Stochastic Taylor Expansions with a Stochastic Volatility Factor

Summary

The document asks how to expand a financial stochastic differential equation when the diffusion coefficient depends on both the underlying price and a stochastic volatility factor, as in a SABR-style model. It contrasts this setup with simpler stochastic Taylor expansions and asks what the first-order approximation for the price process should look like.

It also raises whether an expansion is possible without specifying the volatility factor’s own dynamics, and whether treating that factor as constant is a valid simplification. These are open questions in the document: it supplies no derivation, answer, or numerical evidence. The key limitation is that an expansion generally depends on the joint dynamics and dependence structure of the state variables, so the text alone does not establish a usable approximation.

Key ideas

• The question concerns an SDE whose diffusion depends on both price and a stochastic volatility factor.
• A Taylor expansion may require the dynamics of each stochastic state variable.
• Treating volatility as constant is raised as a simplification but not justified in the document.
• No derivation or conclusion is provided.

Tags

# Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$


# Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$












https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf

In the above document stochastic taylor expansions are nicely explained.

Let us now consider a typical SDE model in finance like SABR. here the form is:

$$ dS_t=b(\sigma_t,S_t)dW_t $$

My point is that because $b$ also depends on another stochastic variable, we cannot follow the simple steps provided in the document. Or can we ....... So my question, how dos the Taylor expansion of $S_t$ look like in this setup? To keep it simple we can just say first order, that is not not important.

And also: can we even derive the Taylor expansion if we dont know the dynamics of $\sigma_t$. I wonder if we can just say $$dS_t=\sigma_t b(S_t)$$ and treat $\sigma_t$ as a "constant". How wrong would that be?

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