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Variance Swaps Alexander Gairat In collaboration with IVolatility.com Variance swaps Introduction The goal of this paper is to make a reader more familiar with pricing and hedging variance swaps and to propose some practical recommendations for quoting variance swaps (see section Conclusions). We give basic ideas of variance swap pricing and hedging (for detailed discussion see [1]) and apply this analyze to real market data. In the last section we discuss the connection with volatility swaps. A variance swap is a forward contract on annualized variance, the square of the realized volatility. n−1 2 2 1 S ti+1 VR =σR = 252 Log n − 2 S ti i=1 i i @ D yy „ j j zz k k @ D {{ Its payoff at expiration is equal to 2 2 PayOff =σR − KR Where S ti - the closing level of stock on ti valuation date. n - number of business days from the Trade Date up to and including the Maturity Date. @ D When entering the swap the strike KR is typically set at a level so that the counterparties do not have to exchange cash flows (‘fair strike’). Variance swap and option delta-hedging. Advantage of trading variance swap rather than buying options is that it is pure play on realized volatility no path dependency is involved. Let us recall how path dependency appears in trading P&L of a delta-hedged option position. If we used implied volatility σi on day i for hedging option then P&L at maturity is sum of daily variance spread weighted by dollar gamma. n−1 1 2 2 2 Final P & L = ri −σi ∆t Γi Si 2 i=1 = Si+1−Si is stock return on day ri S i i ‚ H L 2 2 Γi Si =Γ i, Si Si - dollar gamma on day i. @ D 2 Variance Swaps So periods when dollar gamma is high are dominating in P&L. Market example We use here historical data for the option with one year expiration on SPX500, strike K=1150, observation period from 2004/1/2 to 2004/12/4 2 2 5 days average of ri −σi ∆t 0.00005 4 1 2 4 3 1 4 4 26 4 6 22 4 8 17 4 10 12 4 12 7 -0.00005 ê ê ê ê ê ê ê ê ê ê ê ê ê ê -0.0001 2 5 days average of dollar gamma ΓtSt 10000 8000 6000 4000 2000 4 1 2 4 3 1 4 4 26 4 6 22 4 8 17 4 10 12 4 12 7 5 days average of plain variance spread and variance spread weighted by dollar gamma ê ê ê ê ê ê ê ê ê ê ê ê ê ê 4 1 2 4 3 1 4 4 26 4 6 22 4 8 17 4 10 12 4 12 7 -0.002 ê ê ê ê ê ê ê ê ê ê ê ê ê ê -0.004 -0.006 -0.008 n−1 2 2 2 A∗ i=1 ri −σi ∆t ΓiSi n−1 2 2 i=1 ri −σi ∆t ‚ H L ‚ 2 Where dollar gamma is normalized by factor A = 252 Γi HSi L êH⁄ L Variance Swaps 3 Pricing The fair strike KR can be calculated directly from option prices (under assumptions that the underlying follows a continuous diffusion process, see Appendix 1) ∆ ∆ 2 2 Ki rT 2 Ki rT (1) KR = Put Ki + Call Ki 2 2 T Ki≤FT Ki T Ki>FT Ki where „ @ D „ @ D r−q T FT = is the forward price on the stock at expiration time T, r risk-freeH interestL rate to expiration, q dividend yield. Example: VIX VIX CBOE volatility index it is actually fair strike KR for variance swap on S&P500 index. ∆ ∆ 2 2 2 Ki rT 2 Ki rT 1 FT VIX T = Put Ki + Call Ki − − 1 2 2 T Ki≤FT Ki T Ki>FT Ki T K0 where K is the first strike below the forward index level F . i y 0 @ D „ T@ D „ @ D j z k { The only difference with formula (1) is the correction term 2 1 FT − − 1 T K0 which improvesi the accuracyy of approximation (1). j z k { There two problems in calculation the fair strike of variance swap in from raw market prices: è We need quotes of European options. They are available for indexes but not for stocks. è Number of available option strikes can be not sufficient for accurate calculation. The first problem can be solved by calculating prices of European options from implied volatilities of listed American options. To overcome the second problem we could use additional strikes and interpolate implied volatilities to this set. 4 Variance Swaps Examples In this section we calculate prices of variance swap which starts on n-th trading day counted from 1.1.2004 with the expiration on 31.12.2004. We compare