Volatility Investing with Variance Swaps

Transcription

1 Volatility Investing with Variance Swaps Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and Economics Humboldt-Universität zu Berlin

2 Motivation 1-1 Why investors may wish to trade volatility? DAX 1 month realized volatility DAX Level Jul05 Nov06 Apr08 Aug09 Figure 1: DAX level vs. DAX 1M realized volatility ( )

3 Motivation 1-2 Volatility is an asset "fear indices": VIX, VDAX, VSTOXX 3G volatility derivatives: gamma swaps, corridor variance swaps, conditional variance swaps volatilitiy trading strategies: trading dispersion

4 Motivation 1-3 Research questions How to trade volatility? How to hedge (replicate) volatility? How good can we perform? How does dispersion trading work?

5 Outline 1. Motivation 2. Definition 3. Trading volatility with options 4. Replication and hedging 5. 3G volatility derivatives 6. Dispersion trading strategy 7. Conclusions

6 Definition 2-1 Variance swap Figure 2: Cash flow of a variance swap at expiry

7 Definition 2-2 Variance swap forward contract at maturity pays the difference between realized variance σ 2 R and strike K 2 var (multiplied by notional N var ) (σ 2 R K 2 var ) N var (1) σ R = 252 T T t=1 ( log S t S t 1 ) (2)

8 Definition 2-3 Example 3-month variance swap long Long position in 3-month variance swap. Trade size is 2500 variance notional (represents a payoff of 2500 per point difference between realized and implied variance). If K var is 20% (Kvar 2 = 400) and the realized subsequent variance is (15%) 2 (quoted as σr 2 = 225), the long position makes loss = 2500 ( )

9 Replication and hedging 3-1 Replication and hedging - intuitive approach European option with Black-Scholes (BS) price V BS (S, K, σ τ) variance vega: where V BS σ 2 = S 2σ ϕ(y) (3) τ y = log(s/k) + σ2 τ/2 σ τ ϕ - pdf of a standard normal rv.

10 Replication and hedging 3-2 Variance vega of options with different K Variance vega K= Stock price Figure 3: Dependence of variance on S for vanilla options with K = [50, 200], σ = 0.2, τ = 1

11 Replication and hedging 3-3 Equally-weighted option portfolio Variance vega Stock price Figure 4: Variance vega of option portfolio (red line) with options weighted equally

12 Replication and hedging 3-4 1/K-weighted option portfolio Variance vega Stock price Figure 5: Variance vega of option portfolio (red line) with options weighted proportional to 1/K

13 Replication and hedging 3-5 1/K 2 -weighted option portfolio Independent of stock price changes Variance vega Stock price Figure 6: Variance vega of option portfolio (red line) with options weighted proportional to 1/K 2

14 Replication and hedging 3-6 Replication and hedging - more rigorous approach existence of futures market with delivery dates T T stock price S t (underlying) dynamics: ds t S t = µdt + σdw t (4) all strikes are available (market is complete) continuous trading risk free interest rate r = 0, w.l.o.g.

15 Replication and hedging 3-7 Log contract Define derivatives: and observe f (S 0 ) = 0 f (S t ) = 2 T f (S t ) = 2 T { log S 0 + S } t 1 S t S 0 ( 1 S 0 1 S t f (S t ) = 2 TF 2 t ) (5) (6) (7)

16 Replication and hedging 3-8 Itô s lemma T f (S t ) = f (S 0 ) + f (S t )ds t + 1 T St 2 f (S t )σt 2 dt (8) Substituting (6), (7): 1 T σt 2 dt = 2 ( log S 0 + S ) T 1 (9) T 0 T S T S 0 2 T T 0 ( 1 S 0 1 S t ) ds t

17 Replication and hedging 3-9 Equation (9) gives the value of σ 2 R as a sum of: 2 T ( 1 1 ) ds t T 0 S 0 S t (continuously rebalanced position in underlying stock) and f (S T ) = 2 T ( log S 0 + S ) T 1 S T S 0 (10) (log contract, static position).

18 Replication and hedging 3-10 Carr and Madan (2002) represent any twice differentiable payoff function f (S T ): f (S T ) = f (k) + f (k) { (S T k) + (k S T ) +} (11) k + f (K)(K S T ) + dk 0 + where k is an arbitrary number. k f (K)(S T K) + dk

19 Replication and hedging 3-11 Applying (11) to (10) with k = S 0 gives ( ) S0 log + S T 1 = (12) S T S 0 = S0 0 K 2 (K S T ) + dk + S 0 K 2 (S T K) + dk a portfolio of OTM puts and calls weighted by K 2.

