Numerical Methods for PDE

Abstract

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let $V(S_t, t)$ be the price of an option at time t when the share price of the underlying stock is $S_t$. See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond $B_t$ and a risky asset stock $S_t$. They evolve as where r is the interest rate, $\mu$ is the drift, $\sigma$ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price $B_t^{-1}V(S_t,t)$. By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure . Also, under the risk-neutral measure $\mu =r$. We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.