The Pricing of Variance, Volatility, Covariance, and Correlation Swaps

The Pricing of Variance, Volatility, Covariance, and Correlation Swaps Anatoliy Swishchuk, Ph.D., D.Sc. Associate Professor of Mathematics & Statistics University of Calgary Abstract Swaps are useful for volatility hedging and speculation. Volatility swaps are forward contracts on future realized stock volatility and variance swaps are similar contracts on variance, the square of future volatility. Covariance and correlation swaps are covariance and correlation forward contracts, respectively, of the underlying two assets. Using change of time method, one can model and price variance, volatility, covariance and correlation swaps. Keywords: variance swap, volatility swap, covariance swap, correlation swap, Heston model, stochastic volatility Variance, volatility, covariance and correlation swaps are relatively recent financial products that market participates can use for volatility hedging and speculation. The market for these types of swaps has been growing with many investment banks and other financial institutions are now actively quoting volatility swaps on various assets: stock indexes, currencies, and commodities. A stock s volatility is the simpliest measure of its riskiness or uncertainty. In this entry we describe, model, and price variance, volatility, covariance, and correlation swaps. Description of Swap We begin with a description of the describe kind of swaps that we will be discussing in this entry: variance swaps, volatility swaps, covariance swaps, and correlation swaps. Table 1 provides a summary of studies dealing with these swaps. Variance and Volatility Swaps A stock s volatility is the simplest measure of its risk less or uncertainty. Formally, the volatility σ R is the annualized standard deviation of the stock s 1

returns during the period of interest, where the subscript R denotes the observed or realized volatility. Why trade volatility or variance swaps? As mentioned in Demeterfi et al. (1999, p. 9), just as stock investors think they know something about the direction of the stock market so we may think we have insight into the level of future volatility. If we think current volatility is low, for the right price we might want to take a position that profits if volatility increases. The easiest way to trade volatility is to use volatility swaps, sometimes called realized volatility forward contracts, because they provide only exposure to volatility and not other risk. Variance swaps are similar contracts on variance, the square of the future volatility. As noted by Carr and Madan (1998), both types of swaps provide an easy way for investors to gain exposure to the future level of volatility. A stock volatility swap s payoff at expiration is equal to N(σ R (S) K vol ), where σ R (S) is the realized stock volatility (quoted in annual terms) over the life of contract, 1 T σ R (S) = σ T sds, 2 σ t is a stochastic stock volatility, K vol is the annualized volatility delivery price, and N is the notional amount of the swap in dollar per annualized volatility point. Although options market participants talk of volatility, it is variance, or volatility squared, that has more fundamental significance 1. A variance swap is a forward contract on annualized variance, the square of the realized volatility. Its payoff at expiration is equal to N(σ 2 R(S) K var ), where σr 2 (S) is the realized stock variance(quoted in annual terms) over the life of the contract; that is, σ 2 R(S) = 1 T T σ 2 sds, K var is the delivery price for variance, and N is the notional amount of the swap in dollars per annualized volatility point squared. The holder of variance swap at expiration receives N dollars for every point by which the stock s realized variance σr 2 (S) has exceeded the variance delivery price K var. Therefore, pricing the variance swap reduces to calculating the square of the realized volatility. 1 See Demeterfi, Derman, Kamal, and Zou (1999). 2

Valuing a variance forward contract or swap is no different from valuing any other derivative security. The value of a forward contract P on future realized variance with strike price K var is the expected present value of the future payoff in the risk-neutral world: P var = E{e rt (σ 2 R(S) K var )}, where r is the risk-free interest rate corresponding to the expiration date T, and E denotes the expectation. Thus, for calculating variance swaps we need to know only E{σR 2 (S)}, namely the mean value of the underlying variance. To calculate volatility swaps we need more. Using Brockhaus and Long (2) approximation (which the second-order Taylor expansion for function x) we have 2 E{ σr 2 (S)} E{V } V ar{v } 8E{V }, 3/2 where V = σr 2 V ar{v } (S) and is the convexity adjustment. 8E{V } 3/2 Thus, to calculate the value of volatility swaps P vol = {e rt (E{σ R (S)} K vol )} we need both E{V } and V ar{v }. Later we explicitly solve the Cox-Ingersoll-Ross 3 equation for the Heston model for stochastic volatility 4 using the change of time method and present the formulas to price variance and volaitlity swaps for this model. Covariance and Correlation Swaps Options dependent on exchange rate movements, such as those paying in a currency different from the underlying currency, have an exposure to movements of the correlation between the asset and the exchange rate. This risk can be eliminated by using a covariance swap. A covariance swap is a covariance forward contract of the underlying rates S 1 and S 2 which have a payoff at expiration that is equal to N(Cov R (S 1, S 2 ) K cov ), where K cov is a strike price, N is the notional amount, and Cov R (S 1, S 2 ) is a covariance between two assets S 1 and S 2. Logically, a correlation swap is a correlation forward contract of two underlying rates S 1 and S 2 which payoff at expiration is the following 2 See also Javaheri et al. (22, p.16). 3 See Cox, Ingersoll, and Ross (1985). 4 See Heston (1993). N(Corr R (S 1, S 2 ) K corr ), 3

