Local Volatility from Stochastic Volatility and Option Hedging

Summary

The document asks whether a stochastic volatility model and its Gyöngy local volatility counterpart imply equal option prices and hedging costs. It describes the relationship as matching the terminal distribution of the underlying under the two processes, with local variance expressed as a conditional expectation of stochastic variance given the terminal underlying level. If terminal distributions agree, European call payoffs depending only on the terminal price have the same model value.

The question then extends that pricing equivalence to replication: it asks whether delta and vega hedging under stochastic volatility should cost the same as delta hedging under local volatility. No answer or proof is included, so the hedging claim remains unresolved in the document. In particular, matching terminal distributions and European option prices alone does not demonstrate that dynamic hedge strategies, intermediate paths, or hedge costs coincide.

Key ideas

• Gyöngy’s theorem is invoked to relate stochastic volatility to a local volatility model with matching terminal distributions.
• The local variance is described as a conditional expectation of stochastic variance given the terminal asset level.
• Matching terminal distributions implies equal prices for European options whose payoff depends only on the terminal price.
• The document asks whether this also makes stochastic volatility and local volatility hedge costs equivalent.
• No proof or resolution of the dynamic hedging question is provided.

Tags

# Local volatility from stochastic volatility: implications for hedging


# Local volatility from stochastic volatility: implications for hedging












This is something I've been wondering about:

Given a stochastic volatility model with (stochastic) spot variance $\sigma^2_t$, according to Gyöngy's theorem there exists a local volatility $\sigma^2(K,T) = E_t [\sigma_T^2 | S_T = K ]$ such that the terminal distributions of $S_T$ are equal under $$ dS_t = \sigma_t S_t dW_t, \quad (1) $$ and $$ dS_t = \sigma(S_t,t) S_t dW_t, \quad (2) $$

As the distribution of $S_T$ is the same for both processes, the price of an option $(S_T - K)_+$ is the same for both processes. Since the price is the replicating portfolio, this means that delta-vega hedging the option under (1) should be the same cost as delta-hedging the option under (2) (strictly speaking no vega hedge in local vol).

Is this correct? If so is there a way to prove this other than by the arbitrage argument above?

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