Option on the product of two asset prices

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,

$\frac{d{P}_1(t)}{P_1(t)}=m_{1}dt+{\sigma }_{1}d{W}_t^1$

$\frac{d{P}_2(t)}{P_2(t)}=m_{2}dt+{\sigma }_{2}d{W}_t^2$

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

$dC=[\frac{\partial C}{\partial t}+m_1{P}_1(t)\frac{\partial C}{\partial P_1}+m_2{P}_2(t)\frac{\partial C}{\partial P_2}+\frac{1}{2}{\sigma }_1^2{P}_1(t{)}^2\frac{{\partial }^{2}C}{\partial P_1^2}+\frac{1}{2}{\sigma }_2^2{P}_2(t{)}^2\frac{{\partial }^{2}C}{\partial P_2^2}+{\sigma }_1{\sigma }_2\rho P_1(t)P_2(t)\frac{{\partial }^{2}C}{\partial P_1\partial P_2}]dt+{\sigma }_1\frac{\partial C}{\partial P_1}{P}_1(t)d{W}_t^1+{\sigma }_2\frac{\partial C}{\partial P_2}{P}_2(t)d{W}_t^2$

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,

$d(C-\frac{\partial C}{\partial P_1}{P}_1(t)-\frac{\partial C}{\partial P_2}{P}_2(t))=[\frac{\partial C}{\partial t}+\frac{1}{2}{\sigma }_1^2{P}_1(t{)}^2\frac{{\partial }^{2}C}{\partial P_1^2}+\frac{1}{2}{\sigma }_2^2{P}_2(t{)}^2\frac{{\partial }^{2}C}{\partial P_2^2}+{\sigma }_1{\sigma }_2\rho P_1(t)P_2(t)\frac{{\partial }^{2}C}{\partial P_1\partial P_2}]dt$

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,

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

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

$\frac{{\partial }^{2}C}{\partial P_1^2}=P_2^2\frac{{\partial }^{2}C}{\partial P^2}$

$\frac{{\partial }^{2}C}{\partial P_2^2}=P_1^2\frac{{\partial }^{2}C}{\partial P^2}$

$\frac{{\partial }^{2}C}{\partial P_1\partial P_2}=P_1{P}_2\frac{{\partial }^{2}C}{\partial P^2}+\frac{\partial C}{\partial P}$

Therefore the p.d.e. can be simplified to,

$rC=\frac{\partial C}{\partial t}+mP\frac{\partial C}{\partial P}+\frac{1}{2}{\sigma }^2{P}^2\frac{{\partial }^{2}C}{\partial P^2}$

where,

$m=2r+{\sigma }_1{\sigma }_2\rho$

and,

$\sigma =\sqrt{{\sigma }_1^2+{\sigma }_2^2+2{\sigma }_1{\sigma }_2\rho }$

and boundary condition $C(T)=\max [P(T)/P(0)-1]$. This p.d.e. is the Black-Scholes p.d.e. for a call option and can be solved to give,

$C(0)=\exp [(m-r)T]N(h_1)-\exp [-rT]N(h_2)$

where,

$h_1=\frac{(m+\frac{1}{2}{\sigma }^2)\sqrt{T}}{\sigma }$

$h_2=\frac{(m-\frac{1}{2}{\sigma }^2)\sqrt{T}}{\sigma }$

The same result can be obtained by starting with the risk-neutral processes for the two assets,

$\frac{d{P}_1(t)}{P_1(t)}=rdt+{\sigma }_{1}d{\tilde{W}}_t^1$

$\frac{d{P}_2(t)}{P_2(t)}=rdt+{\sigma }_{2}d{\tilde{W}}_t^2$

Using Itô's lemma, the process for the product of the two prices is,

$\frac{dP(t)}{P(t)}=mdt+\sigma d{W}_t$

and the pricing equation derived using the p.d.e. follows.