon price. Stocks, currencies and bonds, as well as agricultural products (grain, meat) and raw materials (oil, copper, electric energy) are underlying in the contract.

Definition 2.1 (Forward contract)  
A forward contract is an agreement between two parties in which one of the parties assumes a long position (the other party assumes a short position) and obliges to purchase (sell) the underlying asset at a specified future date $ T > t$, (expiration date or maturity) for a specified price $ K$ (delivery price).

At time $ t$, the value $ V_{K,T}(S_t,\tau)$ of such a contract depends on the current value of the underlying $ S_t$, the time to maturity $ \tau = T - t$ and of the parameters $ K$, $ T$ specified in the contract.

Futures contracts closely resemble forward contracts. While the latter do not entail any more payments until maturity once the agreement is signed, futures contracts are traded on an exchange and mark to the market on a daily basis. Under certain assumptions forward and futures prices are identical.

Example 2.1  
An investor enter into a long forward contract on September 1, 2003, which obliges him to buy 1000000 EUR at a specified exchange rate of 1.2 USD/EUR in 90 days. The investor gains if the exchange rate is up to 1.3 USD/EUR on November 30, 2003. Since he can sell the 1000000 EUR for USD 1300000.
In this case $ t$ = September 1, 2003 $ \tau$ = 90 days, $ T$ = November 30, and
$ K$ = USD 1200000.

Definition 2.2 (Spot Price, Forward Price, Future Price)  
The current price of the underlying (stock, currency, raw material) $ S_t$ is often referred to as the spot price. The delivery price giving a forward contract a value of zero is called forward price and denoted $ F_t$. That is, $ F_t$ solves $ V_{F_t,T}(S_t,\tau)$ = 0. The future price is defined accordingly.

Later we will compute the value of a forward contract, which determines the forward price. Since under certain assumptions forward and future contracts have the same value, their prices are equal. When such a contract is initiated in time $ t=0$, often the delivery price is set to $ K=F_0$. The contract has a value of zero for both the seller and the buyer, i.e. no payments occur. In the course of time, as additional transactions take place on the exchange, the delivery price $ K$ and the forward price $ F_t$ can be different.

Contrary to forward and futures contracts where both parties are obligated to carry out the transaction, an option gives one party the right to buy or sell the security. Obviously, it's important to distinguish whether the buyer or seller of the option has the right to exercise. There are two types of options: call and put options. Furthermore, European options are delimited from American options. While European options are like forward contracts, American options can be exercised at any date before maturity. These terms are derived from historical, not geographical roots.

Definition 2.3 (Call Option, Put Option)  
A European call option is an agreement which gives the holder the right to buy the underlying asset at a specified date $ T > t$, (expiration date or maturity), for a specified price $ K$, (strike price or exercise price). If the holder does not exercise, the option expires worthless.

European put option is an agreement which gives the holder the right to sell the underlying asset at a specified date $ T$ for a specified price $ K$.

The holder of an American call or put option has the right to exercise the option at any time between $ t$ and $ T$.

The option types defined above are also called plain vanilla options. In practice, many more complex derivatives exist and numerous new financial instruments are still emerging. Over-the-counter (OTC) derivatives are tailor made instruments designed by banking institutions to satisfy a particular consumer need. A compound option, for example, is such an OTC-derivative. It gives the holder the right to buy or sell at time $ T$ an underlying option which matures in $ T^{\prime}>T$. The mathematical treatment of these exotic options is particularly difficult, since the current value of this instrument does not only depend on the value of the underlying $ S_t$ but also on the entire path of the underlying, $ S_{t^{\prime}}, 0 \leq t^{\prime} \leq t$.

Asian, lookback and knock-out options are path-dependent derivatives. While the delivery price $ K$ of an asian option depends on the average value of the security of a certain period of time, it depends in the case of a lookback option on the minimum or maximum value of the security for a certain period of time. Knock-out options expire worthless if the price level ever hits a specified level.

To get used to forward and futures contracts, plain vanilla options and simple combinations of them, it is convenient to have a look at the payoff of an instrument, i.e. the value of the derivative at maturity $ T$. The payoff of a long position in a forward contract is just $ S_T-K$, with $ S_T$ the security's spot price at expiration date $ T$. The holder of the contract pays $ K$ for the security and can sell it for $ S_T$. Thus, he makes a profit if the value of the security $ S_T$ at expiration is greater than the delivery price $ K$. Being short in a forward contract implies a payoff $ K-S_T$. Both payoff functions are depicted in Figure 1.1.

Fig. 1.1: Value of forward contract at maturity
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The call option payoff function is denoted:

$\displaystyle \max\{S_T - K,0\} = (S_T - K)^+ . $

Thus, the option holder only exercises if the delivery price $ K$ is less than the value of the security $ S_T$ at expiration date $ T$. In this case, he receives the same cash amount as in the case of a forward or future contract. If $ K<S_T$, he will clearly choose not to exercise and the option expires worthless. The put option payoff function is:

$\displaystyle \max\{K - S_T,0\} = (K - S_T)^+ .$

In contrast to forward and future contracts, options have to be bought for a positive amount $ C(S_0,T)$, called the option price or option prime. Often, the options profit function is defined as $ (S_T - K)^+ - C(S_0,T)$. However, this definition adds cash flows of different points in time. The correct profit is obtained by compounding the cash outflow in time $ t=0$ up to time $ t=T$, since the investor could have invested the option option at the risk-free interest rate $ r$. Assuming continuous compounding at a constant interest rate $ r$, the profit function of a call option is denoted: $ (S_{T}-K)^+ - C(S_0,T)
e^{rT}$.

