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.
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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 nonuniqueness is even more pronounced when considering
interest rate derivatives in the context of the HeathJarrowMorton [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
(zerocoupon 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,
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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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 JARROW,ROBERT
 Derivatives, Derivative, Mathematical finance, Aj Aj, …rst layer

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