When Heston Prices Match Its Local Volatility Equivalent

Summary

The document asks whether a Heston stochastic volatility model and a local volatility model derived from it produce identical option prices. It points to a comparison in Gatheral’s book and asks whether the equivalence depends on spot and variance having perfect correlation, which would leave only one source of randomness. The author reports trying to check the relationship with Monte Carlo simulation but not obtaining matching results.

The post provides no answer, derivation, or numerical evidence, so it does not establish when the models should agree or explain the reported discrepancy. Its useful focus is the distinction between a stochastic volatility model and a local volatility representation, and the need to check the assumptions behind any claimed equivalence. The question also leaves the precise local volatility formula and simulation setup unspecified, limiting what can be concluded from the reported mismatch.

Key ideas

• The post asks whether a local volatility model derived from Heston reproduces Heston option prices.
• It raises perfect spot and variance correlation as a possible condition for equivalence.
• A Monte Carlo attempt reportedly did not produce matching results, but its setup is not given.
• The document provides a question rather than a derivation or resolution.

Tags

# Heston Stochastic Vol and the local volatility in Gatheral's Book


# Heston Stochastic Vol and the local volatility in Gatheral's Book












I have been reading Gatheral's Book "The volatility Surface" and in the case of Chapter 4 ( The Heston Nandi model), the author provides the following graph.

It shows that implied volatilities are the same when using the Heston Model Formula and When Using an approximation of the local Volatility in the case of the Heston Model. It seems that the author claim that we have this when spot/variance correlation is equal to one as he obtains only one source of randomness.

And This is the closed formula obtained for local volatility in gatheral's book.

Does this means that given a set of parameters of the heston model, I get the same prices if If I use the Heston Formula and if I solve the PDE using the closed formula of the local volatility that I showed above ? I tried to test this using Monte-Carlo but it doesn't seem to work. Thank you very much.

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