20 Replication and hedging 3-12 What are the costs of this strategy? The strike K 2 var of a variance swap is calculated via the risk-neutral expectation: Kvar 2 = 2 S0 T ert K 2 P 0 (K)dK + 2 T ert 0 S 0 K 2 C 0 (K)dK (13) where P 0 (C 0 ) - value of a put (call) option at t = 0. Problem: vanilla options with a complete strike range (from 0 to ) are not traded. How to replicate a fair future realized variance in reality?

21 Replication and hedging 3-13 Discrete approximation Demeterfi et al. (1998) approximate payoff (10) via piecewise linear approximation. Example: put option with strike K 0 and 2nd closest strike K 1p w(k 0 ) = f (K 1p) f (K 0 ) K 0 K 1p (14) The second segment - combination of puts with strikes K 0 and K 1p : w(k 1p ) = f (K 2p) f (K 1p ) K 1p K 2p w(k 0 ) (15) where w(k) amount of option with strike K in replicating portfolio (the slope of a linear segment at point K, figure 7).

22 Replication and hedging 3-14 Discrete approximation Figure 7: Discrete approximation of a log payoff (10)

23 Replication and hedging 3-15 Simulated payoff of 3M DAX variance swap DAX 3 month variance 1.5 x DAX Level Figure 8: Strike of 3M variance swap, realized 3M variance, payoff of 3M variance swap long,price of underlying asset

24 Replication and hedging M DAX variance swap payoff statistics Min. Max. Mean Median Stdd. Skewn. Kurt. Min. payoff Max. payoff Mean payoff Volatility of payoff Table 1: Summary statistics of 3M variance swap payoff simulation, duration of the strategy - 10 years (2500 days),number of paths , GBM with µ = 0.17, σ = 0.18

25 3G volatility derivatives 4-1 Generalized variance swaps Modify the floating leg of a standard variance swap (1) with a weight process w t to obtain: σ 2 R = 252 T T t=1 w t ( log S t S t 1 ) 2 (16)

26 3G volatility derivatives 4-2 Corridor and conditional variance swaps w t = w(s t ) = I St C defines a corridor variance swap with corridor C. for C = [A, B] the payoff function is defined by f (S T ) = 2 ( log S 0 + S ) T 1 I T S T S ST [A,B] (17) 0 where I is the indicator function. C = [0, B] gives downward variance swap C = [A, ] gives upward variance swap

27 3G volatility derivatives 4-3 Simulated payoff of 3M DAX corridor swap with time-adjusting corridor DAX 3 month variance 15 x DAX Level Figure 9: Strike of 3M corridor swap, realized 3M conditional variance, payoff of 3M corridor swap long,price of underlying asset

28 3G volatility derivatives 4-4 3M DAX corridor swap payoff statistics Min. Max. Mean Median Stdd. Skewn. Kurt. Min. payoff Max. payoff Mean payoff Volatility of payoff Table 2: Summary statistics of 3M corridor swap payoff simulation, duration of the strategy - 10 years (2500 days),number of paths , GBM with µ = 0.17, σ = 0.18

29 3G volatility derivatives 4-5 Gamma swaps w t = w(s t ) = S t /S 0 defines a price-weighted variance swap or gamma swap with realised variance paid at expiry: σ gamma = 252 T ( S t log S ) 2 t 100 (18) T S 0 S t 1 t=1 The payoff function: f (S T ) = 2 T ( ST log S T S ) T + 1 S 0 S 0 S 0 (19)

30 3G volatility derivatives 4-6 Simulated payoff of 3M DAX gamma swap DAX 3 month variance 1.5 x DAX Level Figure 10: Strike of 3M gamma swap, realized 3M gamma-weighted variance, payoff of 3M gamma swap long, price of underlying asset

31 3G volatility derivatives 4-7 Gamma swap vs variance swap DAX 3 month variance x DAX Level Figure 11: Strike of 3M gamma swap, Strike of 3M variance swap, price of underlying asset

32 3G volatility derivatives 4-8 3M DAX gamma swap payoff statistics Min. Max. Mean Median Stdd. Skewn. Kurt. Min. payoff Max. payoff Mean payoff Volatility of payoff Table 3: Summary statistics of 3M gamma swap payoff simulation, duration of the strategy - 10 years (2500 days),number of paths , GBM with µ = 0.17, σ = 0.18