where Corr(S 1, S 2 ) is a realized correlation of two underlying assets S 1 and S 2, K corr is a strike price, N is the notional amount. Pricing covariance swap, from a theoretical point of view, is similar to pricing variance swaps, since Cov R (S 1, S 2 ) = 1/4{σ 2 R(S 1 S 2 ) σ 2 R(S 1 /S 2 )} where S 1 and S 2 are two underlying assets, σr 2 (S) is a variance swap for the underlying assets, and Cov R (S 1, S 2 ) is a realized covariance of the two underlying assets S 1 and S 2. Thus, we need to know the variances for S 1 S 2 and for S 1 /S 2. Correlation Corr R (S 1, S 2 ) is defined as follows: Corr R (S 1, S 2 ) = Cov R (S 1, S 2 ) σ 2 R (S 1 ) σ 2 R (S2 ), where Cov R (S 1, S 2 ) is defined as above and σ 2 R (S1 ) is the realized variance for S 1. Given two assets S 1 t and S 2 t with t [, T ], sampled on days t = < t 1 < t 2 <... < t n = T between today and maturity T, the log-return of each asset is and R j i = log( Sj t i S j t i 1 ), i = 1, 2,..., n, j = 1, 2. Covariance and correlation can be approximated by respectively. Cov n (S 1, S 2 ) = Corr n (S 1, S 2 ) = n (n 1)T n Ri 1 Ri 2 i=1 Cov n (S 1, S 2 ) V arn (S 1 ) V ar n (S 2 ), Table 1: Summary of Studies Dealing with Variance, Volatility, Covariance, and Correlation Swaps Modeling and Pricing of Variance, Volatility, Covariance and Correlation Swaps with Stochastic Volatility In this section, we explicitly solve the Cox-Ingersoll-Ross equation for stochastic volatility Heston model using the change of time method and present the formulas to price variance, volaitlity, covariance, and correlation swaps for this model. 4

Stochastic Volatility: Heston Model Let (Ω, F, F t, P ) be a probability space with filtration F t, t [, T ]. Assume that the underlying asset S t in the risk-neutral world and variance follow the following model (see Heston (1993)): { dst S t = r t dt + σ t dwt 1 dσt 2 = k(θ 2 σt 2 )dt + γσ t dwt 2 (1), where r t is the deterministic interest rate, σ and θ are short and long volatility, k > is the reversion speed, γ > is the volatility (of volatility) parameter, and wt 1 and wt 2 are independent standard Wiener processes. The Heston asset process has a variance σt 2 that follows a Cox-Ingersoll- Ross process, described by the second equation in (1). If the volatility σ t follows Ornstein-Uhlenbeck process (see, for example, Øksendal (1998)), then Itô s lemma shows that the variance σt 2 follows the process described exactly by the second equation in (1). Note that if 2kθ 2 > γ 2, then σt 2 > with P = 1 (see Heston (1993)). Solving the equation for variance σt 2 in (1) explicitly using change of time method gives and takes the following form: dσ 2 t = k(θ 2 σ 2 t )dt + γσ t dw 2 t (2) σ 2 t = e kt (σ 2 θ 2 + w 2 (φ 1 t )) + θ 2, (3) where w 2 (t) is an F t -measurable one-dimensional Wiener process, and φ 1 t an inverse function to φ t : t φ t = γ 2 {e kφs (σ 2 θ 2 + w 2 (s)) + θ 2 e 2kφs } 1 ds. (4) This result simply follows from the following substitution v t = e kt (σ 2 t θ 2 ) (5) into the equation (2) instead of σ 2 t. Note that if 2kθ 2 > γ 2, then σ 2 t > with P = 1 (see, for example, Heston (1993)). From (5) it follows that v t e kt + θ 2 is strictly positive too. If we take the integrand in the last integral we obtain [e kφs (σ 2 θ 2 + w 2 (t)) + θ 2 e 2kφs ] 1 = [e 2kφs (e kt (σ 2 θ 2 + w 2 (t))) + θ 2 )] 1 = [e kφs e kt (σ 2 θ 2 + w 2 (t))) + θ 2 ] 2 = [e kφs e kt v t + θ 2 ] 2, since v t = σ 2 θ 2 + w 2 (t)). The expression under square root sign is positive above and square root is well-defined. Hence, the last expression and therefore, the integrand in the integral in (4), are strictly positive. It means that φ t is monotone function and there exists inverse function φ 1 t in (3). 5 is

