Why Stochastic Volatility Adds Value Beyond Local Volatility
Summary
The document explains why stochastic volatility models can be useful even when a local volatility surface has been calibrated to vanilla option prices. Matching those prices does not by itself establish that a model captures how markets evolve or how options behave when hedged. One proposed diagnostic is whether stochastic volatility changes option deltas in a way that reduces the variance of a hedged portfolio compared with local volatility.
The answer also highlights exotic option pricing. Distinct models can reproduce the same vanilla prices while implying different future volatility behavior, leading to different values for products that depend on forward volatility, such as cliquets and Bermudan options. These are conceptual examples, not empirical tests or proof that stochastic volatility is always superior. Model choice depends on the dynamics and products being assessed.
Key ideas
- Matching vanilla option prices does not uniquely determine a model’s market dynamics.
- Stochastic volatility can change option deltas and may improve hedging if it reduces hedged portfolio variance.
- Models that fit the same vanilla prices can produce different forward volatility behavior.
- Exotic option values can vary across models because they depend on future volatility dynamics.
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# What's the point of stochastic volatiliy models if you can use local volatility? # What's the point of stochastic volatiliy models if you can use local volatility? Given known call option prices, there is a unique local volatility function consistent with those prices. So why use stochastic volatility models? We can use the market to find local volatility, and then that's our model, no? Why do we need to complicate things by introducing a stochastic volatility model? Doesn't that also mean that we need to find a model that produces the same local volatility given by Dupire's equation, since otherwise it would not match the market prices. How is there any guarantee it does that? ## Answer by dm63 (score 7) https://quant.stackexchange.com/a/46376 Well, what you find is that the introduction of stochastic vol changes the delta of your options. So what does this mean? If the new delta reduces the variance of your hedged portfolio versus the pure local vol model , then it means that the introduction of stochastic vol has resulted in a better description of market dynamics versus the pure local vol model. Secondly, what you also find is that you can have different models all of which reprice the vanilla options, but that some exotic options have very different prices in the different models. For example , the introduction of stochastic vol can be done in a way that preserves the vanilla option prices , but it lowers the value of forward implied volatilities in the model versus a simple local vol model. Thus, exotics that depend on forward vols ( cliques, Bermudan etc) are priced very differently. Hence another reason to introduce stochastic vol is to improve the pricing of exotics, given the vanilla market.
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