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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 F_{s} and F_{ss} denote Δ and the Γ for X and let G_{s} and G_{ss} 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 G_{s}= F_{s},

cG_{ss}= F_{ss},

for which we easily obtain the solution

c=F_{ss}/G_{ss},

a= F_{s}-F_{ss}/G_{ss}.

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

a+c G

cG

for which we easily obtain the solution

c=F

a= F

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.

A. Makhlouf and E Gobet. (2009). The tracking error rate of the Delta-Gamma hedging strategy. Preprint

Questions: Magnus Wiktorsson

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