Quadratic Variation with Stochastic Volatility in Itô Processes

Summary

The document addresses why an asset price process with stochastic volatility has quadratic variation equal to the integral of price squared times instantaneous variance. For a process whose proportional change is volatility times a Brownian increment, the volatility itself may be stochastic without invalidating the quadratic variation expression.

The answer states the conditions needed for Itô calculus: volatility must be adapted to the filtration, so its value at a given time is measurable using information available by then, and it must be square integrable over finite intervals. The discussion gives conditions rather than a derivation, numerical example, or comparison of stochastic volatility models. Its scope is the mathematical validity of applying Itô's lemma under these assumptions.

Key ideas

• A stochastic volatility process can appear in an Itô process when it satisfies suitable regularity conditions.
• Volatility must be adapted to the information filtration.
• The squared volatility must have a finite time integral over the interval considered.
• Under these conditions, the quadratic variation of price accumulates at a rate determined by price squared and instantaneous variance.

Tags

# Ito's lemma in stochastic volatility models


# Ito's lemma in stochastic volatility models












I couldn't help but notice that in all stochastic volatility models articles I consulted, whenever Ito lema is applied with a process of the sort $$\frac{d S_t}{S_t} = \sigma_t d W_t $$ With $(\sigma_t)$ being a stochastic process.

It's considered that $$d<S_t> = S_t^2 \sigma_t^2 dt$$ Is this justified? Given that $\sigma_t$ is stochastic?

You can find such statment for instance in the original Heston's article (page 14 of the pdf document). https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.139.3204&rep=rep1&type=pdf

Thank you

## Answer by Daneel Olivaw (score 2)

https://quant.stackexchange.com/a/72085

If we work on a probability space $(\Omega,\mathfrak{F},\mathbb{R})$ endowed with a filtration $\mathbb{F}=(\mathfrak{F}_t)_{t\geq0}$, Itô's Lemma is applicable to Itô processes, requiring the stochastic process $(\sigma_t)_{t\geq0}$ to be:

- Adapted to the filtration $\mathbb{F}$ i.e. $\sigma_t$ is mesurable w.r.t to $\mathfrak{F}_t$ for any $t\geq0$; and

- Integrable i.e. $\int_{[0,t]}\sigma^2_s\text{d}s<\infty$ for any $t\geq0$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.