Gyöngy’s Theorem and Local-to-Stochastic Volatility Models

Summary

The document presents Gyöngy’s theorem as a way to construct a local volatility process whose one-time probability distributions match those of a given stochastic process. In the stated result, the local drift and variance are conditional expectations of the stochastic process’s drift and variance given its current value. This connects stochastic volatility models to local volatility models through distribution matching.

It then poses the reverse problem: whether a given local volatility model can be represented by a stochastic volatility model with the same marginal distributions, especially when the stochastic volatility follows its own dynamics, such as a CIR process. The text raises the possibility that some local volatility models may not admit such a representation and asks what properties would distinguish them. It does not supply a construction, proof, or characterization, so it is best read as a research question motivated by the theorem rather than as a complete modeling procedure. The equivalence under discussion concerns distributions at each time, not necessarily identical paths or joint distributions across times.

Key ideas

• Gyöngy’s theorem links a stochastic process to a local volatility process with matching distributions at each time.
• The local drift and variance in the theorem are conditional expectations given the current process value.
• The document asks whether the construction can be reversed for a specified local volatility model.
• A proposed reverse representation may require volatility factors with their own stochastic dynamics.
• Matching marginal distributions alone does not establish that two models have matching paths or joint distributions across time.

Tags

# Construction of stochastic volatility model from a given local volatility model


# Construction of stochastic volatility model from a given local volatility model












The Gyongy's theorem:

> Let $X_t$ be a stochastic process satisfying $$dX_t = \mu_t dt+\sigma_tdW_t$$ where $\mu_t, \sigma_t$ are bounded stochastic process adapted to the filtration $\mathcal{F}_t$. Then there exists a stochastic differential equation $$dY_t = b(t,Y_t) +s(t,Y_t)dW_t$$ such that $Y_t$ and $X_t$ have the same probability distribution for every $t$. In addition, the two functions $b(t,y)$ and $s(t,y)$ satisfy $$b(t,y) =\mathbb{E}(\mu_t|X_t=y)$$ $$s^2(t,y) = \mathbb{E}(\sigma_t^2|X_t=y)$$

This beautiful theorem gives us a method to construct an equivalent local volatility model (i.e. the model of the process $Y_t$) from a given stochastic volatility model (i.e. the model of the process $X_t$).

I would like to know whether there exists a method to construct an equivalent stochastic volatility model from a given local volatility model. In other words, given the process $Y_t$, can we construct the process $X_t$ such that $X_t$ and $Y_t$ have the same probability distribution for every $t$?

If there are LVMs that can not be generated from any SVM, I would like to know their characteristics?

PS: we may have $(\mu_t, \sigma_t) = (b(t,X_t), s(t,X_t))$, but this result is too trivial. Is there a way to contruct $(\mu_t, \sigma_t)$ such that they follow some stochastic differential equations that are not directly related to $X_t$? (for example, $\sigma_t$ follows a CIR model)

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