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1 Introduction

The key to using the general framework of the previous chapter is to interpret the excess return of each underlying factor to its market price of risk and its volatility

$$\displaystyle{ m_{i} + q_{i} - r =\sum _{ j=1}^{n}\lambda _{ j}s_{\mathit{ij}}, }$$
(11.1)

where (m i + q i − r) is the excess return of factor i. In calculating the excess return we must account for any income, or costs, associated with holding the asset underlying price x i . This is captured by the q i term which may be either a continuously compounded dividend or cost. In the next two sections we show how the general pricing structure of Sect. 10.4 may be applied, once Eq. (11.1) has been appropriately interpreted for the situation at hand. Section 11.2 considers some standard one-factor examples, whilst Sect. 11.3 considers some two-factor examples.

2 One-Factor Examples

2.1 Stock Options

In this case we have one underlying factor, the price x of the stock, which we assume here pays no dividend. It follows the diffusion process

$$\displaystyle{\frac{\mathit{dx}} {x} = m\mathit{dt} + \mathit{sdw},}$$

and from (11.1) the excess expected return from holding the stock is

$$\displaystyle{m - r =\lambda s,}$$

from which

$$\displaystyle{ m -\lambda s = r. }$$
(11.2)

Substituting this last expression into the general pricing Eq. (10.90) with n t  = n = 1, n n  = 0 we obtain

$$\displaystyle{ \frac{\partial f} {\partial t} + \mathit{rx}\frac{\partial f} {\partial x} + \frac{1} {2}s^{2}x^{2}\frac{\partial ^{2}f} {\partial x^{2}} = \mathit{rf }, }$$
(11.3)

which is of course the Black–Scholes partial differential equation which we obtained earlier.

If the stock does pay a continuously compounded dividend q then the expected excess return relation becomes

$$\displaystyle{m + q - r =\lambda s}$$

so that (11.2) is replaced by

$$\displaystyle{ m -\lambda s = r - q. }$$
(11.4)

The partial differential equation (11.3) then becomes

$$\displaystyle{ \frac{\partial f} {\partial t} + (r - q)x\frac{\partial f} {\partial x} + \frac{1} {2}s^{2}x^{2}\frac{\partial ^{2}f} {\partial x^{2}} = \mathit{rf }. }$$

2.2 Foreign Currency Options

Here we have one factor x which is the exchange rate (domestic currency/unit of foreign currency) that is the price of a unit of the foreign currency. The foreign currency yields continuously the risk free rate in the foreign country, which we denote by r f . If the diffusion process followed by the exchange rate is written as

$$\displaystyle{\frac{\mathit{dx}} {x} = \mathit{mdt} + \mathit{sdw},}$$

the expected return from holding the foreign currency is (m + r f ) and hence the relationship (11.1) for expected excess return as it applies to foreign currency becomes

$$\displaystyle{(m + r_{f}) - r =\lambda s,}$$

and hence

$$\displaystyle{ m -\lambda s = r - r_{f}. }$$
(11.5)

Upon substituting this into the general pricing Eq. (10.90) with n = 1 yields

$$\displaystyle{ \frac{\partial f} {\partial t} + (r - r_{f})x\frac{\partial f} {\partial x} + \frac{1} {2}s^{2}x^{2}\frac{\partial ^{2}f} {\partial x^{2}} = \mathit{rf }, }$$
(11.6)

which is the equation obtained by Garman and Kohlhagen (1983) for the pricing of a foreign currency option. In the case of a European call option with exercise exchange rate E on the foreign currency its solution turns out to be

$$\displaystyle{ f(x,t) = \mathit{xe}^{-r_{f}(T-t)}\mathcal{N}(d_{ 1}) -\mathit{Ee}^{-r(T-t)}\mathcal{N}(d_{ 2}), }$$
(11.7)

where

$$\displaystyle\begin{array}{rcl} d_{1}& =& \frac{\ln (x/E) + (r - r_{f} + s^{2}/2)(T - t)} {s\sqrt{T - t}}, {}\\ d_{2}& =& d_{1} - s\sqrt{T - t}. {}\\ \end{array}$$

It should also be noted that from (10.92) the foreign currency option value can also be expressed as

$$\displaystyle{ f(x,t) =\tilde{ \mathbb{E}}_{t}[f(x_{T},T)]. }$$
(11.8)

From (10.94) with q i  = r f the dynamics for the exchange rate under the equivalent measure \(\tilde{\mathbb{P}}\) is given by

$$\displaystyle{ \frac{\mathit{dx}} {x} = (r - r_{f})\mathit{dt} + \mathit{sd}\tilde{w} }$$
(11.9)

It is the conditional transition density function associated with (11.9) that is required to calculate the \(\tilde{\mathbb{E}}_{t}\) in (11.8).

