How do finite difference methods solve the Black-Scholes PDE?

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AcadiFi · Accepted Answer

Finite difference methods discretize the BSM PDE dV/dt + 0.5*sigma^2*S^2*d2V/dS2 + r*S*dV/dS - r*V = 0 on a grid of stock prices and times. You create M price nodes (S_0 to S_max) and N time nodes (0 to T), approximate derivatives with differences: dV/dS ~ (V_i+1 - V_i-1)/(2*dS), d2V/dS2 ~ (V_i+1 - 2V_i + V_i-1)/dS^2, dV/dt ~ (V_j+1 - V_j)/dt. Terminal condition V(S, T) = payoff, boundary conditions at S=0 (V=0 for call) and S=S_max (V=S-K*exp(-r*(T-t)) for call). Step backward in time solving for grid values. For a put on Bridgeport Chemicals strike $45, S_max=$90, dS=$1, dT=0.002, you build a 91x500 grid and roll back from T to 0. For American options, at each backward step check max(V_computed, payoff) at every node. FD handles arbitrary payoffs and early exercise, unlike closed-form BSM. ```mermaid graph TD A[Terminal payoff] --> B[Time step
backward] B --> C[Apply
boundary] C --> D{American?} D -->|Yes| E[Max exercise] ```

How do finite difference methods solve the Black-Scholes PDE?

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