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Delta-gamma hedge

A Delta-gamma hedge or Δ-Γ hedge (See also Greeks for general discussion and definition of sensitivity derivatives) is a self financing portfolio consisting of the underlying asset,S say, and a standard bond, B say, and some additional derivative (usually some frequently traded European put or call option on S), Y say, that replicates (or in some cases approximates) the value of a derivative, X say. Let F(t,s) be price at time t of the derivative X given that S(t)=s and G(t,s) be the price of the derivative Y. Let V(t) be the value of the self-financing portfolio at time t, i.e. V(t)=a(t)S(t)+b(t)B(t)+c(t)G(t,S(t)), where a(t) is the number of the underlying asset S and b(t) is the number of bonds B and c(t) is the number of the derivative Y in the portfolio. We now want to find a,b,c such that the Δ and the Γ of the portfolio V, matches Δ and the Γ for X. Let Fs and Fss denote Δ and the Γ for X and let Gs and Gss denote Δ and the Γ for Y. In order to match Δ and the Γ we get the following linear system of equations for a and c:
a+c Gs= Fs,
cGss= Fss,
for which we easily obtain the solution
c=Fss/Gss,
a= Fs-Fss/Gss.
For the market consisting of S,B we have that b(t)=(F(t,S(t))-a(t)S(t)-c(t)G(t,S(t)))/B(t). If the market is not complete b(t) will not equal (F(t,S(t))-a(t)S(t)-c(t)G(t,S(t)))/B(t) and the Delta-gamma hedge will not exactly replicate X. For most cases however it will at least be a better approximation than a standard Delta-hedge even in the case where the trading is done discretely.Sometimes it is better to use a Delta-vega hedge instead of a Delta-gamma hedge e.g. if the volatility changes sufficiently fast. In fact if S follows the Heston model and if we trade continuously the Delta-vega hedge will in fact replicate the contract X.

References

M. Brode'n and M. Wiktorsson. (2009) On the Convergence of Higher Order Hedging Schemes. Preprint
A. Makhlouf and E Gobet. (2009). The tracking error rate of the Delta-Gamma hedging strategy. Preprint

 

Questions: Magnus Wiktorsson
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