Why Local Volatility Cannot Match Stochastic Volatility Path Distributions
Summary
The document contrasts local volatility models, where price diffusion is driven by a volatility function of price and time, with stochastic volatility models that add a random volatility state coupled to the price process. It asks whether a local volatility model can reproduce the full joint distribution of prices across dates generated by a stochastic volatility model, even when vanilla option prices agree.
The response says local volatility has enough flexibility to match individual price distributions at each time, but not the joint distributions across dates in general. For a stochastic volatility model with nonzero volatility of volatility, path behavior differs: under local volatility, instantaneous variance is fixed by the current price and time, while under stochastic volatility it remains random when conditioned on the current price. Full path distributions therefore distinguish the models. The argument is conceptual and applies under the stated nontrivial volatility-of-volatility condition; the document does not provide a formal proof or discuss specific calibration details.
Key ideas
- Local volatility specifies diffusion as a function of price and time, while stochastic volatility includes an additional random state.
- A local volatility model can match individual price distributions at separate times.
- Matching marginal distributions does not generally ensure matching joint distributions across dates.
- With nonzero volatility of volatility, instantaneous variance remains random conditional on price in a stochastic volatility model.
- The path distribution can therefore distinguish the two model classes.
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# A Difference between Local Vol and Stochastic Vol Models # A Difference between Local Vol and Stochastic Vol Models For the purpose of this question a local vol model is a 1d SDE which specifies the price process and we have a contingent claim that depends on those prices (in general, at multiple times). e.g. $dX_t = \sigma(X_t, t)dW_t$. A stochastic vol model is an at least 2d SDE where one of the equations is for the aforementioned prices process, but the additional equations specify other variables that the price process is not independent of. e.g. $dX_t = X_t Y_t dW^1_t, dY_t = \nu Y_t dW^2_t$. Do you agree that in general, given a stochastic vol model, there is no equivalent local vol model in the following sense: The joint density across all times of the price process $X_t$ in both models can be made the same. In other words, given a set of prices of contingent claims on $X_t$(that depend on multiple dates) a stochastic vol model determines, there is no local vol model that gives the same set of prices. If so, can you point me in the direction of a proof of this? If no proof is available, a nice counterexample will suffice. A local vol and stochastic vol model which give the same vanilla options price, but have at least one different joint density. ## Answer by q.t.f. (score 8) https://quant.stackexchange.com/a/17926 The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,X_s).$ When you ask about the joint density across the continuum of times $t \in [0,T]$ it is pretty easy to show that any local vol model differs from any stochastic vol model provided the stochastic volatility model has nonzeeo vol-of-vol. That is because the joint density along the continuum of times gives full information about the measure on paths $X_t$. For a local vol model, the instantaneous variance of a path $X_t$ at time $t$ is almost surely $\sigma^2(X_t,t).$ For a stochastic volatility model with nontrivial vol-of-vol this is not true for $t>0$: the instantaneous variance at time $t$ conditioned on $X_t$ is a random variable with mean $\sigma^2(X_t,t)$ but posive variance.
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