I need to write the Black-Scholes formula for option $V = (S_1, S_2, t)$, where: $$d S_1 = \mu_1 S_1 dt + \sigma_1 S_1 d W_1,$$ $$ d S_2 = \mu_2 S_2 dt + \sigma_2 S_2 d W_2.$$

We know that $W_1$ and $W_2$ are Brownian motions. Generally, I tried to use a multidimensional version of Ito's lemma. I used this formula:

$$dV = \Bigg(\frac{\partial V}{\partial t} + \frac12 \sum_{i=1}^d \sum_{j=1}^d \sigma_i \sigma_j \rho_{ij} S_i S_j \frac{\partial^2 V}{\partial S_i \partial S_j}\Bigg)dt + \sum_{i=1}^d \frac{\partial V}{\partial S_i} d S_i.$$

I got:

$$dV = \Bigg( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma_1^2 S_1^2 \frac{\partial^2 V}{\partial S_1^2} + \sigma_1 \sigma_2 S_1 S_2 \frac{\partial^2 V}{\partial S_1 \partial S_2} + \frac12 \sigma_2^2 S_2^2 \frac{\partial^2 V}{\partial S_2^2} \Bigg)dt + \frac{\partial V}{\partial S_1} dS_1 + \frac{\partial V}{\partial S_2} dS_2.$$

Is it a proper way to determine the Black-Scholes formula for this option? What should I do next?

  • $\begingroup$ Hello Dracks and welcome to StackExchange. Could you specify more what you are trying to achieve? Are you trying to derive the Black Scholes PDE for two underlyings $S_1, S_2$? or price an option on both underlyings $S_1, S_2$, if that's the case then what type of option (spread? basket? etc.) $\endgroup$ – byouness May 14 at 8:26
  • $\begingroup$ I'm sorry, it could have been a bit unclear. I'm interested in deriving Black-Scholes PDE. $\endgroup$ – Dracks May 14 at 9:31
  • $\begingroup$ @Dracks Does this answer help? Granted, the explanation there is short, but the PDE can be directly derived from no-arbitrage pricing (or using standard hedging arguments - that's a bit lengthy though). If you look at the last example for elliptic PDEs, you find the pricing PDE for you problem, you only have to include a time derivative, $\frac{\partial V}{\partial t}$ (and replace $K_t$ by $S_2$). In general, you probably want to look at Margrabe's Formula. Does that make sense? $\endgroup$ – Kevin May 14 at 11:08
  • $\begingroup$ You should try to remove the $dS_1$ and $dS_2$ terms by forming a basket of the option and the two underlying assets (i.e. compute $w_1$ and $w_2$ so that $dV + w_1 dS_1 + w_2 dS_2 = (\dots) dt + 0 dW_1 + 0 dW_2$) like you would do in the single asset case. Then, you know that $(\dots) = r$ by absence of arbitrage and this would give you the PDE. $\endgroup$ – byouness May 14 at 11:17

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