Multiple Asset Option
Definition
Multi asset option’s value is depending on the multiple asset value. For example, there are two assets denoted as $S_1$ and $S_2$, the payoff of one possible two-asset option is $max(S_1,S_2)$.
Two-Asset General Example
Suppose there are two assets follow belowing SDE:
\[dS_1 = \mu S_1dt + \sigma_1S_1dW_1\] \[dS_2 = \mu S_2dt + \sigma_1S_1dW_2\]And $dW$ is normal distributed random variable with mean $0$ and variance $t$.
There is a correlation between $dW_1$ and $dW_2$, defined as below:
\[Cov(dW_1,dW_2) = E(dW_1dW_2) - E(dW_1)E(dW_2) = E(dW_1dW_2) = \rho dt\] \[Corr(dW_1,dW_2) = \frac{Cov(dW_1,dW_2)}{\sigma_{dW_1}\sigma_{dW_2}} = \rho\]Denote the value of two-asset option as $V$ and it’s a function of $S_1$, $S_2$ and $t$.
\[V(S_1,S_2,t)\]Take derivative on $V$:
\[dV = \frac{\partial V}{\partial S_1}dS_1 + \frac{1}{2}\frac{\partial^2 V}{\partial S_1^2}dS_1dS_1+\frac{\partial V}{\partial S_2}dS_2 + \frac{1}{2}\frac{\partial^2 V}{\partial S_2^2}dS_2dS_2 +\frac{1}{2}\frac{\partial^2 V}{\partial S_1S_2}dS_1dS_2+\frac{\partial V}{\partial t}dt\]
- $dS_1dS_1 = \sigma^2_1S_1^2dt$
- $dS_2dS_2 = \sigma^2_2S_2^2dt$
- $dS_1dS_2 = \sigma_1\sigma_2S_1S_2\rho dt$
