El voluntariado como proceso educativo

where ˜d1(f, t) and ˜d2(f, t) are given by (3.92).

Example 3.3.1 Suppose that the call option considered in Example 3.2.1 is a futures option. This means, in particular, that the price is now interpreted as the futures price. Using (3.91), one finds that the arbitrage price of a futures call option equals (approximately) C₀^f = 1.22. Moreover, the portfolio that replicates the option is composed at time 0of φ¹₀ futures contracts and φ²₀ invested in risk-free bonds, where φ¹₀= 0.75 and φ²₀ = 1.22. Since the number φ¹₀ is positive, it is clear that an investor who assumes a short option position needs to enter φ¹₀(long) futures contracts. Such a position, commonly referred to as the long hedge, is also a generally accepted practical strategy for a party who expects to purchase a given asset at some future date. To find the arbitrage price of the corresponding put futures option, we make use of the put-call parity relationship (3.100). We find that P₀^f = 0.23; moreover, for the replicating portfolio of the put option we have φ¹₀ =−0.25 and φ²₀= 0.23. Since now φ¹₀< 0, we deal here with the short hedge – a strategy typical for an investor who expects to sell a given asset at some future date.

3.3.4 Options on Forward Contracts

We adopt the classic Black-Scholes framework of Sect. 3.2.1. We will consider a forward contract with delivery date T^∗> 0written on a non-dividend-paying stock S. Recall that the forward price at time t of a stock S for the settlement date T^∗equals

F_S(t, T^∗) = S_te^r(T∗−t), ∀ t ∈ [0, T^∗].

This means that the forward contract, established at time t, in which the delivery price is set to be equal to F_S(t, T^∗) is worthless at time t. It should be stressed that the value of such a contract at time u∈ [t, T^∗] is no longer zero, in general. It is intuitively clear that the value V^F(t, u, T^∗) of such a contract at time u equals the discounted value of the difference between the current forward price of S at time u and its value at time t, that is

V^F(t, u, T^∗) = e^{−r(T∗−u)}

S_ue^r(T∗−u)− Ste^r(T∗−t)

= S_u− Ste^r(u−t)

for every u ∈ [t, T^∗]. The last equality can also be derived by applying directly the risk-neutral valuation formula to the claim X = S_T∗ − FS(t, T^∗), which settles at time T^∗. Indeed, we have

V^F(t, u, T^∗) = BuEP^∗

B_T⁻¹∗ST^∗− B_T⁻¹∗Ste^r(T∗−t)| Fu



= BuEP^∗(S_T^∗∗| Fu)− Ste^r(T∗−t)e^{−r(T∗−u)}

= S_u− Ste^r(u−t)= S_u− FS(t, u),

since the random variable St is Fu-measurable. It is worthwhile to observe that V^F(t, u, T^∗) is in fact independent of the settlement date T^∗, therefore we may and do write V^F(t, u, T^∗) = V^F(t, u) in what follows. By definition,¹⁰a call option written at time t on a forward contract with the expiry

10Since options on forward contracts are not traded on exchanges,the definition of an option written on a forward contract is largely a matter of convention.

3.3. FUTURES MARKET 73 date t < T < T^∗ is simply a call option, with zero strike price, which is written on the value of the underlying forward contract. The terminal option’s payoff thus equals

C_T^F =

V^F(t, T )+

=

ST − Ste^r(T−t)+

.

It is clear that the call option on the forward contract purchased at time t gives the right to enter at time T into the forward contract on the stock S with delivery date T^∗ and delivery price FS(t, T^∗).

If the forward price at time T is less than it was at time t, the option is abandoned. In the opposite case, the holder exercises the option, and either enters, at no additional cost, into a forward contract under more favorable conditions than those prevailing at time T, or simply takes the payoff of the option. Assume now that the option was written at time 0, so that

C_T^F = (V^F(0, T ))⁺= (S_T − S0e^rT)⁺.