33 Dispersion trading strategy 5-1 Basket volatility replace then ρ = ( dispersion ). σ 2 Basket = N i=1 w 2 i σ 2 i + 2 N N i=1 j=i+1 w i w j σ i σ j ρ ij 1 ρ 12 ρ 1N 1 ρ ρ ρ 21 1 ρ 2N with ρ 1 ρ......, ρ N1 ρ N2 1 ρ ρ 1 σ2 Basket N i=1 w i 2σ2 i 2 N N i=1 j=i+1 w is the basket correlation iw j σ i σ j

34 Dispersion trading strategy 5-2 Dispersion Strategy ρ = σ2 Basket N i=1 w 2 i σ2 i 2 N i=1 N j=i+1 w iw j σ i σ j Long: Variance of basket (index) Short:Variance of basket constituents Long: Dispersion How to implement?

35 Dispersion trading strategy 5-3 Dispersion Strategy For a basket of i = 1,..., N stocks payoff of direct dispersion strategy is sum of: and of short position in where (σ 2 R,i K 2 var,i) N i (K 2 var,index σ2 R,index ) N index N i = N index w i notional amount of the i-th stock.

36 Dispersion trading strategy 5-4 Dispersion Strategy Overall payoff: ( n ) N index w i σr,i 2 σ2 R,Index ResidualStrike (20) i=1 ( n ) ResidualStrike = N index w i Kvar,i 2 Kvar,Index 2 i=1

37 Dispersion trading strategy 5-5 Simulated payoff of 3M DAX dispersion strategy DAX 3 month dispersion Figure 12: 3M strike dispersion, 3M realized dispersion, 3M direct dispersion strategy (dispersion long)

38 Dispersion trading strategy 5-6 3M DAX dispersion strategy statistics Min. Max. Mean Median Stdd. Skewn. Kurt. Min. payoff Max. payoff Mean payoff Volatility of payoff Table 4: Summary statistics of 3M dispersion strategy simulation, duration of the strategy - 10 years (2500 days),number of paths

39 Dispersion trading strategy 5-7 Conclusions Volatility can be traded as an asset Future realized volatility can be replicated with option portfolios With linear interpolation replication performs well The success of the volatility dispersion strategy lies in determining: Direction of the strategy (GARCH volatility forecasts) Constituents for the offsetting variance basket (PCA, DSFM) Proper weights of the constituents (vega-flat strategy, gamma-flat strategy, theta-flat strategy)

40 Volatility Investing with Variance Swaps Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and Economics Humboldt-Universität zu Berlin

41 Bibliography 6-1 Bibliography Bossu, S., Strasser, E. and Guichard, R. Just what you need to know about Variance Swaps Equity Derivatives Investor Marketing, Quantitative Research and Development, JPMorgan - London, (May 2005) Canina, L. and Figlewski, S. The Informational Content of Implied Volatility The Review of Financial Studies, 6 (3), (1993) Carr, P. and Lee, R. Realized Volatility and Variance: Options via Swaps RISK, 20 (5), (2007)

42 Bibliography 6-2 Bibliography Carr, P. and Lee, R. Robust Replication of Volatility Derivatives PRMIA award for Best Paper in Derivatives, MFA 2008 Annual Meeting, (14 April 2008) Carr, P. and Wu, L. A Tale of Two Indices The Journal of Derivatives, (Spring, 2006) Carr, P. and Madan D. Towards a theory of volatility trading Volatility, pages , (2002)

43 Bibliography 6-3 Bibliography Chriss, N. and Moroko, W. Market Risk for Volatility and Variance Swaps RISK, (July 1999) Demeterfi, K., Derman, E., Kamal, M. and Zou, J. More Than You Ever Wanted To Know About Volatility Swaps Goldman Sachs Quantitative Strategies Research Notes, (1999) Franke, J., Härdle, W. and Hafner, C. M. Statistics of Financial Markets: An Introduction (Second ed.) Springer Berlin Heidelberg (2008)

44 Bibliography 6-4 Bibliography Hull, J. Options, Futures, and Other Derivatives (7th revised ed.) Prentice Hall International (2008) Neil, C. and Morokoff, W. Realised volatility and variance: options via swaps RISK, (May, 2007) Sulima, C. L. Volatility and Variance Swaps Capital Market News, Federal Reserve Bank of Chicago, (March 2001)