2.3 Futures Options

In this case the factor x is the price of a futures contract on an underlying asset whose price is S. The derivative security in this case is an option on the futures contract. Simple arbitrage arguments can be used to show that the relationship between the futures price x and the price S of the asset underlying the futures contract, is

$$\displaystyle{ x = \mathit{Se}^{\alpha (T^{{\ast}}-t) }, }$$
(11.10)

where T ∗ is the maturity date of the futures contract. Here

$$\displaystyle{ \alpha = \mbox{ risk free rate} -\mbox{ yield on the asset}. }$$
(11.11)

For example if the underlying asset were a commodity, then α would be, the risk-free rate plus storage costs minus the convenience yield. If the price of the underlying asset follows the diffusion process

$$\displaystyle{ \frac{\mathit{dS}} {S} =\mu \mathit{dt} +\sigma \mathit{dw}, }$$
(11.12)

then a straight forward application of Ito’s lemma reveals that x follows the diffusion process

$$\displaystyle{ \frac{\mathit{dx}} {x} = (\mu -\alpha )\mathit{dt} +\sigma \mathit{dw} \equiv \mu _{x}\mathit{dt} +\sigma \mathit{dw}. }$$
(11.13)

It is well known that a futures price can be regarded as the price of a security paying a continuous dividend yield at the risk free rate r which means we set α = r in (11.13) (see Hull 2000). Thus applying (11.1) here yields (r +μ x ) − r = λ σ, i.e.

$$\displaystyle{ \mu _{x}-\lambda \sigma = 0, }$$

and so the pricing equation becomes

$$\displaystyle{ \frac{\partial f} {\partial t} + \frac{1} {2}\sigma ^{2}x^{2}\frac{\partial ^{2}f} {\partial x^{2}} = \mathit{rf }. }$$

In the case of a European futures option the solution to this partial differential equation is

$$\displaystyle{ f = e^{-r(T-t)}[x\mathcal{N}(d_{ 1}) - E\mathcal{N}(d_{2})], }$$

where

$$\displaystyle{ d_{1} = \frac{\ln (x/E) + (\sigma ^{2}/2)(T - t)} {\sigma \sqrt{T - t}},\qquad d_{2} = d_{1} -\sigma \sqrt{T - t}, }$$

which is Black (1976) model. Note that formally Black’s model could be obtained from the foreign currency option model by setting r f  = r. From (10.92) the futures option value can also be expressed as

$$\displaystyle{ f(x,t) =\tilde{ \mathbb{E}}_{t}[f(x_{T},T)], }$$
(11.14)

where by use of (10.94) with q i  = r, the dynamics of the futures price under the equivalent measure \(\tilde{\mathbb{P}}\) are

$$\displaystyle{ \frac{\mathit{dx}} {x} =\sigma \mathit{d\tilde{w}}. }$$
(11.15)

The transition probability density function associated with (11.15) is used to calculate \(\tilde{\mathbb{E}}_{t}\) in (11.14). Incidentally we note that (11.15) indicates that the futures price is a Martingale under the equivalent measure \(\tilde{\mathbb{P}}\).

3 Options on Two Underlying Factors

As an application of options on two underlying assets, both of which are traded, we consider exchange options which were first studied by Margrabe (1978) . These are the most basic of a class of multi-asset options (digital options , quotient options , foreign equity options , quanto options etc.) which derive their value from the correlation structure between two underlying traded assets. For much more information and details about multi-asset options, we refer the reader to Zhang (1997).

We consider the framework and notation of Sect. 10.3.1 dealing with two traded underlying assets. A European exchange option to pay the second asset in exchange for the first has payoff given by

$$\displaystyle{ f_{T} =\max [x_{1}(T) - x_{2}(T),0]. }$$

This payoff differs from that of a standard European call option on x 1 in that the exercise price E is replaced by x 2(T), the value at maturity of the second asset. Alternatively, it could be regarded as a European put option on x 2 with exercise price equal to x 1(T). An investor might be interested in this type of option if x 1 and x 2 were negatively correlated and the investor wished to have some insurance against x 2 performing “badly”.

