Multi-Asset Options
Question:
Derive the Option Pricing Partial Differential Equation for Multi-Asset options.
Solution:
Given a universe of \( N \) assets \( S_{1}, \cdots, S_{N} \) that follow a log-normal random walk based on \( N \) brownian motions \( W_{1}, \cdots, W_{N}\)
\[ dS_{i,t} = \mu_{i}S_{i,t}dt + \sigma_{i}S_{i,t}dW_{i} \]
with correlation \( \rho_{ij} \) between brownian two motions \( W_{i} \& W_{j} \), i.e
\[ dW_{i}dW_{j} = \rho_{ij}dt \]
For an option \( V(S_{1,t}, \cdots, S_{N,t}, t) \) on the \( N \) assets, we can use the multi-dimensional Ito to get
\[ dV(S_{1,t}, \cdots,S_{N,t}) = (\frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^2V}{\partial S_{i}\partial S_{j}})dt + \sum_{i=1}^{N}\frac{\partial V}{\partial S_{i,t}}dS_{i,t} \]
\( \)
Delta Hedging:
The delta hedging for a multi-asset option follows the same dynamics as the concet of delta hedging used in deriving Black Scholes PDE for single asset options
Let us consider a portfolio \( \Pi \) consisting long an option and short \( \Delta_{i,t} \) of asset \( S_{i,t} \) for all the \( N \) assets.
\[ \Pi = V(S_{1,t}, \cdots, S_{N,t}) - \sum_{i=1}^{N}\Delta_{i,t}S_{i,t} \]
A change in this portfolio would be
\[ d\Pi = dV(S_{1,t, \cdots, S_{N,t}}) - \sum_{i=1}^{N}\Delta_{i,t}dS_{i,t} \]
\[ \implies d\Pi = (\frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^2V}{\partial S_{i}\partial S_{j}})dt + \sum_{i=1}^{N}(\frac{\partial V}{\partial S_{i,t}} - \Delta_{i,t})dS_{i,t} \]
By choosing
\[ \Delta_{i,t} = \frac{\partial V}{\partial S_{i,t}}, 1 \le i \le N \]
we can eliminate the random/diffusion term from the equation and be left with only the deterministic/drift term making
\[ d\Pi = (\frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^2V}{\partial S_{i}\partial S_{j}})dt \]
Now since the randomness in the portfolio is removed, the portfolio should grow at a risk-free rate. i.e
\[ d\Pi = r\Pi dt = r(V - \sum_{i=1}^{N}\Delta_{i,t}S_{i,t})dt \]
From the two equations for \( d\Pi \) above, we have
\[ r(V - \sum_{i=1}^{N}\Delta_{i,t}S_{i,t}) = \frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^2V}{\partial S_{i}\partial S_{j}} \]
\[ \frac{\partial V}{\partial t} + r\sum_{i=1}^{N}S_{i,t}\frac{\partial V}{\partial S_{i,t}} + \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^2V}{\partial S_{i}\partial S_{j}} - rV = 0 \]
Each asset with a continuous dividend yield \( D_{i} \), the corresponding equation becomes
\[ \frac{\partial V}{\partial t} + \sum_{i=1}^{N}(r - D_{i})S_{i,t}\frac{\partial V}{\partial S_{i,t}} + \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^2V}{\partial S_{i}\partial S_{j}} - rV = 0 \]