Different aspects in correlation products pricing Pascal DELANOE, - PowerPoint PPT Presentation

Different aspects in correlation products pricing Pascal DELANOE, Structured Equity Derivatives HSBC 2nd March 2012 Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 1 / 46

First part : Local correlation calibration Outline 1 First part : Local correlation calibration References Dupire general formula Calibration results : local volatility Extension to Stochastic Volatility Calibration results : stochastic volatility Products Study 2 Second part : study of correlation swap product Purpose of the study RCS Risk analysis : Hedge with Basket and stock vanillas Results PnL decomposition RCS Risk analysis : Hedge with Var Swaps Transaction costs Other Basket Main conclusions Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 2 / 46

First part : Local correlation calibration References Presentation of the main used methodologies introducing local correlation 1 Langnau : Interesting intuitions, copula method 2 Sbai-Jourdan : no explicit local correlation => complex framework : cf. deduce stocks vols from index and not index from stocks 3 Reghai : convergence using fixed point algorithm, but slow convergence 4 Guyon/PHL : iterative approach and Dupire Formula (also available in Avellaneda et al. ) Objectives : Investigate (4) and present alternative ideas Extend to Stochastic Volatility and applications to products Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 3 / 46

First part : Local correlation calibration References Our framework = Reghai’s Local Correlation introduced through the use of an overomega approach. What is Overomega ? ρ Pricing = ( 1 − ω ) ρ Histo + ω i , j i , j First Model = Simple Local Vol Model with continuous dividends (mix of prop and cash dividends). � � dS i t = ( r t S i t − Q i t − q i t S i t ) dt + σ ( t , S i t ) S i t ) dW i t ) dW ⊥ 1 − ω ( t , I S ω ( t , I S t ( t + t ) with Q i t and q i t deterministic and : N I S � w i S i = t t i = 1 < dW i t , dW j ρ 0 = i , j dt t > < dW i t , dW ⊥ = 0 ∀ i > t Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 4 / 46

First part : Local correlation calibration Dupire general formula Local Correlation formula (general case) Second Model = simple local vol model written on the index(continuous divs) : dI t = ( r t I t − Q t − q t I t ) dt + I t σ ( t , I t ) dW t with Q t and q t deterministic. Same Basket Call prices in both models (Specific set of w i ) : � t E Q ( exp ( − r s ds )( I t − K ) + ) C ( K , t ) = 0 � t E Q ( exp ( − r s ds )( I S t − K ) + ) ∀ t , K = 0 but : � t ∂ C 1 E Q ( exp ( − r s ds )(( dI S t − r t ( I S d < I S > t 1 IS dt = t − K ) dt ) 1 IS t > K + t = K )) in basket model ∂ t 2 0 � t ∂ C 1 E Q ( exp ( − dt = r s ds )(( dI t − r t ( I t − K ) dt ) 1 It > K + d < I > t 1 It = K )) in index model ∂ t 2 0 Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 5 / 46

First part : Local correlation calibration Dupire general formula Local Correlation formula(2) 1 1 E Qt t = K ) = E Qt d < I S > t 1 IS w i ( Q i t + q i t S i � (( − t ) + r t Kdt ) 1 IS t > K + (( − ( Q t + q t I t ) + r t Kdt ) 1 It > K + d < I > t 1 It = K ) 2 2 i but : E Qt ( d < I > t | I t = K ) ∂ 2 C E Qt ( d < I > t 1 It = K ) = ∂ K 2 B ( 0 , t ) and also : E Qt E Qt 1 ∂ C ( I S ( I t 1 It > K ) = t 1 IS t > K ) = ( C − K ) B ( 0 , t ) ∂ K E Qt E Qt 1 ∂ C ( 1 It > K ) = ( 1 IS t > K ) = ( − ) B ( 0 , t ) ∂ K Condition on the forward ( K = 0): � w i Q i Q t = t i E Qt ( � i w i q i t S i t ) q t = E Qt ( I t ) Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 6 / 46

First part : Local correlation calibration Dupire general formula Local Correlation formula(3) � � �� E Qt (( � i w i q i t S i E Qt (( q t I t − r t K ) 1 It > K ) t − r t K ) 1 IS t > K ) K 2 σ ( t , K ) 2 + 2 ∂ C − ∂ K E Qt ( 1 It > K ) E Qt ( 1 IS ∂ 2 C t > K ) ∂ K 2 ω ( t , K ) = E Q t ( � t S j t ) σ ( t , S j i , j w i w j S i t σ ( t , S i t )( 1 − ρ 0 i , j ) | I S t = K ) E Q t ( � t S j t ) σ ( t , S j i , j w i w j S i t σ ( t , S i t ) ρ 0 i , j | I S t = K ) − E Q t ( � t S j t ) σ ( t , S j i , j w i w j S i t σ ( t , S i t )( 1 − ρ 0 i , j ) | I S t = K ) Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 7 / 46

