Martingale Conditions for Discounted Black–Scholes Prices

Summary

The document examines whether a discounted option value in the Black–Scholes model is a martingale. Applying Itô’s formula and the Black–Scholes partial differential equation leaves a stochastic integral whose integrand contains the stock price and the option’s delta. The question is whether that integral satisfies a square-integrability condition.

The response invokes a regularity result for Itô processes and argues that, over a finite interval, continuity of the stock path and continuity of the option’s spatial derivative make the delta term bounded along each path. It says this suffices for finiteness given assumptions on the stock process. The explanation is brief and does not spell out the precise integrability assumptions or establish a uniform bound across paths. Its claim about arbitrary options therefore depends on the stated smoothness and model conditions, rather than showing that every option delta is globally bounded.

Key ideas

• Discounting and applying the Black–Scholes equation leaves a stochastic integral involving the option delta.
• Showing the integral is a martingale requires suitable integrability conditions.
• Under continuity assumptions, the delta term is bounded along each stock path over a finite interval.
• The response relies on regularity assumptions and does not prove a uniform bound for every option delta.

Tags

# Discounted asset price is martingale in BS model


# 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.

## Answer by Ezy (score 1)

https://quant.stackexchange.com/a/43294

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.

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