# Why is the stochastic process of the volatility of a stock price square integrable?

I am taking a course in financial mathematics(Ito-Integrals, Black-Scholes,...) and there is something that is not immediately clear to me. When constructing our stock price model, the integral $$\int_0^t \sigma_s\;dW_s$$, $$\sigma_t$$ being the volatility stochastic process, was the reason we had to construct the Ito-Integral. So we set up all the theory and here in my course notes it is stated that $$\int_0^t \sigma_s\;dW_s$$ really is an Ito-Integral which means that $$E\big[\int_0^t\sigma_s^2 ds\big] < \infty$$. So far so good.

Right now we are dealing with Ito-processes because our stock price model seems to be an Ito-process. According to my lecture notes for the stock price model to really be an Ito-process the volatility $$\sigma_t$$ needs to be square integrable meaning $$\int_0^t\sigma_s^2 ds < \infty$$. Is this normally the case? In my lecture notes it isn't stated anywhere so maybe this is a trivial implication from some of our assumptions which I overlooked.

You can define the Ito integral without square integrability but this makes working with applications like pricing more complicated, so the assumption is typically made in practice.

The question of whether this actually holds is pretty complicated and boils down to an old question of whether stock return variance is finite or not. Famously, Mandelbrot (https://www.jstor.org/stable/168611) argued it is not, implying that square integrability is unrealistic. However, I think most people nowadays believe variance is actually finite so the condition would be reasonable.

Another issue is that these types of models are typically used for pricing and applied under the risk neutral measure. Hence the goal is often to provide a realistic model for option pricing rather than model stock prices as accurately as possible.

• If you don't assume square-integrability, then the mathematics needed are considerably above my level. Which is a very good reason why I am in favour of making this assumption ;) Commented Jul 6, 2023 at 9:36
• Risk neutral pricing has an effect on variance?
– SBF
Commented Jul 7, 2023 at 5:25
• @SBF It can with stochastic volatility models, e.g. the Heston model. But the broader point of the paragraph is that practicioners use these models to match option prices rather than actual stock price dynamics.
– fes
Commented Jul 7, 2023 at 5:57