The price of the exchange option satisfies the partial differential equation (10.21) subject to the terminal condition

$$\displaystyle{ f(x_{1},x_{2},T) =\max [x_{1}(T) - x_{2}(T),0]. }$$

Alteratively, we may use the martingale representation (10.31) so that

$$\displaystyle{ f(x_{1},x_{2},t) = e^{-r(T-t)}\tilde{\mathbb{E}}_{ t}[\max (x_{1}(T) - x_{2}(T),0)], }$$
(11.16)

where we recall that \(\tilde{\mathbb{E}}_{t}\) is calculated using the distribution generated by the stochastic differential equations (10.28), which may be re-expressed as (see Sect. 6.3.2 and Problem 6.7)

$$\displaystyle\begin{array}{rcl} d(\ln x_{1})& =& [r - q_{1} -\frac{1} {2}(s_{11}^{2} + s_{ 12}^{2})]\mathit{dt} + s_{ 11}d\tilde{w}_{1} + s_{12}d\tilde{w}_{2}, \\ d(\ln x_{2})& =& [r - q_{2} -\frac{1} {2}(s_{21}^{2} + s_{ 22}^{2})]\mathit{dt} + s_{ 21}d\tilde{w}_{1} + s_{22}d\tilde{w}_{2}.{}\end{array}$$
(11.17)

We shall concentrate on the case represented by the stochastic differential equations (10.7) so that (11.17) becomes

$$\displaystyle\begin{array}{rcl} d(\ln x_{1})& =& (r - q_{1} -\frac{s_{1}^{2}} {2} )\mathit{dt} + s_{1}d\tilde{w}_{1}, {}\\ d(\ln x_{2})& =& (r - q_{2} -\frac{s_{2}^{2}} {2} )\mathit{dt} +\rho s_{2}d\tilde{w}_{1} + \sqrt{1 -\rho ^{2}}s_{2}d\tilde{w}_{2}. {}\\ \end{array}$$

Thus, the probability density function for the joint distribution of ln[x 1(T)∕x 1(t)], ln[x 2(T)∕x 2(t)] isFootnote 1

$$\displaystyle{ \tilde{\pi }[v_{1}(T),v_{2}(T),T] = \frac{1} {2\pi s_{1}s_{2}\sqrt{1 -\rho ^{2}}(T - t)}\exp \bigg[-\frac{u_{1}^{2} - 2\rho u_{1}u_{2} + u_{2}^{2}} {2(1 -\rho ^{2})} \bigg], }$$
(11.18)

where \(v_{i}(T) \equiv \ln \left [x_{i}(T)/x_{i}(t)\right ]\) (as we are expressing the distribution of relative prices we have dropped the notation for conditioning on time t) and

$$\displaystyle{ u_{i} \equiv \frac{v_{i}(T) - (r - q_{i} -\frac{1} {2}s_{i}^{2})(T - t)} {s_{i}\sqrt{T - t}},\qquad \mbox{ for }\qquad i = 1,2. }$$

Equation (11.18) is the bivariate normal distribution for the logarithm of the relative prices (see Fig. 11.1). Equation (11.18) may also be expressed as

$$\displaystyle{ \tilde{\pi }[v_{1}(T),v_{2}(T),T] =\tilde{\pi } _{1}[v_{1}(T),T]\tilde{\pi }_{2}[v_{2}(T),T\mid v_{1}(T),T], }$$

where

$$\displaystyle{ \tilde{\pi }_{1}[v_{1}(T),T] = \frac{1} {s_{1}\sqrt{2\pi (T - t)}}\exp \left (-\frac{u_{1}^{2}} {2} \right ), }$$
(11.19)

and

$$\displaystyle{ \tilde{\pi }_{2}[v_{2}(T),T\mid v_{1}(T),T] = \frac{1} {s_{2}\sqrt{1 -\rho ^{2}}\sqrt{2\pi (T - t)}}\exp \left (-\frac{(u_{2} -\rho u_{1})^{2}} {2(1 -\rho ^{2})} \right ), }$$
(11.20)

or, alternatively as

$$\displaystyle{ \tilde{\pi }[v_{1}(T),v_{2}(T),T] =\tilde{\pi } _{2}[v_{2}(T),T]\tilde{\pi }_{1}[v_{1}(T),T\mid v_{2}(T),T], }$$

where now (with a slight abuse of notation)

$$\displaystyle{ \tilde{\pi }_{2}[v_{2}(T),T] = \frac{1} {s_{2}\,\sqrt{2\pi (T - t)}}\exp \left (-\frac{u_{2}^{2}} {2} \right ), }$$

and

$$\displaystyle{ \tilde{\pi }_{1}[v_{1}(T),T\mid v_{2}(T),T] = \frac{1} {s_{1}\sqrt{1 -\rho ^{2}}\sqrt{2\pi (T - t)}}\exp \left (-\frac{(u_{1} -\rho u_{2})^{2}} {2(1 -\rho ^{2})} \right ). }$$

