# Discounted asset price is martingale in BS model

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.

• If you are interested in the discounted stock price, then $V \left( S_t, t \right) = S_t$..? As for your related question, see quant.stackexchange.com/questions/30177 - the delta is bounded by the slope of the payoff function in the B/S model. – LocalVolatility Jan 1 at 22:46
• Are you interested in the discounted option price, $V(S_t, t)$ or the discounted stock price, $S_t$? Note the difference... – Nap D. Lover Jan 2 at 0:00
• You can check the technical point on my answer, where I use the Black-Scholes formula and the fact that delta is a probability. – Daneel Olivaw Jan 2 at 10:38

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
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.