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dual Black-Scholes partial differential equation

The value function for vanilla options can be written as

\begin{displaymath} v(x,K,T,t,\sigma,r_d,r_f,\phi)=e^{-r_d(T-t)}I\!\!E[F\vert S_t=x]. \end{displaymath} (42)

Consequently, the process $v(t,S_t)e^{-r_dt}=e^{-r_dT}I\!\!E[F\vert S_t]$ is a martingale, whence the $dt$-coefficient of its differential must vanish. Therefore $v(x,K,T,t,\sigma,r_d,r_f,\phi)$ satisfies the Black-Scholes partial differential equation
\begin{displaymath} v_t-r_dv+(r_d-r_f)xv_x+\frac{1}{2}\sigma^2x^2v_{xx}=0. \end{displaymath} (43)

This can easily be remembered by noting that the derivatives have the same sign.

Viewing $v$ as a function of $T$ and $K$, one can verify by direct computation that the so-called dual Black-Scholes partial differential equation

\begin{displaymath} -v_T-r_fv+(r_f-r_d)Kv_K+\frac{1}{2}\sigma^2K^2v_{KK}=0 \end{displaymath} (44)

also holds. We note that the Black-Scholes equation holds for all options, whereas its dual is a particularity of put and call options. More details on this issue can be found in [9] and [1].



2000-06-11
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