We shall use \(\tilde{\pi }[x_{1}(T),x_{2}(T),T]\) to denote the corresponding probabilities in terms of the original asset prices x 1, x 2. In terms of these distributions, Eq. (11.16) becomes

$$\displaystyle{ f(x_{1},x_{2},T) = e^{-r(T-t)}\int _{ 0}^{\infty }\left (\int _{ 0}^{x_{1} }(x_{1} - x_{2})\tilde{\pi }(x_{1},x_{2},T)\frac{\mathit{dx}_{2}} {x_{2}} \right )\frac{\mathit{dx}_{1}} {x_{1}}, }$$
(11.21)

where the region of integration is illustrated in Fig. 11.2.

Fig. 11.1
figure 1

Bivariate normal distribution

Full size image
Fig. 11.2
figure 2

Region of integration for the exchange option

Full size image

Since the integration with respect to x 2 is performed, whilst x 1 is held constant, it is more convenient to use the conditional distributions (11.19), (11.20) (after converting to lnS). Thus, we may express (11.21) as

$$\displaystyle\begin{array}{rcl} f(x_{1},x_{2},T)& =& e^{-r(T-t)}\int _{ 0}^{\infty }\left (\int _{ 0}^{x_{1} }(x_{1} - x_{2})\tilde{\pi }_{2}(x_{2},T\mid x_{1},T)\frac{{\it \text{dx}}_{2}} {x_{2}} \right )\tilde{\pi }_{1}(x_{1},T)\frac{{\it \text{dx}}_{1}} {x_{1}} \\ & \equiv & e^{-r(T-t)}\left (\int _{ 0}^{\infty }J_{ 1}(x_{1},T)\tilde{\pi }_{1}(x_{1},T){\it \text{dx}}_{1}\right. \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \left.-\int _{0}^{\infty }J_{ 2}(x_{1},T)\tilde{\pi }_{1}(x_{1},T)\frac{{\it \text{dx}}_{1}} {x_{1}} \right ), {}\end{array}$$
(11.22)

where

$$\displaystyle{ J_{1}(x_{1},T) \equiv \int _{0}^{x_{1} }\tilde{\pi }_{2}(x_{2},T\mid x_{1},T)\frac{\mathit{dx}_{2}} {x_{2}}, }$$
(11.23)

and

$$\displaystyle{ J_{2}(x_{1},T) \equiv \int _{0}^{x_{1} }\tilde{\pi }_{2}(x_{2},T\mid x_{1},T)\mathit{dx}_{2}. }$$
(11.24)

Our integration task is simplified by noting that the integrals J 1, J 2 are essentially the integrals A 1, A 2 that we evaluated in Appendix 3.1. The same changes of variables, completing the square etc., need to be applied to the evaluation of J 1 and J 2. In Appendix 11.1 we show that (setting τ = T − t)

$$\displaystyle{ J_{1}(x_{1},T) = \mathcal{N}\left (\frac{(1 -\overline{\rho })\ln x_{1} +\ln \left (\frac{x_{1}(t)^{\overline{\rho }}} {x_{2}(t)} \right ) -\left (r -\gamma -\frac{s_{2}^{2}(1-\rho ^{2})} {2} \right )\tau } {s_{2}\sqrt{1 -\rho ^{2}}\sqrt{\tau }} \right ), }$$
(11.25)

when

$$\displaystyle{ \overline{\rho } =\rho \frac{s_{2}} {s_{1}},\qquad \gamma \equiv q_{2} + \frac{s_{2}^{2}\rho ^{2}} {2} + \overline{\rho }\left (r - q_{1} -\frac{s_{1}^{2}} {2} \right ), }$$
(11.26)

and

$$\displaystyle\begin{array}{rcl} J_{2}(x_{1},T)& =& x_{2}(t)( \frac{x_{1}} {x_{1}(t)})^{\overline{\rho }}e^{(r-\gamma )\tau } \\ & & \qquad \mathcal{N}\left (\frac{(1 -\overline{\rho })\ln x_{1} +\ln \left (\frac{x_{1}(t)^{\overline{\rho }}} {x_{2}(t)} \right ) -\left (r -\gamma +\frac{s_{2}^{2}(1-\rho ^{2})} {2} \right )\tau } {s_{2}\sqrt{1 -\rho ^{2}}\sqrt{\tau }} \right ).{}\end{array}$$
(11.27)

