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modelerror6 - Hedging Derivatives with Model Error Robert A...

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Hedging Derivatives with Model Error Robert A. Jarrow March 21, 2010 Abstract The current derivatives pricing technology enables users to hedge deriv- atives with the underlying asset or any other traded derivative. In theory, there is no reason to prefer one hedging instrument to another. However, given model errors, this is not true. Imposing some simple assumptions on the structure of model errors, this paper shows that to maximize hedg- ing accuracy, there is an ordering to the hedging instruments utilized. Holding constant market illiquidities, one should always hedge °rst with "like" derivatives, next with derivatives one layer down the hierarchy of derivatives, and lastly using the underlying. 1 Introduction The modern theory of derivatives pricing and hedging is in a mature state as evidenced by the existence of numerous textbooks on the subject (e.g. Baxter and Rennie [2], Chance and Brooks [3], Jarrow and Turnbull [9], Detemple [4], Hull [6], Jarrow [8], Musiela and Rutkowski [11], and Rebonato [9]). This technology enables one to price and hedge various layers of derivatives. For example, considering equity as the underlying, one can price and hedge plain vanilla calls and puts, various exotic options, and swaps. When hedging, the theory is indi/erent among the hedging instruments. For our example, one can hedge an exotic option using the underlying equity, calls and puts, or other exotic options. This non-uniqueness is even more pronounced when considering interest rate derivatives in the context of the Heath-Jarrow-Morton [5] model. Indeed, in this pricing methodology, given a °nite number of Brownian motions generating the evolution, there is even more choice in which underlying securities (zero-coupon bonds) to hedge with. But, the theory is only an approximation to market structures, and there is model error. Given the existence of model error, it is no longer true that there is an indi/erence among hedging instruments. Model error can arise because the theory ignores market frictions, or because markets are not perfectly competi- tive, or because the evolution of the underlying security is misspeci°ed. Indeed, 1
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a common sense hedging procedure for market illiquidities is already standard practice. Everything else constant, it is well known that one should hedge deriv- atives with the most liquid hedging instrument available. But, everything else is not always constant. As noted above, other model errors are also present, for example, using the wrong stochastic process for the evolution of the underlying asset. This paper studies hedging in the presence of these secondary model errors. As such, one can interpret our analysis as solving the following question: given various layers of traded derivatives with equal market liquidity, which hedging instrument is preferred. Given some simple and intuitive assumptions on the structure of these secondary model errors, we show that to maximize hedging accuracy, there does exist a secondary ordering of the hedging instruments.
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