Deriving a Two-Asset Black–Scholes PDE with Correlated Brownian Motion

Summary

The document works through the Itô expansion for an option whose value depends on two asset prices and time. It identifies the second-order terms for each asset and the mixed partial derivative, which accounts for covariance between the assets’ Brownian motions. The author asks whether this is the right starting point for deriving a Black–Scholes formula and what steps should follow.

The excerpt provides the stochastic dynamics and the proposed differential expansion, but no derivation of the pricing equation or closed-form payoff value. The mixed term must include the correlation between the Brownian motions; the displayed two-asset expression implicitly assumes correlation of one. To proceed, specify the correlation and option payoff, then use a risk-neutral or hedged-portfolio argument to obtain the pricing PDE and its boundary conditions. A general two-asset payoff need not have a simple Black–Scholes closed form.

Key ideas

• The value of a two-asset option depends on both asset prices and time.
• Itô’s lemma includes a mixed second derivative when the asset returns are correlated.
• The covariance term must reflect the correlation between the two Brownian motions.
• A pricing PDE also requires a pricing argument, payoff, and suitable boundary conditions.

Tags

# Multidimentional Black Scholes Formula


# Multidimentional Black Scholes Formula












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?

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