hard · Quantitative Finance prob-stats
A European option value V(S,t) on a non-dividend-paying underlying solves the Black–Scholes PDE partial_tV+rSpartial_SV+tfrac12σ^2S^2partial_SSV-rV=0 with terminal condition V(S,T)=payoff(S_T).
By the Feynman–Kac theorem, V(S,t)=e^-r(T-t)E[payoff(S_T)mid S_t=S], where the expectation is taken over which measure and dynamics for S?
- A measure under which dS_t=μ S_t,dt+σ S_t,dW_t, the real-world drift μ, since Feynman–Kac only pins down the diffusion term.
- A measure Q under which dS_t=rS_t,dt+σ S_t,dW_t^Q, matching the PDE's own first-order drift coefficient rS exactly.
- A measure under which dS_t=(r+tfrac12σ^2)S_t,dt+σ S_t,dW_t, adding the usual Itô convexity term on top of r.
- A measure under which dS_t=(r-tfrac12σ^2)S_t,dt+σ S_t,dW_t, borrowing the terminal log-price drift as the SDE's own drift.
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