A variance swap is a forward contract on annualized variance, the square of the realized volatility. n 1. 1 n 2 i=1.

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1 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 []) 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. V R =σ R = 5 n n i= i jlog i k k j S@ti+D y z y S@t i D {{ z Its payoff at expiration is equal to Where S@t i D - the closing level of stock on t i valuation date. PayOff =σ R n - number of business days from the Trade Date up to and including the Maturity Date. When entering the swap the strike 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 σ i on day i for hedging option then P&L at maturity is sum of daily variance spread weighted by dollar gamma. Final P & L = n Hri σ i tl Γ i S i i= r i = S i+ S i is stock return on day i S i Γ i S i =Γ@i, S i D S i - dollar gamma on day i.

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=50, observation period from 004// to 004// days average of r i σ i t 4êê 4ê3ê 4ê4ê6 4ê6ê 4ê8ê7 4êê 4êê days average of dollar gamma Γ t S t 4êê 4ê3ê 4ê4ê6 4ê6ê 4ê8ê7 4êê 4êê7 5 days average of plain variance spread and variance spread weighted by dollar gamma 4êê 4ê3ê 4ê4ê6 4ê6ê 4ê8ê7 4êê 4êê A n i= Hr i σ i tlγ i S i n i= Hr i σ i tl Where dollar gamma is normalized by factor A = 5 êh Γ i S i L

3 Variance Swaps 3 Pricing The fair strike can be calculated directly from option prices (under assumptions that the underlying follows a continuous diffusion process, see Appendix ) where () = T K i F T K i K rt Put@K i D + i F T = Hr ql T is the forward price on the stock at expiration time T, r risk-free interest rate to expiration, q dividend yield. T K i >F T K i K rt Call@K i D i Example: VIX VIX CBOE volatility index it is actually fair strike for variance swap on S&P500 index. VIX@TD = T K i F T K i K rt Put@K i D + i T K i >F T K i K rt Call@K i D i T z i j F T y k K 0 { where K 0 is the first strike below the forward index level F T. The only difference with formula () is the correction term T z i j F T y k K 0 { which improves the accuracy of approximation (). 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 4 Variance Swaps Examples In this section we calculate prices of variance swap which starts on n-th trading day counted from..004 with the expiration on We compare two values: The first is calculated based on 40 European options with standardized moneyness x x = Log@K ê F T D ë Iσ è!!! t M x in range from - to with step 0.. The second value HRawL 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. These strikes we call valid strikes. The number of valid strikes we denote by N[K]. SPX The differences here are quite small less than 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 HRawL

5 Variance Swaps 5 HRawL EBAY INC This example is different. Only American options are available. But still deference are not too large. And we see that divergence happens due to the small number of strikes available for calculation of HRawL. Still the number of valid strikes is relatively large close to HRawL

6 6 Variance Swaps HRawL In the following two examples we will see that the quality of approximation () with raw market prices drops dramatically due to small number of valid strikes. ADVANCED MICRO DEVICES HRawL

7 Variance Swaps 7 HRawL Time Warner INC 5 0 HRawL

8 8 Variance Swaps HRawL Hedging Replication strategy for variance V[T] follows from the relation (for details see Appendix) n T V@TD T i= i jlog i k k j S@ti+D y z y S@t i D {{ z T i n j S@t i+ D S@t i D ki= S@t i D Log@S T ê S 0 D y z { The first term in the brackets n S@t i+ D S@t i D S@t i D i= can be thought as P&L of continuous rebalancing a stock position so that it is always long ê S t shares of the stock. The second term Log@S T ê S 0 D represent static short position in a contract which pays the logarithm of the total return. Example As an example we take the stock prices of PFIZER INC, for one year period from //004 to //005.

