Option on the Product of Two Asset Prices

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The growth of the financial sector has resulted in products which are covered under the broad term "exotic derivatives". These derivatives are often written on indices which are derived from traded prices but which themselves are not traded. Depending on investor preferences an index can be a function of more than one asset prices and can be determined from the value of these asset prices from a single or a series of observations. Exotic derivatives can either be priced using analytic methods or numerical techniques. The framework used to price all exotic derivatives is based on the Black-Scholes option pricing theory, in which dynamic hedging is used to obtain an arbitrage-free equation for the option price. Although we can always obtain a p.d.e. for all exotic derivatives, an analytic solution cannot always be obtained. However, there exists a large range of exotics where an analytic solution is possible. An option on the product of two asset prices has an analytic solution.

Given two traded assets, an index can be created where the value of the index at some time $t$ is defined as,

$S(t)=\frac{P_1(t)P_2(t)}{P_1(0)P_2(0)}$

where $t = 0$ is the time at which the index is created and $S (0) = 1$ . An option can be written on this index with payoff at expiry $T$ ,

$C (T) = max[S (T) - 1,0]$

Since the option is only a function of $P 1$ , $P 2$ and $t$ , given the s.d.e.s for the prices of the two assets,

$\frac{dP_1(t)}{P_1(t)}=m_1 dt + \sigma_1 dW_t^1$

$\frac{dP_2(t)}{P_2(t)}=m_2 dt + \sigma_2 dW_t^2$

(where $dW_t^1 dW_t^2 = \rho dt$ ) Itô's lemma can be applied the price of the option to give,

$dC=\left[ \frac{\partial C}{\partial t} + m_1 P_1(t)\frac{\partial C}{\partial P_1} + m_2 P_2(t)\frac{\partial C}{\partial P_2} + \frac{1}{2} \sigma_1^2 P_1(t)^2 \frac{\partial^2 C}{\partial P_1^2} + \frac{1}{2} \sigma_2^2 P_2(t)^2 \frac{\partial^2 C}{\partial P_2^2} + \sigma_1 \sigma_2 \rho P_1(t) P_2(t) \frac{\partial^2 C}{\partial P_1\partial P_2}\right] dt + \sigma_1 \frac{\partial C}{\partial P_1} P_1(t) dW_t^1+ \sigma_2 \frac{\partial C}{\partial P_2}P_2(t) dW_t^2$

A portfolio consisting of $1 of the option, $-\partial C / \partial P_1$ of asset 1 and $-\partial C / \partial P_2$ of asset 2 must therefore have an s.d.e. given by,

$d\left( C - \frac{\partial C}{\partial P_1} P_1(t) - \frac{\partial C}{\partial P_2} P_2(t)\right)=\left[ \frac{\partial C}{\partial t} + \frac{1}{2} \sigma_1^2 P_1(t)^2 \frac{\partial^2 C}{\partial P_1^2} + \frac{1}{2} \sigma_2^2 P_2(t)^2 \frac{\partial^2 C}{\partial P_2^2} + \sigma_1 \sigma_2 \rho P_1(t) P_2(t) \frac{\partial^2 C}{\partial P_1\partial P_2}\right] dt$

Since this portfolio has no sources of risk, in the absence of arbitrage it must have an instantaneous return equal to the risk-free rate $r$ . Therefore the last equation gives rise to the following p.d.e.:

$rC= \frac{\partial C}{\partial t} + r P_1\frac{\partial C}{\partial P_1} + r P_2\frac{\partial C}{\partial P_2} + \frac{1}{2} \sigma_1^2 P_1^2 \frac{\partial^2 C}{\partial P_1^2} + \frac{1}{2} \sigma_2^2 P_2^2 \frac{\partial^2 C}{\partial P_2^2} + \sigma_1 \sigma_2 \rho P_1 P_2 \frac{\partial^2 C}{\partial P_1\partial P_2}$

From the payoff function of this option we can deduce that the pricing equation can be transformed into a two-dimensional one with variables $t$ and $P = P 1 P 2$ . Note that,

$\frac{\partial C}{\partial P_1}=P_2\frac{\partial C}{\partial P}$