Heston and Implied Volatility: European Payoffs Versus Exotics
Summary
The document asks whether pricing options from a Black–Scholes volatility surface implied by calibrated Heston parameters gives the same values as direct Heston pricing. It distinguishes vanilla options, used to calibrate the model, from other payoffs such as binaries and path-dependent contracts. The question’s intuition is that matching terminal stock-price distributions may not match the distributions along the path, which matter for barriers and similar exotics.
The answer frames the comparison through the local volatility model built from Heston’s vanilla prices using Dupire’s method. It states that this local volatility model and Heston give the same prices for European payoffs, while exotic-option prices will generally differ. The note gives no derivation, numerical example, or details about particular exotic structures. Its conclusion therefore describes a general model distinction rather than a payoff-by-payoff pricing recipe; it does not establish that all path-dependent or early-exercise products behave identically.
Key ideas
- A volatility surface implied from Heston vanilla prices reproduces those vanilla prices by construction.
- A local volatility model derived from Heston vanilla prices matches Heston prices for European payoffs.
- Exotic-option prices generally differ between the Heston and derived local volatility models.
- Path-dependent products depend on more than the terminal distribution of the underlying.
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Full text
# Answer by Antoine Conze (score 5) # Is pricing options using the volatility surface implied by the Heston model equivalent to pricing using the Heston model directly for all options? Given Heston model parameters calibrated from vanilla put/call options it is possible to imply a volatility surface by pricing calls or puts for different strikes and maturities and solving the inverse BS equation to find the corresponding black volatilities. Clearly, by construction, using this volatility in the BS formula for puts/calls will yield the same price as pricing the same option directly with the Heston model. To what extent does this hold for other options? My intuition tell me that for path dependent options like barrier options this would not be the same as the black vol and Heston parameters only imply the same distribution of the stock price at maturity, but not the same distributions along the path there. Would it however be the same thing for non-path dependent instruments, such as a binary call option? Meaning that you would retrieve the same price for the binary call option by using the black-vol from your Heston-implied vol surface in a BS model as you would by pricing the binary call option directly with the Hesotn model? A different way to view this I suppose is: Would you retrieve the same volatility surface by pricing options with the Heston model and solving for the corresponding black vol regardless of the type of option, or would you get a different surface for e.g. puts/call and binary options? What about path dependent options like American or barriers? ## Answer by Antoine Conze (score 5) https://quant.stackexchange.com/a/61461 Consider the Heston model and the Local Volatility model with local volatility built (using Dupire) from the Heston reconstructed vanilla options implied volatility. The price of any European payoff will be the same under both models. The price of exotic options will usually not be the same.
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