Valuing of Variance and Volatility Swaps From previous results we get the following expression for the price of a variance and volatility swap. The value (or price) P var of a variance swap is P var = e rt [ 1 e kt (σ 2 θ 2 ) + θ 2 K var ] (6) kt and the value (or price) P vol of volatility swap is approximately P vol e rt {( 1 e kt (σ 2 kt θ 2 ) + θ 2 ) 1/2 ( γ2 e 2kT [(2e 2kT 4e kt kt 2)(σ 2 2k 3 T 2 θ 2 ) + (2e 2kT kt 3e 2kT + 4e kt 1)θ 2 ])/[8( 1 e kt (σ 2 kt θ 2 ) + θ 2 ) 3/2 ]] K vol }. (7) The same expressions for E[V ] and for V ar[v ] may be also found in Brockhaus and Long (2). Valuing of Covariance and Correlation Swaps To value a covariance swap must be calculated the following P = e rt (ECov(S 1, S 2 ) K cov ). (8) To calculate ECov(S 1, S 2 ) we need to calculate E{σR 2 (S1 S 2 ) σr 2 (S1 /S 2 )} for the given two underlying assets S 1 and S 2. Let St, i i = 1, 2, be two strictly positive Ito s processes given by the following model { dst i = µ i S tdt + σtdw i t, i t i d(σ i ) 2 t = k i (θi 2 (σ i ) 2 t )dt + γ i σtdw i j t, i = 1, 2, j = 3, 4, where µ i t, i = 1, 2, are deterministic functions, k i, θ i, γ i, i = 1, 2, are defined in a similar way as in (1), standard Wiener processes w j t, j = 3, 4, are independent, [w 1 t, w 2 t ] = ρ t dt, ρ t is deterministic function of time, [, ] means the quadratic covariance, and standard Wiener processes w i t, i = 1, 2, and w j t, j = 3, 4, are independent. We note that where and d ln S i t = m i tdt + σ i tdw i t, m i t := (µ i t (σi t) 2 2 ), Cov R (S 1 T, S 2 T ) = 1 T [ln S1 T, ln S 2 T ] = 1 T [ T 6 T σt 1 dwt 1, σt 2 dwt 2 ] = 1 T T ρ t σ 1 t σ 2 t dt.

and where and Let us show that First, note that [ln S 1 T, ln S 2 T ] = 1 4 ([ln(s1 T S 2 T )] [ln(s 1 T /S 2 T )]). (9) d ln(s 1 t S 2 t ) = (m 1 t + m 2 t )dt + σ + t dw + t, d ln(s 1 t /S 2 t ) = (m 1 t m 2 t )dt + σ t dw t, (σ ± t ) 2 := (σ 1 t ) 2 ± 2ρ t σ 1 t σ 2 t + (σ 2 t ) 2, dw ± t := 1 σ ± t (σ 1 t dw 1 t ± σ 2 t dw 2 t ). Processes w t ± above are standard Wiener processes by the Levi-Kunita- Watanabe theorem and σ t ± are defined above. In this way, we obtain that and [ln(s 1 t S 2 t )] = [ln(s 1 t /S 2 t )] = t t (σ + s ) 2 ds = (σ s ) 2 ds = t t From (9)-(11) we have directly formula (8): ((σ 1 s) 2 + 2ρ t σ 1 sσ 2 s + (σ 2 s) 2 )ds, (1) ((σ 1 s) 2 2ρ t σ 1 sσ 2 s + (σ 2 s) 2 )ds. (11) [ln S 1 T, ln S 2 T ] = 1 4 ([ln(s1 T S 2 T )] [ln(s 1 T /S 2 T )]). (12) Thus, from (12) we obtain that Cov R (S 1, S 2 ) = 1/4(σ 2 R(S 1 S 2 ) σ 2 R(S 1 /S 2 )). Returning to the valuation of the covariance swap in (8) we have P = E{e rt (Cov(S 1, S 2 ) K cov } = 1 4 e rt (Eσ 2 R(S 1 S 2 ) Eσ 2 R(S 1 /S 2 ) 4K cov ). The problem now has reduced to the same problem as above, but instead of σ 2 t we need to take (σ + t ) 2 for S 1 S 2 and (σ t ) 2 for S 1 /S 2 (with (σ ± t ) 2 = (σ 1 t ) 2 ± 2ρ t σ 1 t σ 2 t + (σ 2 t ) 2 ), and proceed with the similar calculations as for the variance and volatility swaps. 7

