In Stein-Shakarchi's book Fourier Analysis (p. 170), the solution of the Black-Scholes equation $$\frac{\partial V}{\partial t}+rs\frac{\partial V}{\partial s}+\frac{\sigma^2s^2}{2}\frac{\partial^2 V}{\partial s^2}-rV=0$$ for $0\lt t\lt T$

with boundary condition $V(s,T)=F(s)$, is given as

$$V(s,t)=\frac{e^{-r(T-t)}}{\sqrt{2\pi\sigma^2(T-t)}}\int_0^\infty e^{-\frac{(\log(s/s^*)+(r-\sigma^2/2)(T-t))^2}{2\sigma^2(T-t)}} F(s^*)\,ds^*.$$

My own derivation shows that $ds^*$ in the formula should be replaced by $$\frac{ds^*}{s^*}.$$

Is this a typo in the book?

  • $\begingroup$ That is not really the Black and Scholes equation, which is stochastic. $\endgroup$ – AD. Nov 13 '10 at 6:11
  • $\begingroup$ That is what the book calls it. $\endgroup$ – TCL Nov 15 '10 at 14:31

I have checked my derivation thoroughly, and I believe now that it is a typo.

  • $\begingroup$ Agreed, I got the same result. $\endgroup$ – process91 Oct 13 '14 at 18:47

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