Substituting (11.25) and (11.29) into (11.22) we have

$$\displaystyle{ f(x_{1},x_{2},T) = e^{-r\tau }B_{ 1} - e^{-\gamma \tau }B_{ 2}, }$$
(11.28)

where

$$\displaystyle\begin{array}{rcl} & & B_{1} \equiv \frac{1} {\sqrt{2\pi \tau }s_{1}}\int _{0}^{\infty }\mathcal{N}\left (\frac{(1 -\overline{\rho })\ln x_{1} +\ln \left (\frac{x_{1}(t)^{\overline{\rho }}} {x_{2}(t)} \right ) -\left (r -\gamma -\frac{s_{2}^{2}(1-\rho ^{2})} {2} \right )\tau } {s_{2}\sqrt{1 -\rho ^{2}}\sqrt{\tau }} \right ) {}\\ & & \qquad \qquad \qquad \qquad \qquad \exp \left [-\frac{\left \{\ln \left ( \frac{x_{1}} {x_{1}(t)}\right ) - (r - q_{1} -\frac{s_{1}^{2}} {2} )\tau \right \}^{2}} {2s_{1}^{2}\tau } \right ]{\it \text{dx}}_{1}, {}\\ \end{array}$$

and we show in Appendix 11.2 that

$$\displaystyle{ B_{1} = x_{1}(t)e^{(r-q_{1})\tau }\int _{ -\infty }^{\infty }\frac{e^{-\frac{1} {2} z^{2} }} {\sqrt{2\pi }} \mathcal{N}(\alpha _{1} +\beta z)\mathit{dz}, }$$

where

$$\displaystyle\begin{array}{rcl} \alpha _{1}& =& \frac{\ln \left (\frac{x_{1}(t)} {x_{2}(t)}\right ) + (1 -\overline{\rho })(r - q_{1} + \frac{s_{1}^{2}} {2} )\tau -\left (r -\gamma -\frac{s_{2}^{2}(1-\rho ^{2})} {2} \right )\tau } {s_{2}\sqrt{1 -\rho ^{2}}\sqrt{\tau }}, {}\\ & & \qquad \qquad \qquad \qquad \beta = \frac{s_{1}} {s_{2}} \frac{1 -\overline{\rho }} {\sqrt{1 -\rho ^{2}}}, {}\\ \end{array}$$

and that

$$\displaystyle{ B_{2} = x_{2}(t)e^{\overline{\rho }(r-q_{1}+\frac{(\overline{\rho }-1)s_{1}^{2}} {2} )\tau }\int _{-\infty }^{\infty }\frac{e^{-z^{2}/2 }} {\sqrt{2\pi }} \mathcal{N}(\alpha _{2} +\beta z)\mathit{dz}, }$$

where

$$\displaystyle{\alpha _{2} = \frac{\ln \left (\frac{x_{1}(t)} {x_{2}(t)}\right ) + (1 -\overline{\rho })(r - q_{1} + s_{1}^{2}(\overline{\rho } -\frac{1} {2}))\tau -\left (r -\gamma +\frac{s_{2}^{2}(1-\rho ^{2})} {2} \right )\tau } {s_{2}\sqrt{1 -\rho ^{2}}\sqrt{\tau }}.}$$

Finally we use the result that (see Appendix 11.2)

$$\displaystyle{ \int _{-\infty }^{\infty }\frac{e^{-z^{2}/2 }} {\sqrt{2\pi }} \mathcal{N}(\alpha +\beta z)\mathit{dz} = \mathcal{N}\left ( \frac{\alpha } {\sqrt{1 +\beta ^{2}}}\right ), }$$

from which we obtain

$$\displaystyle\begin{array}{rcl} B_{1}& =& x_{1}(t)e^{(r-q_{1})\tau }\mathcal{N}\left ( \frac{\alpha _{1}} {\sqrt{1 +\beta ^{2}}}\right ), {}\\ B_{2}& =& x_{2}(t)e^{\overline{\rho }(r-q_{1}+\frac{(\overline{\rho }-1)s_{1}^{2}} {2} )\tau }\mathcal{N}\left ( \frac{\alpha _{2}} {\sqrt{1 +\beta ^{2}}}\right ). {}\\ \end{array}$$

The value of the exchange option (see (11.28)) is given by

$$\displaystyle{f(x_{1},x_{2},T) = x_{1}(t)e^{-q_{1}\tau }\mathcal{N}\bigg( \frac{\alpha _{1}} {\sqrt{1 +\beta ^{2}}}\bigg) - x_{2}(t)e^{-q_{2}\tau }\mathcal{N}\bigg( \frac{\alpha _{2}} {\sqrt{1 +\beta ^{2}}}\bigg).}$$