Numerical Example: Volatility Swap for S&P 6 Canada Index In this section, we apply the analytical provided solutions from above to price a swap on the volatility of the S&P6 Canada index for five years (January 1997-February 22). 5 Suppose that at the end of February 22 we wanted to price the fixed leg of a volatility swap based on the volatility of the S&P6 Canada index. The statistics on log returns for the S&P 6 Canada Index for the five years covering January 1997-February 22 is presented in Table 2. Table 2 Statistics on Log Returns S&P 6 Canada Index Series: LOG RETURNS S&P 6 CANADA INDEX Sample: 1 13 Observations: 13 Mean.235 Median.593 Maximum.51983 Minimum -.1118 Std. Dev..13567 Skewness -.665741 Kurtosis 7.787327 From the statistical data for the S&P 6 Canada index log returns for the 5-year historical period (1,3 observations from January 1997 to February 22) it may be seen that the data exhibits leptokurtosis. If we take a look at the S&P 6 Canada index log returns for the 5-year historical period, we observe volatility clustering in the return series. These facts indicate the presence of conditional heteroscedasticity. A GARCH(1,1) regression is applied to the series and the results are obtained as in Table 3. This table allows one to generate different input variables for the volatility swap model. Table 3 5 These data were supplied by Raymond Théoret (Université du Québec à Montréal, Montréal, Québec, Canada) and Pierre Rostan (Analyst at the R&D Department of Bourse de Montréal and Université du Québec à Montréal, Montréal, Québec, Canada). They calibrated the GARCH parameters from five years of daily historic S&P 6 Canada Index from January 1997 to February 22. See Theoret, Zabre and Rostan (22). 8

Estimation of the GARCH(1,1) process Dependent Variable: Log returns of S&P 6 Canada Index Prices Method: ML-ARCH Included Observations: 13 Convergence achieved after 28 observations - Coefficient: Std. error: z- statistic: Prob. C.617.338 1.824378.681 Variance Equation C 2.58E-6 3.91E-7 6.597337 ARCH(1).6445.7336 8.238968 GARCH(1).927264.6554 141.4812 R-squared -.791 Mean dependent -.235 var Adjusted R- -.318 S.D. dependent var -.13567 squared S.E. of regression.13588 Akaike info criterion - - 5.928474 Sum squared resid.239283 Schwartz criterion - - 5.912566 Log likelihood 3857.58 Durbin-Watson stat - 1.88628 We use the following relationships: θ = V, k = 1 α β ξ 1, γ = α dt dt dt to calculate the following discrete GARCH(1,1) parameters: ARCH(1,1) coefficient α =.6445; GARCH(1,1) coefficient β =.927264; the Pearson kurtosis (fourth moment of the drift-adjusted stock return) ξ = 7.787327; long volatility θ =.5289724; k = 3.9733; γ = 2.499827486; short volatility σ =.1. Parameter V may be found from the expression V = 2.58 1 6 is defined in Table 3. Thus, V =.2991; dt = 1/252 =.3968254. Applying the analytical solutions (6) and (7) for a swap maturity T of.91 years, we find the following values: C, where C = 1 α β and E{V } = 1 e kt kt (σ 2 θ 2 ) + θ 2 =.33641835, V ar(v ) = γ2 e 2kT 2k 3 T 2 [(2e 2kT 4e kt kt 2)(σ 2 θ 2 ) + (2e 2kT kt 3e 2kT + 4e kt 1)θ 2 ] =.551649969. The convexity adjustment V ar{v } 8E{V } 3/2 is equal to.353374855. 9

