Multivariate Option Pricing Using Copulae Carole Bernard (University of Waterloo) & Claudia Czado (Technische Universit¨ at M¨ unchen) Bologna, September 2010. Carole Bernard Multivariate Option Pricing Using Copulae 1
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Introduction Multivariate Options ∙ Financial products linked to more than one underlying ∙ Most are over-the-counter ∙ Some are listed on the New York Stock Exchange. Carole Bernard Multivariate Option Pricing Using Copulae 2
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Introduction Multivariate Options Pricing ∙ Multivariate Black and Scholes model. ∙ Stochastic correlation model (Galichon (2006), Langnau (2009)). ∙ Non-parametric estimation of the marginal risk neutral densities and of the risk neutral copula (Rosenberg (2000), Cherubini and Luciano (2002)). ∙ Parametric approach of dynamic copula modelling with GARCH(1,1) processes (Van den Goorbergh, Genest and Werker (2005)). Carole Bernard Multivariate Option Pricing Using Copulae 3
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Underlying Indices Modeling ▶ Daily returns ∙ S i ( t ) : closing price of index i for the trading day t ∙ r i , t +1 = log ( S i ( t + 1) / S i ( t )) ▶ GARCH(1,1) ⎧ r i , t +1 = 휇 i + 휂 i , t +1 , ⎨ 휎 2 i , t +1 = w i + 훽 i 휎 2 i , t + 훼 i ( r i , t +1 − 휇 i ) 2 , ∙ ⎩ 휂 i , t +1 ∣ℱ t ∼ P N ( 0 , 휎 2 i , t ) where w i > 0 , 훽 i > 0 and 훼 i > 0 ∙ Standardized innovations for 3 indices (for example) ( 휂 1 , s ) , 휂 2 , s , 휂 3 , s ( Z 1 , s , Z 2 , s , Z 3 , s ) s ⩽ t := 휎 1 , s 휎 2 , s 휎 3 , s Carole Bernard Multivariate Option Pricing Using Copulae 4
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Risk-Neutral Dynamics for each Index Following Duan (1995), the log-returns under the risk neutral probability measure Q are given as follows ⎧ r i , t +1 = r f − 1 2 휎 2 i , t + 휂 ∗ i , t +1 , ⎨ 휎 2 i , t +1 = w i + 훽 i 휎 2 i , t + 훼 i ( r i , t +1 − 휇 i ) 2 , ⎩ 휂 ∗ i , t +1 ∣ℱ t ∼ Q N ( 0 , 휎 2 i , t ) where r f is the (constant) daily risk-free rate on the market. Carole Bernard Multivariate Option Pricing Using Copulae 5
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Pricing formula Initial Price = e − r f T E Q [ g ( S 1 ( T ) , S 2 ( T ) , S 3 ( T ))] , where ∙ T denotes the number of days between the issuance date and the maturity of the option. ∙ r f is the risk-free rate. ⇒ We need to understand the dependence under Q between S 1 , S 2 and S 3 . Carole Bernard Multivariate Option Pricing Using Copulae 6
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Pricing formula Initial Price = e − r f T E Q [ g ( S 1 ( T ) , S 2 ( T ) , S 3 ( T ))] , where ∙ T denotes the number of days between the issuance date and the maturity of the option. ∙ r f is the risk-free rate. ⇒ We need to understand the dependence under Q between S 1 , S 2 and S 3 . Carole Bernard Multivariate Option Pricing Using Copulae 6
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Dependence Structure ▶ Need for a multivariate distribution with more than 2 dimensions: there are many bivariate copulae but a limited number of multivariate copulae ▶ Use of pair-copula construction (Aas, Czado, Frigessi and Bakken (2009) and Czado (2010)) ▶ This method involves only bivariate copulae ▶ Example with 3 dimensions Carole Bernard Multivariate Option Pricing Using Copulae 7
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Pair-copula Construction ∙ Joint density f ( x 1 , x 2 , x 3 ) ∙ A possible decomposition by conditioning f ( x 1 , x 2 , x 3 ) = f ( x 1 ∣ x 2 , x 3 ) × f 2 ∣ 3 ( x 2 ∣ x 3 ) × f 3 ( x 3 ) . ∙ By Sklar’s theorem f ( x 2 , x 3 ) = c 23 ( F 2 ( x 2 ) , F 3 ( x 3 )) f 2 ( x 2 ) f 3 ( x 3 ) therefore f 2 ∣ 3 ( x 2 ∣ x 3 ) = c 23 ( F 2 ( x 2 ) , F 3 ( x 3 )) f 2 ( x 2 ) . ∙ Similarly we have f 1 ∣ 2 ( x 1 ∣ x 2 ) = c 12 ( F 1 ( x 1 ) , F 2 ( x 2 )) f 1 ( x 1 ) . Carole Bernard Multivariate Option Pricing Using Copulae 8
