Hedge backtesting for model validation

Hedge backtesting for model validation

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Pricing and hedging is easy, in theory. The existence of arbitrage-free prices is equivalent to the existence of martingale measures, and there are ways of constructing replication strategies via the martingale representation theorem and the Clark-Ocone formula (see, for example, Harrison & Pliska, 1981, and Musiela & Rutkowski, 2005). Arbitrage-free prices can also be obtained as solutions to certain partial differential equations (PDEs) (see, for example, Harrison & Pliska, 1981, and Black & Scholes, 1973). But one cannot deem a model as appropriate solely by observing that it satisfies theory; one needs to introduce the vital information coming from the market, in particular the dynamics of the hedging instruments and traded prices of a derivative. For this, one can perform a hedge backtesting exercise in which one looks at the quality of the dynamic hedging performance of the model over a historical period. All things being equal, one would prefer a model that gave a good hedging performance to one that didn’t. But then surely one could get a good hedging performance simply by using a simplistic model, for example a model that is linear in the hedging instruments? Closely related issues, among others, were looked at in Bakshi, Cao & Chen (2000), where it was found that the best model in terms of hedging is not necessarily the most advanced or realistic. Many such questions can be asked. How should you mark-to-market? What if a model gives very good hedging performances but its price differs from market quotes for the derivative? Should one then start using the market price, even though we are confident that we can hedge the value as given by the model? The purpose of this article is to answer these questions, provide a rigorous definition of a hedge backtest, explain what precisely a backtest tells us about a model, and relate hedge backtesting to martingale and PDE pricing theories.

Hedge backtesting for model validation

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