Example 2.2  
Consider a long call option with delivery price $ K$ and option price $ C_0$ in time $ t=0$. The payoff and profit function are given in Figure 1.2 and 1.3, respectively.

Fig. 1.2: Payoff of a short position in a call option
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Fig. 1.3: Profit of a short position in a call option
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Example 2.3  
Combining a long call and a long put with the same delivery price $ K$ is called a straddle. Figure 1.4 shows the straddle profit function. $ C_0$ and $ P_0$ denote the call and put option option respectively.

Fig. 1.4: Profit of a straddle
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Another fundamental financial instrument which is used in option pricing is a bond. Apart from interest yields, the bond holder possibly receives coupon payments at fixed points in time. In particular, we will consider zero-coupon bonds, i.e. bonds which promise a single payment at a fixed future date.

Definition 2.4 (Zero coupon Bond, Discount Bond)  
A zero coupon bond or discount bond is a bond without coupon payments which pays an interest $ r$. The investor pays in time 0 an amount $ B_0$ and receives at maturity $ T$ the amount $ B_T$ which is the sum of $ B_0$ and the interest earned on $ B_0$. The bonds' value at maturity is termed face value.

Buying a zero-coupon bond corresponds to lending money at a fixed interest rate for a fixed period of time. Conversely, selling a zero-coupon bond is equivalent to borrowing money at rate $ r$. Since bonds are traded on an exchange, they can be sold prior to maturity at price $ B_t$, i.e. $ B_0$ plus accrued interest up to time $ t$.

In practice, interest rates are compounded at discrete points in time, for example annually, semiannually or monthly. If the interest rate $ r$ is compounded annually, the initial investment $ B_0$ has $ n$ years later a value of $ B_n^ {(1)} = B_0 (1+r)^ n.$ If it is compounded $ k$ times per annum (p.a.), the investment pays an interest rate of $ \frac{r}{k}$ each $ \frac{1}{k}$ years, and has a terminal value of $ B_n^ {(k)} = B_0 (1 + \frac{r}{k})^
{nk}$ after $ n$ years. However, when options and other complex derivatives are priced, continuous compounding is used, which denoted for $ k\rightarrow \infty$. In this case, the initial investment $ B_0$ grows in $ n$ years to $ B_n = B_0 \cdot e ^ {nr},$ and $ r$ is called short rate. The difference between discrete and continuous compounding is small when $ k$ is large. While an investment of $ B_0=1000$ EUR at a yearly rate $ r=10$% grows to 1100 EUR within a year when annually compounded, it grows to 1105.17 EUR when continuously compounded.

In light of this, the continuous compounded rate $ r$ can be modified to account for these deviations. Assuming annual compounding at rate $ r_1$, for both continuous and annual compounding, a continuous compounded rate
$ r = \log
(1+r_1)$ has to be applied, in order to obtain the same terminal value $ B_n = B_n^ {(1)}.$

If not stated otherwise, continuous compounding will be assumed from here on. For comparing cash flows occurring at different points in time, they have to be compounded or discounted to the same point in time. That is, interest payments are added or subtracted. With continuous compounding, an investment of $ B$ in time $ t$ in $ \Delta t > 0$ is

compounded
to time $ t + \Delta t$: $ B\, e^ {r\Delta t}$
discounted
to time $ t - \Delta t$: $ B\, e^ {-r\Delta t}$.

Before finishing the chapter, some more financial terms will be introduced. A portfolio is a combination of one or more financial instruments - its value is considered as an individual financial instrument. One element of a portfolio is also called a position. An investor assumes a long position when he buys an instrument, and a short position when he sells it. A long call results from buying a call option, a long put from buying a put option, and a short forward from selling a forward contract.

An investor closes out a position of his portfolio by making the future portfolio performance independent of the instrument. If the latter is traded on an exchange, he can sell (e.g. a stock or a bond) or buy (e.g. borrowed money) it. Should the instrument not be traded, however, the investor can close out the position by adding to the portfolio the inverse instrument. Thus, both sum up to zero, and do not influence the portfolio performance any more.

Example 2.4  
Consider an investor who bought on February 1 a 1000000 USD forward contract with a delivery price of 1200000 EUR and which matures in one year. On June 1, he wishes to close out the position. He can sell another forward contract of the same size with the same delivery price and the maturity date, namely January 31. The long and the short positions sum up to zero at any time.

Short selling is a trading strategy that involves selling financial instruments, for example stocks, which he does not own. At a later point in time, he buys back these objects. In practice, this requires the intervention of a broker who mediates another client owing the objects and willing to lend them to the investor. The short selling investor commits to pay to the client any foregone income, as dividends for example, that would be received in the meantime.

Example 2.5  
An investor selling short 1000 stocks, lends them from the owner and sells them immediately for 1000 $ S_0$ in the market ($ S_t$ denotes the stock price at time $ t$). Later, at time $ t>0$, he closes out the position, by buying back the stocks for 1000 $ S_t$ and returning them to the owner. The strategy is profitable if $ S_t$ is clearly below $ S_0$. If in time $ t_0, \, \, 0 < t_0 < t,$ a dividend $ D$ per share is paid, the investor pays 1000 $ D$ to the owner. Short selling is in practice subject to numerous restrictions. In the following, it is only the possibility of short selling that will be of interest.