Variance Swap - frouah.com

Variance Swapby Fabrice Douglas this Note we present a detailed derivation of the fair value of variancethat is used in pricing a Variance swap. We describe the approach describedby Demeter et al.[2] and others. We also show how a simpler version can bederived, using the forward price as the threshold in the payo decompositionthat is used in the derivation. The Variance swap has a payo equal toNvar 2R Kvar (1)whereNvaris the notional, 2 Ris the realized annual Variance of the stock overthe life of the swap, andKvar=E[ 2R]is the delivery (strike) Variance . Theobjective is to nd the value ofKvar:1 Stock Price SDEThe Variance swap starts by assuming a stock price evolution similar to Black-Scholes, but with time-varying volatility parameter tdStSt= dt+ tdWt:Considerf(S) = lnSand apply It o s LemmadlnSt= 12 2t dt+ tdWtso that12 2t=dStSt dlnSt:(2)2 The VarianceIn equation (2) take the average Variance fromt= 0tot=TVT=1 TZT0 2tdt=2T"ZT0dStSt ZT0dlnSt#(3)=2T"ZT0dStSt lnSTS0#:1 The Variance swap rateKvaris the fair value of the Variance ; that is, it is theexpected value of the average Variance under the risk neutral measure.

The variance swap rate K var is the fair value of the variance; that is, it is the expected value of the average variance under the risk neutral measure. Hence K var = E[V T] = E 1 T Z T 0 ˙2 t dt # (4) = 2 T E "Z T 0 dS t S t ln S T S # = 2

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  Variance, Swaps, Variance swap

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