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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 ΔX= Fs and ѴX=Fσ denote Δ and the Ѵ for X and ΔY= Gs and ѴY=Gσ 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+cΔYX,
YX,
for which we easily obtain the solution
c=ѴXY,
a=ΔX- ΔYѴXY.
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
Last update: 2010 Feb 22 13:45:28. Validate: HTML CSS

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