If the non-adjusted strike is equal to 18.7751%, then the adjusted strike is equal to 18.7751%.353374855% = 18.73976259%. This is the fixed leg of the volatility swap for a maturity T =.91. Repeating this approach for a series of maturities up to 1 years, we obtain the result shown in see Figure 2 for S&P 6 Canada Index Volatility Swap. Figure 1 illustrates the non-adjusted and adjusted volatility for the same series of maturities (see formula (7)). Key Points Variance, volatility, covariance, and correlation swaps are useful for volatility hedging and speculation. Volatility swaps are forward contracts on future realized stock volatility. Variance swaps are similar contracts on variance, the square of the future volatility. Covariance and correlation swaps are covariance and correlation forward contracts, respectively, of the underlying two assets. Using change of time one can model and price variance, volatility, covariance, and correlation swaps for stochastic volatility Heston model. References Brockhaus, O. and Long, D. (2). Volatility swaps made simple. Risk, January: 92-96. Carr, P. and Madan, D. (1998). Towards a Theory of Volatility Trading. Volatility, Risk book publications. Cheng, R., Lawi, S., Swishchuk, A., Badescu, A., Ben Mekki, H., Gashaw, A., Hua, Y., Molyboga, M., Neocleous, T. and Petrachenko, Y. (22). Price Pseudo-Variance, Pseudo-Covariance, Pseudo-Volatility, and Pseudo- Correlation Swaps-In Analytical Closed-Forms, Proceedings of the Sixth PIMS Industrial Problems Solving Workshop, PIMS IPSW 6, University of British Columbia, Vancouver, Canada, May 27-31, 22, pp. 45-55. Editor: J. Macki, University of Alberta, June. Chernov, R., Gallant, E., Ghysels, E. and Tauchen, G. (23). Alternative models for stock price dynamics. Journal of Econometrics, 116: 225-257. Demeterfi, K., Derman, E., Kamal, M. and Zou, J. (1999). A guide to volatility and variance swaps. The Journal of Derivatives, Summer: 9-32. Elliott, R. and Swishchuk, A. (27). Pricing options and variance swaps in Markov-modulated Brownian markets. Hidden Markov Models in Finance. New York: Springer. 1

Javaheri, A., Wilmott, P. and Haug, E. (22). GARCH and volatility swaps. Wilmott Magazine, January: 17p. Kallsen, J. and Shiryaev, A. (22). Time change representation of stochastic integrals. Theory Probability and Its Applications, 46, 3: 522-528. Swishchuk, A. (211). Varinace and volatility swaps in energy markets. Journal of Energy Markets, forthcoming. Swishchuk, A. and Malenfant, K. (211). Pricing of variance swaps for Lévy-based stochastic volatility with delay. International Review of Applied Financial Issues and Economics, forthcoming. Swishchuk, A. and Li, X. (211). Variance swaps for stochastic volatility with delay and jumps. International Journal of Stochastic Analysis, 211 (211), Article ID 435145, 27 pages. Swishchuk, A. and Couch, M. (21). Volatility and variance swpas for COGARCH(1,1) model. Wilmott Magazine, 2, 5: 231-246. Swishchuk, A. and Manca, R. (21). Modeling and pricing of variance swaps for local semi-markov volatility in financnial engineering. Mathematical Models in Engineering, New York: Hindawi Publications, 21 (21), 1-17. Swishchuk, A. (29a). Pricing of Variance and Volatility Swaps with semi-markov Volatilities, Canadian Applied Mathematics Quaterly, 18, 4. Swishchuk, A. (29b). Variance swaps for local stochastic volatility with delay and jumps. Working Paper, Calgary: University of Calgary. Swishchuk, A. (27). Change of time method in mathematical finance. Canadian Applied Mathematics Quarterly, 15, 3: 299-336. Swishchuk, A. (26). Modeling and Pricing of Variance Swaps for Multi- Factor Stochastic Volatilities with Delay. Canadian Applied Mathematics Quarterly, 14, 4. Swishchuk, A. (25). Modeling and Pricing of Variance Swaps for Stochastic Volatilities with Delay. Wilmott Magazine, 19, September: 63-73. Swishchuk, A. (24). Modelling and valuing of variance and volatility swaps for financial markets with stochastic volatilites. Wilmott Magazine, 2, September: 64-72. Théoret, R., Zabré, L., and Rostan, P. (22). Pricing volatility swaps: Empirical testing with Canadian data. Working paper. Centre de Recherche en Gestion, Université du Québec á Montréal, Document 17-22, July 22, 2 pages. 11

Figure 1: Convexity Adjustment. Figure 2: S&P 6 Canada Index Volatility Swap. 12