Coupling Index and Stocks Mohamed Sbai Joint work with Benjamin - PowerPoint PPT Presentation

Coupling Index and Stocks Mohamed Sbai Joint work with Benjamin Jourdain Universit´ e Paris-Est, CERMICS (now Soc. Gen.) Modeling and managing financial risks 10 th to 13 th January, 2011 Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 1 / 37

Outline Introduction 1 Model Specification 2 Calibration 3 Numerical experiments 4 Conclusion 5 Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 2 / 37

Handling both an Index and its composing stocks is still a challenging task. Standard approach : a model for the stocks (with smile) + a correlation matrix. Then, reconstruct the index local/implied vol. ( Avellaneda Boyer-Olson, Busca, Friz [2002] , Lee, Wang, Karim [2003] , . . .) ◮ Difficulty to retrieve the the index smile ( steeper than stock smile) by historical estimation of the correlation matrix . ◮ Adjusting the correlation matrix is tedious (keep it positive definite ? implied correlation matrix ?). Our objective : a new modeling approach allowing for a good fit of both Index and stocks. Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 3 / 37

Another viewpoint : a factor model (the index represents the market and influences the stocks). (In discrete time) Cizeau, Potters and Bouchaud [2001] show that it is possible to capture the essential features of stocks cross-correlations by a simple non-Gaussian one factor model, specially in extreme market S j ( t ) conditions : the daily return r j ( t ) = S j ( t − 1 ) − 1 of stock j is given by r j ( t ) = β j r m ( t ) + ǫ j ( t ) where r m ( t ) is the market daily return and ǫ j ( t ) is a Student random variable. The regression coefficients β j are narrowly distributed around 1. Our model can be seen as an extension in continuous time. Calibration to both index and stocks is feasible and leads to a new correlation structure. Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 4 / 37

Outline Introduction 1 Model Specification 2 Calibration 3 Numerical experiments 4 Conclusion 5 Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 5 / 37

Consider an Index composed of M stocks ( S j , M ) 1 ≤ j ≤ M : M � w j S j , M I M = t t j = 1 where the w j are positive weights assumed to be constant. In a risk-neutral world, we specify the following dynamics for the stocks : ∀ j ∈ { 1 , . . . , M } , dS j , M R M �→ ( s j σ ( t , � M t ) dB t + η j ( t , S j , M ) dW j t = ( r − δ j ) dt + β j σ ( t , I M (1) t t S j , M t r is the short interest rate. δ j ∈ [ 0 , ∞ [ incorporates both repo cost and dividend yield of the stock j . β j is the usual beta coefficient of the stock j . ( B t ) t ∈ [ 0 , T ] , ( W 1 t ) t ∈ [ 0 , T ] , . . . , ( W M t ) t ∈ [ 0 , T ] are independent BMs. We assume that the functions ( s 1 , . . . , s M ) ∈ j = 1 w j s j ) , s j η j ( t , s j )) 1 ≤ j ≤ M are Lipschitz continuous and have linear growth unif. in t . ⇒ Existence and trajectorial uniqueness for the SDE (1). Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 6 / 37

∀ j ∈ { 1 , . . . , M } , dS j , M t ) dB t + η j ( t , S j , M ) dW j t = ( r − δ j ) dt + β j σ ( t , I M t t S j , M t M -dimensional SDE driven by M + 1 sources of noise B , W 1 , . . . , W M : incomplete market. The dynamics of a given stock depends on all the other stocks composing the index through the volatility term σ ( t , I M t ) . The cross-correlations between stocks are not constant but stochastic : β i β j σ 2 ( t , I M t ) ρ ij ( t ) = � � i ( t , S i , M j ( t , S j , M β 2 i σ 2 ( t , I M t ) + η 2 β 2 j σ 2 ( t , I M t ) + η 2 ) ) t t Note that they depend not only on the stocks but also on the index. Nice feature : when the systemic volatility σ ( t , I M t ) raises, so does the correlation ρ ij ( t ) . Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 7 / 37

= � M j = 1 w j S j , M The index I M satisfies the following SDE t t �� M � j = 1 δ j w j S j , M dI M = rI M t dt − dt t t   � M � M  β j w j S j , M  w j S j , M η j ( t , S j , M ) dW j σ ( t , I M + t ) dB t + t t t t j = 1 j = 1 � �� � � �� � Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 8 / 37

= � M j = 1 w j S j , M The index I M satisfies the following SDE t t �� M � j = 1 δ j w j S j , M dI M = rI M t dt − dt t t   � M � M  β j w j S j , M  w j S j , M η j ( t , S j , M ) dW j σ ( t , I M + t ) dB t + t t t t j = 1 j = 1 � �� � � �� � ≃ I M t Our model is inline with Cizeau, Potters and Bouchaud [2001] : The beta coefficients are narrowly distributed around 1 ⇒ � M j = 1 β j w j S j , M ≃ I M t . t Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 8 / 37

