Discounted asset price is martingale in BS model

Question

I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that

$$ \mathrm{d}\mathrm{e}^{-r(T-t)}V(S_t,t)= \mathrm{e}^{-r(T-t)}\sigma S_t\frac{\partial V}{\partial S}(S_t,t)\mathrm{d}W_t $$

The Ito integral is a martingale if

$$ \mathbb{E}\left[\int_0^T\left(S_t\frac{\partial V}{\partial S}(S_t,t)\right)^2\right]<\infty $$

Unfortunately, I am not able to show this as I cannot apply Jensen, Hölder or Cauchy-Schwartz to eliminate the square. How do I get around this issue. A related question is whether the delta for an arbitrary option is bounded in the BS-model.

Answer 1

First of all, it is part of the Ito formula theorem that if $S_t$ is an Ito process then $V(t,S_t)$ is also an Ito process. This includes the regularity property you mention for the integral you mention and only assumes $V$ is continuous in time and $C^2$ in space. See Oksendal theorem 4.1.2

http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf

In order to give a bit more color notice that since $S_t$ is continuous a.s. then $\frac{\partial V}{\partial S}$ is bounded over the interval $[0,T]$ which is enough to ensure the integral is finite following assumption made on $S_t$ itself.