On robustness of the Black-Scholes partial differential equation model (Q2800055)

This paper deals with the Black-Scholes partial differential equation model. It is assumed that \(S=S(t)\) follows a geometric Brownian motion \(dS(t)=\mu Sdt+\sigma SdW\), where \(\mu\) is the expected annual drift rate, \(\sigma\) the volatility and \(W(t)\) a Brownian motion. If at time \(t\) a portfolio consists of a long position in the option and short position in \(N(t)\) units of stock \(S\), then the portfolio value \(\Pi\) at time \(t\) is equal to \(\Pi=V-N(t)S\), and the return to the portfolio value over the interval \([t,t+\Delta t]\) is then equal to \(\Delta\Pi=\Delta V-N(t)\Delta S\). If the amount \(\Pi\) is invested in a riskless asset with an interest rate \(r\), then over the interval of length \(\Delta t\) the return to the riskless investment is equal to \(\Pi r\Delta t\). The hedging error is defined by \(\Delta H=\Delta\Pi-\Pi r\Delta t\). The author obtains that the mean and the variance of the hedging error are of order \(O(\Delta t^2)\). It is proved that the mean absolute value of the hedging error would be lower than that for the standard continuous-time delta. The adjusted partial differential equation describing the option value \(V(t,S)\) is derived by \(V_{t}(t,S)+{1\over2}\sigma^2 S^2 V_{SS}(t,S)+rSV_{S}(t+\alpha\Delta t,S)-rV(t,S)=0\). Then, using the change of variable, the equation \(u_{t}(t,x)=u_{xx}(t,x)+au_{x}(t-h,x)+bu(t-h,x)\) is derived, and the solution of this equation is compared to that of the Black-Scholes partial differential equation.