Problem bank

Problem 210 of 333HardFinanceP210

Reduce Black-Scholes to the heat equation

  1. The Black-Scholes PDE is

    Vt+rSVS+12σ2S2VSS=rV,V(S,T)=max⁡(S−K,0).V_t + rSV_S + \tfrac12\sigma^2S^2V_{SS} = rV, \qquad V(S,T) = \max(S - K, 0).

    Find a change of variables that turns it into the heat equation wτ=D wzzw_\tau = D\,w_{zz} with an initial condition. State DD, the initial condition, and VV in terms of ww.