First part : Local correlation calibration Dupire general formula Local Correlation formula : understand each term Particular cases : no dividends, deterministic interest rates K 2 σ ( t , K ) 2 − E Q ( � t S j t ) σ ( t , S j i , j w i w j S i t σ ( t , S i t ) ρ 0 i , j | I S t = K ) ω ( t , K ) = E Q ( � t S j t ) σ ( t , S j i , j w i w j S i t σ ( t , S i t )( 1 − ρ 0 i , j ) | I S t = K ) Dupire/Avellaneda/Piterbarg/Guyon-PHL formula. Case where constant vol and null correlation: I − 1 ω = σ 2 N σ 2 S S ( 1 − 1 σ 2 N ) Well known formula : see Bossu for example. Idea : (Implied Index Covariance - Minimum Covariance)/(Maximum Covariance - Minimum Covariance) Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 8 / 46

First part : Local correlation calibration Dupire general formula Local Correlation formula : understand each term(2) E Qt (( � E Qt (( q t I t − r t K ) 1 It > K ) i w i q i t S i t − r t K ) 1 IS t > K ) − E Qt ( 1 It > K ) E Qt ( 1 IS t > K ) Stochastic rate term + Dividend term. Deterministic interest rates : first term vanishes since r t in factor and E Qt ( 1 It > K ) = E Qt ( 1 IS B ( 0 , t ) ( − ∂ C 1 t > K ) = ∂ K ) Residual term : cf. no arbitrage condition in case of discrete dividends : E Qt (( I t − K ) + ) − E Qt E Qt (( I t − − K ) + ) ≃ (( I t − I t − ) 1 It > K ) E Qt t − K ) + ) − E Qt E Qt (( I S (( I S t − − K ) + ) (( I S t − I S ≃ t − ) 1 IS t > K ) but : I t − I t − = − ( Q t + q t I t − ) I S t − I S − w i ( Q i t + q t S i � t − = t − ) i leads to (first order in dividend level) : E Qt ( q t I t 1 It > K ) = E Qt � w i q i t S i ( t 1 IS t > K ) i Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 9 / 46

First part : Local correlation calibration Dupire general formula Local Correlation formula : understand each term(3) If discrete dividends : impossible to achieve for each K if q t constant(except in particular cases : null volatility or q t = q i t ∀ i ) => two models are generally inconsistent. => Need to use continuous div model One more derivation in K + Same ∂ 2 C ∂ K 2 gives : � E Q t ( q t I t | I t = K ) = E Q t ( w i q i t S i t | I S t = K ) i cf. Markovian projection : sufficient but not necessary condition Other possible conditions : E Q ( � i wi qi t Si  t ) q t =   E Q ( It )       E Q (( � i wi qi t Si   t ) 1 IS t > K )  2 ( C − K ∂ C  ∂ K ) t Sj t ) σ ( t , Sj  K 2 σ ( t , K ) 2 −  − E Q ( � i , j wi wj Si t σ ( t , Si t ) ρ 0 i , j | IS      qt − t = K )  ∂ 2 C  E Q (( � i wi Si   t ) 1 IS t > K )     ∂ K 2   ω ( t , K ) =   t Sj t ) σ ( t , Sj E Q ( � i , j wi wj Si t σ ( t , Si t )( 1 − ρ 0 i , j ) | IS  t = K ) ω helps recover from the generated error. Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 10 / 46

First part : Local correlation calibration Dupire general formula Implement the convergence algorithm Ideas proposed in Guyon/PHL. Formula available for ω ( t , K ) Fixed point algorithm (cf. Reghai also but mixes implied/local) Hybrid-LSV model : "Kernelize" the drift term introducing Malliavin calculus K 2 σ ( t , K ) 2 − E Q ( � i , j w i w j S i , ( n ) S j , ( n ) σ ( t , S i , ( n ) ) σ ( t , S j , ( n ) i , j | I S , ( n ) ) ρ 0 = K ) t t t t t ω ( n + 1 ) ( t , K ) = i , j w i w j S i , ( n ) S j , ( n ) σ ( t , S i , ( n ) ) σ ( t , S j , ( n ) i , j ) | I S , ( n ) )( 1 − ρ 0 E Q ( � = K ) t t t t t Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 11 / 46

First part : Local correlation calibration Dupire general formula Implement the convergence algorithm Alternative ideas: "Kernelize" the drift term by introducing a division by E Q ( 1 I S t > K ) or E Q ( I S t 1 I S t > K ) : "natural" kernel for more stability replace non parametric regression by a parametric regression for the dirac term ex : Case of hybrid model.Use the following formula : � � � t ∂ C E Q ( exp ( − 0 ( r s − r 0 s ) ds )( r t − r 0 t ) 1 S t > K ) σ Dup ( t , K ) 2 + 2 σ ( t , K ) 2 ∂ K = � t K ∂ 2 C 0 ( r s − r 0 E Q ( exp ( − s ) ds ) 1 S t > K ) ∂ K 2 Evaluate numerator and denominator using same paths + Use a fast exponential => no need for Malliavin calculus (multiple diffusions) and numeraire changes. Non parametric regression : complexity in O ( Nln ( N )) operations if sorted performances. Structured Equity Research (HSBC) Different aspects of Correlation 2nd March 2012 12 / 46