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Pair-copula Construction By Sklar’s theorem for the conditional bivariate density f ( x 1 , x 3 ∣ x 2 ) = c 13 ∣ 2 ( F 1 ∣ 2 ( x 1 ∣ x 2 ) , F 3 ∣ 2 ( x 3 ∣ x 2 )) f 1 ∣ 2 ( x 1 ∣ x 2 ) f 3 ∣ 2 ( x 3 ∣ x 2 ) and therefore f ( x 1 ∣ x 2 , x 3 ) = c 13 ∣ 2 ( F 1 ∣ 2 ( x 1 ∣ x 2 ) , F 3 ∣ 2 ( x 3 ∣ x 2 )) f 1 ∣ 2 ( x 1 ∣ x 2 ) . It follows that f ( x 1 , x 2 , x 3 ) = c 12 ( F 1 ( x 1 ) , F 2 ( x 2 )) c 23 ( F 2 ( x 2 ) , F 3 ( x 3 )) × c 13 ∣ 2 ( F 1 ∣ 2 ( x 1 ∣ x 2 ) , F 3 ∣ 2 ( x 3 ∣ x 2 )) f 1 ( x 1 ) f 2 ( x 2 ) f 3 ( x 3 ) . The corresponding copula density is therefore given by c 123 ( u 1 , u 2 , u 3 ) = c 12 ( u 1 , u 2 ) c 23 ( u 2 , u 3 ) . c 13 ∣ 2 ( F 1 ∣ 2 ( u 1 ∣ u 2 ) , F 3 ∣ 2 ( u 3 ∣ u 2 )) It is called a D-vine in three dimensions and involves only bivariate copulae. Carole Bernard Multivariate Option Pricing Using Copulae 9
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Example IIL : “Capital Protected Notes Based on the Value of a Basket of Three Indices”, issued by Morgan Stanley. The notes IIL are linked to ∙ S 1 : the Dow Jones EURO STOXX 50 SM Index, ∙ S 2 : the S&P 500 Index, ∙ S 3 : the Nikkei 225 Index Issue date: July 31st, 2006. Maturity date: July 20, 2010. Initial price $10. Their final payoff is given by ( m 1 S 1 ( T ) + m 2 S 2 ( T ) + m 3 S 3 ( T ) − 10 ) $10 + $10 max , 0 10 10 where m i = 3 S i (0) such that m 1 S 1 (0) + m 2 S 2 (0) + m 3 S 3 (0) = 10 and the % weighting in the basket is 33.33% for each index. Carole Bernard Multivariate Option Pricing Using Copulae 10
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions GARCH(1,1) parameters The table next slide: Estimated parameters of GARCH(1,1) ∙ 3 indices: ∙ the STOXX50 , ∙ the S&P500 , ∙ the NIK225 . ∙ ¯ 휎 i , t denotes the average of the daily volatilities over the period under study (full period is July 2006 to November 2009). The table highlights different regimes of the economy (time varying parameters for the GARCH(1,1) model) and changes in volatility. Carole Bernard Multivariate Option Pricing Using Copulae 11
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Full sample period 1 period 2 period 3 휇 1 ˆ 0.000414 0.000664 -0.000513 0.000593 휔 1 ˆ 1.85e-06 2.72e-06 8.95e-06 5.42e-06 훼 1 ˆ 0.0932 0.0338 0.0513 0.119 ˆ 훽 1 0.900 0.903 0.899 0.876 휇 2 ˆ 0.000350 0.000907 -0.000494 0.000743 휔 2 ˆ 3.76e-06 9.88e-06 1.027e-05 7.57e-06 훼 2 ˆ 0.1343 0.1598 0.1482 0.1062 ˆ 훽 2 0.8575 0.7275 0.8063 0.8854 휇 3 ˆ 0.000107 0.000525 -0.000594 0.0000213 휔 3 ˆ 4.63e-06 4.75e-06 6.09e-06 1.83e-05 훼 3 ˆ 0.127 0.0643 0.142 0.197 ˆ 훽 3 0.863 0.896 0.851 0.782 √ ¯ 휎 1 , t 250 24.8% 14.5% 21.2% 38.7% √ ¯ 휎 2 , t 250 23.2% 10.3% 20.6% 38.8% √ ¯ 휎 3 , t 250 27.4% 17.5% 25.8% 39.1% Carole Bernard Multivariate Option Pricing Using Copulae 12
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Methodology ∙ Example with 3 indices Identify the 2 couples that have the most dependence. 1 Identify the family of copula using empirical contour plots and 2 Cramer von Mises Goodness of Fit test. Generate the conditional data and identify the copula of the 3 conditional data. ∙ Illustration with the contract IIL Note that the Pair-Copula Construction depends on the order of the indices (item 1 is arbitrary). Carole Bernard Multivariate Option Pricing Using Copulae 13
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions S 1 − S 2 S 1 − S 3 S 2 − S 3 Full 0.404 0.202 0.079 Period 1 0.314 0.197 0.104 Period 2 0.384 0.239 0.075 Period 3 0.495 0.181 0.062 Overall dependence measured by the Kendall’s Tau for the full sample and then for each of the 3 periods Carole Bernard Multivariate Option Pricing Using Copulae 14
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Contour plots ∙ We draw the contour plots for S 1 − S 2 and S 1 − S 3 ∙ The empirical contours are compared with theoretical contours. ∙ All parameter estimates are obtained by maximum likelihood estimation. We only present the 1st and 2nd period. Carole Bernard Multivariate Option Pricing Using Copulae 15
Setting Dependence Pricing Example Risk Neutral Parameters Conclusions Conditional copula We compute the copula C 23 ∣ 1 between the conditional distributions of S 2 given S 1 and S 3 given S 1 as follows F 2 ∣ 1 ,휃 12 ( u 2 s ∣ u 1 s , ˆ 휃 P u 2 ∣ 1 s = 12 ) F 3 ∣ 1 ,휃 13 ( u 3 s ∣ u 1 s , ˆ 휃 P u 3 ∣ 1 s = 13 ) where the conditional distribution F 2 ∣ 1 ,휃 P 12 is obtained by ∂ F ( u 2 ∣ u 1 , ˆ C 12 ( u 2 ∣ u 1 , ˆ 12 ) =: h ( u 2 , u 1 , ˆ 휃 P 휃 P 휃 P 12 ) = 12 ) ∂ u 1 and F 3 ∣ 1 ,휃 P 13 similarly. (For the second period, we assume a Clayton copula between S 1 and S 2 and a Gaussian copula between S 1 and S 3 .) Carole Bernard Multivariate Option Pricing Using Copulae 18
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