two values: The first KR is calculated based on 40 European options with standardized moneyness x x = Log K FT σ t è!!!! x in range@ fromê D-2ë toI 2 withM step 0.1. The second value KR Raw is calculated based on the raw market prices (either they are European or American). We use only the strikes of options with valid implied volatilities IV, i.e. IV is in range from 0 to 1. These strikes we call valid strikes. The number of valid strikes we denote by N[K]. H L SPX The differences here are quite small less than 1 volatility point. It is natural to expect small difference for SPX because in both cases we used European option and, what is more important, the number of valid strikes is rather large more than 40. 25 22.5 20 17.5 50 100 150 200 12.5 KR KR Raw H L Variance Swaps 5 KR−KR Raw H L 50 100 150 200 -0.2 -0.4 -0.6 -0.8 N K 50 45@ D 40 50 100 150 200 EBAY INC This example is different. Only American options are available. But still deference are not too large. And we see that diver- gence happens due to the small number of strikes available for calculation of KR Raw . Still the number of valid strikes is relatively large close to 20. 42.5 H L 40 37.5 35 32.5 30 27.5 50 100 150 200 250 KR KR Raw H L 6 Variance Swaps KR−KR Raw 2.5 2 H L 1.5 1 0.5 50 100 150 200 250 -0.5 -1 N K 20 18 16@ D 14 12 10 50 100 150 200 250 In the following two examples we will see that the quality of approximation (1) with raw market prices drops dramatically due to small number of valid strikes. ADVANCED MICRO DEVICES 70 65 60 55 50 45 50 100 150 200 250 KR KR Raw H L Variance Swaps 7 KR−KR Raw 15 10 H L 5 50 100 150 200 250 -5 -10 -15 -20 N K 14 12 @ D 10 8 50 100 150 200 250 Time Warner INC 30 25 20 50 100 150 200 250 KR KR Raw H L 8 Variance Swaps KR−KR Raw 10 H L 5 50 100 150 200 250 -5 N K 10 8 @ D 6 4 50 100 150 200 250 Hedging Replication strategy for variance V[T] follows from the relation (for details see Appendix) n−1 2 n−1 1 1 S ti+1 2 S ti+1 − S ti V T ≈ Log ≈ − Log ST S0 T T S ti T i=1 S ti i=1 @ D i @ D @ D y ji ji zyzy j z The first term in the brackets@ D „ j j zz j‚ @ ê Dz k k @ D {{ k @ D { n−1 S ti+1 − S ti i=1 S ti @ D @ D can be thought as P&L of continuous rebalancing a stock position‚ so that it is always long 1 St shares of the stock. @ D The second term ê −Log ST S0 represent static short position in a contract which pays the logarithm of the total return. @ ê D Example As an example we take the stock prices of PFIZER INC, for one year period from 1/1/2004 to 1/1/2005. Variance Swaps 9 St PFIZER INC 38 36 34 32 30 28 26 Trading days from 1 1 2004 50 100 150 200 250 ê ê For this stock prices we calculate realized variance Vt and its replication Πt for each trading day Πt = 2 S ti+1 − S ti S ti − 2 Log St S0 ti<t ‚ H @ D @ DL ê @ D @ ê D Difference of realized variance and replication in volatility points 100 Vt t − Πt t è!!!!!!!!!!!! è!!!!!!!!!!!! 0.1 H ê ê L 0.05 50 100 150 200 250 -0.05 -0.1 The typical differences in this plot are less than 0.1 volatility point. Large differences in the first 10 days are explained by large ratio of expiration period to time step - 1 day. Large difference around 240-th trading day appears due to the jump of the stock price. 10 Variance Swaps We see that combination of dynamic trading on stock and log contract can efficiently replicate variance swap. The payoff of log contract can be replicated by linear combination of puts and calls payoffs (see Appendix) ST − S0 ∆Ki ∆Ki −Log ST S0 ≈− + Ki − ST + + ST − Ki + 2 2 S0 Ki≤S0 Ki Ki>S0 Ki @ ê D „ H L „ H L Hence log contract can be replicated by standard market instruments: è short position in 1/S forward contracts strike at S0 2 è long position in ∆Ki Ki put options strike at K, for all strikes Ki from 0 to S0, 2 è long position in ∆Ki Ki call options strike at K, for all strikes Ki > S0 ê So this portfolio (with doubled positions) and dynamic trading on stock replicates variance swap.
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