To value such an option at time t ≤ T, we can make use of the Black-Scholes formula with the (fixed) strike price K = S₀e^rT. After simple manipulations, we find that the option’s value at time t is

C_t^F = S_tN

d1(S_t, t)

− S0e^rtN

d2(S_t, t)

, (3.101)

where

d1,2(S_t, t) = ln S_t− ln(S0e^rt)±¹₂σ²(T− t) σ√

T− t .

Alternatively, we can make use of Black’s futures formula. Since the futures price fS(T, T ) coincides with ST, we have

C_T^F = (f_S(T, T )− S0e^rT)⁺. An application of Black’s formula yields

C_t^F = e^{−r(T −t)}



ftNd˜1(ft, t)

− S0e^rTNd˜2(ft, t)

, (3.102)

where f_t= f_S(t, T ), and

d˜1,2(f_t, t) = ln f_t− ln(S0e^rT)±¹₂σ²(T− t) σ√

T− t .

Since in the Black-Scholes setting the relationship fS(t, T ) = Ste^r(T−t) is satisfied, it is apparent that expressions (3.101) and (3.102) are equivalent.

Chapter 4

Foreign Market Derivatives

In this chapter, an arbitrage-free model of the domestic security market is extended by assuming that trading in foreign assets, such as foreign risk-free bonds and foreign stocks (and their derivatives), is allowed. We shall work within the classic Black-Scholes framework. More specifically, both domestic and foreign risk-free interest rates are assumed throughout to be non-negative constants, and the foreign stock price and the exchange rate are modelled by means of geometric Brownian motions.

This implies that the foreign stock price, as well as the price in domestic currency of one unit of foreign currency (i.e., the exchange rate) will have lognormal probability distributions at future times. Notice, however, that in order to avoid perfect correlation between these two processes, the underlying noise process should be modelled by means of a multidimensional, rather than a one-dimensional, Brownian motion. Our main goal is to establish explicit valuation formulae for various kinds of currency and foreign equity options. Also, we will provide some indications concerning the form of the corresponding hedging strategies. It is clear that foreign market contracts of certain kinds should be hedged both against exchange rate movements and against the fluctuations of relevant foreign equities.

4.1 Cross-currency Market Model

All processes considered in what follows are defined on a common filtered probability space (Ω, F, P), where the filtration F is assumed to be the P-augmentation of the natural filtration generated by a d-dimensional Brownian motion W = (W¹, . . . , W^d). The domestic and foreign interest rates, rd and rf, are assumed to be given real numbers. Consequently, the domestic and foreign savings accounts satisfy

B^d_t = exp(rdt), B_t^f= exp(rft), ∀ t ∈ [0, T^∗], (4.1) where B^d_t and B_t^f are denominated in units of domestic and foreign currency, respectively. The exchange rate process Q, which is used to convert foreign payoffs into domestic currency, is modelled by the following stochastic differential equation

dQt= Qt

µQdt + σQ· dWt

, Q0> 0, (4.2)

where µ_Q ∈ R is a constant drift coefficient and σQ ∈ R^d denotes a constant volatility vector. As usual, the dot “· ” stands for the Euclidean inner product in R^d, for instance

σ_Q· dWt= d i=1

σ_Qⁱ dW_tⁱ.

Also, we write| · | to denote the Euclidean norm in R^d. Using this notation, we can make a clear distinction between models which are based on a one-dimensional Brownian motion, and those

75

models in which the multidimensional character of the underlying noise process is essential. Let us make clear that we adopt here the convention that the exchange rate process Q is denominated in units of domestic currency per unit of foreign currency; that is, Q_t represents the domestic price at time t of one unit of the foreign currency. It should be stressed, however, that the exchange rate process Q cannot be treated on an equal basis with the price processes of domestic assets;

put another way, the foreign currency cannot be seen as just an additional traded security in the domestic market model, unless the impact of the foreign interest rate is taken into account. The process Q plays an important role as a tool which allows the conversion of foreign market cash flows into units of domestic currency. Moreover, it can also play the role of an option’s underlying “asset”.