9 Variance Swaps 9 S t PFIZER INC Trading days from êê004 For this stock prices we calculate realized variance V t and its replication Π t for each trading day Π t = HS@t i+ D S@t i DL ê S@t i D Log@S t ê S 0 D t i <t Difference of realized variance and replication in volatility points 0H è!!!!!!!!!!! V t ê t è!!!!!!!!!!! Π t ê t L The typical differences in this plot are less than 0. volatility point. Large differences in the first days are explained by large ratio of expiration period to time step - day. Large difference around 40-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) T ê S 0 D S T S 0 S 0 + K i S 0 K i K i HK i S T L + + K i >S 0 K i K i HS T K i L + Hence log contract can be replicated by standard market instruments: è short position in /S forward contracts strike at S 0 è long position in K i ê K i put options strike at K, for all strikes K i from 0 to S 0, è long position in K i ê K i call options strike at K, for all strikes K i > S 0 So this portfolio (with doubled positions) and dynamic trading on stock replicates variance swap. The replication also gives the fair strike value of volatility swap at time t i T j S@t i+ D S@t i D kt i+ <t S@t i D Hdynamic tradingl + i S 0 F j + T k S 0 K i S 0 where r is interest rate. K i >S 0 + Log@F t D y z { K i K i r HT tl Put@S t,k i,t td + K i K r HT tl Call@S t,k i,t td y z Hstatic replicationl i { Market examples In this section we test replication performance on a set of market data. For each day we calculate price of volatility swap è!!!!!!!!!!!!!!!!!!!! V started on..004 with expiration on..005 and the price of replication At time t the volatility swap consists of the realized variance and the expected future variance T V t@td = T i jlog i t i+ <t k k j S@ti+D y z y S@t i D {{ z Æ realized variance + T r HT tl i K i j k K i F T K Put@S K i t,k i,t td + i K i >F T K Call@S t,k i,t td y z i { Æ expected variance

11 Variance Swaps For replication we use 40 European options with strikes K such that the standardized moneyness x x = Log@K ê F ë Iσ è!!! T M x is in n range from - to with step 0.. The implied volatilities for these strikes we get by linear interpolation of the market implied volatilities. Then we calculate option prices and get the fair strike from Eq.(). To estimate the performance of replication we define the replication error i Error@tD = 0 j $%%%%%%%%%%%%%%%%% y T V z k { We also plot the number of valid market strikes N[K] (strikes with well defined implied volatilities).

12 Variance Swaps Replication Error SPX Error

13 Variance Swaps 3 PFIZER INC Error

14 4 Variance Swaps EBAY INC Error

15 Variance Swaps 5 ADVANCED MICRO DEVICES Error

16 6 Variance Swaps GENERAL MOTORS CORP Error

17 Variance Swaps 7 Time Warner Inc Error

18 8 Variance Swaps Citigroup Inc Error

19 Variance Swaps 9 EXXON MOBIL CORP Error

20 0 Variance Swaps MORGAN STANLEY DEAN WITTER CORP Error

21 Variance Swaps FORD MOTOR CORP Error

22 Variance Swaps GUIDANT CORP 34 3 Error

23 Variance Swaps 3 TENET HEALTHCARE CORP Error

24 4 Variance Swaps MICRON TECHNOLOGY INC Error

25 Variance Swaps 5 Conclusion In section Pricing we tested two methods for estimation variance swap price. One method is based on the raw option prices. Another is based on a virtual set of European options with implied volatilities extracted from the market data. We saw that the both methods give very close results in case of large number of valid market strikes. But if this number is small (less than 5-) the second method gives more regular historical prices. Under condition that implied volatilities are available this method is simple, fast and can be recommended for quoting variance swaps. In section Hedging we tested replication strategy for hedging variance swaps. The results confirm that described replication is quite robust, the error of replication is typically less than one volatility point. However practical attractiveness of this method can be restricted by small number of listed options on underlying. In this case trader needs to use OTC market to construct replication portfolio.

