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# Delta-vega hedge

A **Delta-vega hedge** or **Δ-Ѵ 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 Ѵ of the portfolio
V, matches Δ and the Γ for X. Let Δ_{X}= F_{s} and Ѵ_{X}=F_{σ} denote Δ and the Ѵ for X and Δ_{Y}= G_{s} and Ѵ_{Y}=G_{σ} denote Δ and the Ѵ for Y. In order to match Δ and the Ѵ we get the following linear system of equations for a and c:

a+cΔ_{Y}=Δ_{X},

cѴ_{Y}=Ѵ_{X},

for which we easily obtain the solution

c=Ѵ_{X}/Ѵ_{Y},

a=Δ_{X}- Δ_{Y}Ѵ_{X}/Ѵ_{Y}.

If the market consisting of S,B and Y is complete (and it might be complete even if the market consisting only of S and B is not complete e.g. in the Heston model if Y is the price of a European call or put option) we have that b(t)=(F(t,S(t))-a(t)S(t)-c(t)G(t,S(t)))/B(t). In fact if S follows the Heston model and if we trade continuously the Delta-vega hedge will in fact replicate the contract X. 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-vega 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.

a+cΔ

cѴ

for which we easily obtain the solution

c=Ѵ

a=Δ

If the market consisting of S,B and Y is complete (and it might be complete even if the market consisting only of S and B is not complete e.g. in the Heston model if Y is the price of a European call or put option) we have that b(t)=(F(t,S(t))-a(t)S(t)-c(t)G(t,S(t)))/B(t). In fact if S follows the Heston model and if we trade continuously the Delta-vega hedge will in fact replicate the contract X. 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-vega 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.

Questions: Magnus Wiktorsson

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