26 6 Variance Swaps Variance and Volatility swaps It is interesting to compare fair strike level for variance swap with ATM Implied Volatility (IV ATM ). This topic is extensively discussed in [] (see also [3] ) with examples of VIX and VXO. This comparison is interesting if we think of IV ATM as an estimate of future volatility σ R. In this case it is natural to expect that there is a positive spread between and IV ATM. It follows from convexity adjustment argument (see Appendix). Let us recall it. Suppose the future variance V@TD has mean V and variance W (under risk-neutral measure) Then V = EV@TD, W = E HV@TD WL. = è!!!!!!!!!!!!! EV@TD = "### V And σ R W 8 3 We plot here 5 days moving average of fair strike level calculated for the period (t,t) calculated by Eq. () and ATM implied volatility of the same period. Indeed we see quite stable positive spread between and ATM implied volatility. If interpretation of ATM implied volatility as expected future volatility really holds then from this spread we can estimate new parameter W variance of variance. This parameter can be used for price estimates of volatility or variance derivatives., σ R and σ R SPX IV ATM IV ATM

27 Variance Swaps 7 PFIZER INC IV ATM IV ATM EBAY INC 40 0 IV ATM IV ATM ADVANCED MICRO DEVICES IV ATM IV ATM GENERAL MOTORS CORP IV ATM IV ATM

28 8 Variance Swaps Time Warner INC IV ATM IV ATM Citigroup INC IV ATM IV ATM EXXON MOBIL CORP IV ATM IV ATM MORGAN STANLEY DEAN WITTER IV ATM IV ATM

29 Variance Swaps 9 FORD MOTOR 40 0 IV ATM IV ATM GUIDANT CORP IV ATM IV ATM TENET HEALTHCARE CORP IV ATM IV ATM MICRON TECHNOLOGY INC IV ATM IV ATM

30 Variance Swaps References [] K. Demeterfi, E. Derman, M. Kamal, J. Zou More Than You Ever Wanted To Know About Volatility Swaps. Quantitative Strategies: Research Notes, Goldman Sachs 999 [] P.Carr and L.Wu A Tale of Two Indices [3] P.Carr and R. Lee Robust replication of volatility derivatives Appendix Variance Replication If stock price is follows continuous diffusion with volatility σ t S t S t =µ t t +σ t W t Then by Ito's lemma HLog@S t DL =µ t t +σ t W t σ t t Or Hence for total variance from 0 to T we get σ t t = S t S t Log@S t D T T S t (A.) V@TD = σ t t = Log@S T ê S 0 D 0 0 S t Log payoff decomposition It is easy to check the following identity for any S>0 (A.) Log@S T ê SD = S HS T SL 0 S K HK S TL + K S K HS T KL + K The fair strike value of variance swap van be calculated by taking expectation of future variance under risk-neutral measure at time t T i j k 0 t S = T EV@TD = T + Hr ql HT tl y z S τ { T E Log@S T ê S 0 D i T S τ je E Log@S T ê S 0 D y z = k 0 S τ {

31 Variance Swaps 3 Using (A.) with S = S 0 E Log@S T ê S 0 D = S 0 HF S 0 L S 0 0 K r HT tl Put@K, T td K S 0 K r HT tl Put@K, T td K where Finally we have F = Hr ql HT tl. T i j k S 0 HF S 0 L + Pricing at initial moment, = T S 0 0 i j k 0 t S τ + Hr ql HT tl y z S + τ { K r HT tl Put@K, T td K + Expression for the fair strike at time-0 can be simplified in the following way Using (A.) with S = F T K R = T EV@TD = T i T S t je E Log@S T ê S 0 D y k 0 S t { = T E Log@S T ê F T D S 0 K y r HT tl Put@K, T td K z { z = T HHr ql T E Log@S T ê S 0 DL E Log@S T ê F T D = F T 0 K rt Put@KD K + F T K rt Call@KD K Hence i F T K R = rt T j K Put@KD K + k 0 F T K y rt Call@KD K z { Convexity Adjustment

32 3 Variance Swaps σ s R convexity adjustment The expected value of future volatility σ R is equal to expectation of square root of future variance σ@td = è!!!!!!!!!! V@TD Taking second order approximation of square root in V And hence è!!!!!!!!!! V@TD "### V + V σ R = Eσ@TD = E è!!!!!!!!!! V@TD V@TD V HV@TD V L V ê 8 V 3ê σ R = E è!!!!!!!!!! V@TD "### V W 8 V 3ê