= � M j = 1 w j S j , M The index I M satisfies the following SDE t t �� M � j = 1 δ j w j S j , M dI M = rI M t dt − dt t t   � M � M  β j w j S j , M  w j S j , M η j ( t , S j , M ) dW j σ ( t , I M + t ) dB t + t t t t j = 1 j = 1 � �� � � �� � ≃ 0 ≃ I M t Our model is inline with Cizeau, Potters and Bouchaud [2001] : The beta coefficients are narrowly distributed around 1 ⇒ � M j = 1 β j w j S j , M ≃ I M t . t For large M , we will show that the term � M j = 1 w j S j , M η j ( t , S j t ) dW j t can be t neglected. Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 8 / 37

= � M j = 1 w j S j , M The index I M satisfies the following SDE t t �� M � j = 1 δ j w j S j , M dI M = rI M t dt − dt t t   � M � M  β j w j S j , M  w j S j , M η j ( t , S j , M ) dW j σ ( t , I M + t ) dB t + t t t t j = 1 j = 1 � �� � � �� � ≃ 0 ≃ I M t Our model is inline with Cizeau, Potters and Bouchaud [2001] : The beta coefficients are narrowly distributed around 1 ⇒ � M j = 1 β j w j S j , M ≃ I M t . t For large M , we will show that the term � M j = 1 w j S j , M η j ( t , S j t ) dW j t can be t neglected. ⇒ r j = β j r I M + η j ∆ W j + drift where r j (resp. r I M ) is the log-return of the stock j (resp. the index). The return of a stock is decomposed into a systemic part driven by the index, which represents the market, and a residual part. Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 8 / 37

N ∗ . If A simplified Model We look at the asymptotics for a large number M of underlying stocks. Consider the limit candidate ( I t ) t ∈ [ 0 , T ] solution of dI t I 0 = I M = ( r − δ ) dt + βσ ( t , I t ) dB t ; (2) 0 I t E Theorem 1 Let p ∈ ( H 1) ∃ K b s.t. ∀ ( t , s ) , | σ ( t , s ) | + | η j ( t , s ) | ≤ K b ∃ K σ s.t. ∀ ( t , s 1 , s 2 ) , | s 1 σ ( t , s 1 ) − s 2 σ ( t , s 2 ) | ≤ K σ | s 1 − s 2 | . then, there exists a constant C T depending on β, δ, K b , K σ but not on M such that � � ��� M � 2 p C T max 1 ≤ j ≤ M | S j , M | I M t − I t | 2 p 0 | 2 p ≤ j = 1 w j | β j − β | sup 0 ≤ t ≤ T � 2 p � �� M � p �� M j = 1 w 2 + + j = 1 w j | δ j − δ | j Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 9 / 37

Replacing I M by I in the dynamics of the j -th stock, one obtains dS j = ( r − δ j ) dt + β j σ ( t , I t ) dB t + η j ( t , S j t ) dW j t t S j t E Theorem 2 Under the assumptions of Theorem 1 and if ( H 2) ∃ K η s.t. ∀ j ≤ M , ∀ ( t , s 1 , s 2 ) , | s 1 η j ( t , s 1 ) − s 2 η j ( t , s 2 ) | ≤ K η | s 1 − s 2 | ∃ K Lip s.t. ∀ ( t , s 1 , s 2 ) , | σ ( t , s 1 ) − σ ( t , s 2 ) | ≤ K Lip | s 1 − s 2 | C j then, ∀ j ∈ { 1 , . . . , M } , there exists a constant � T depending on β, δ, β j , δ j , K b , K σ , K η , K Lip and max 1 ≤ j ≤ M S j , M but not on M s.t. 0 � � ��� M � 2 p | S j , M − S j C j � t | 2 p ≤ j = 1 w j | β j − β | sup t T 0 ≤ t ≤ T � 2 p � �� M � p �� M j = 1 w 2 + + j = 1 w j | δ j − δ | j Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 10 / 37

Three different indexes = � M j = 1 w j S j , M where ( S j , M ) 1 ≤ j ≤ M solves (1). Original index : I M t t Simplified limit index : I t solving dI t I t = ( r − δ ) dt + βσ ( t , I t ) dB t Beware, in general I t � = � M j = 1 w j S j t where E dS j t = ( r − δ j ) dt + β j σ ( t , I t ) dB t + η j ( t , S j t ) dW j t t S j = � M Reconstructed index : I M def j = 1 w j S j t t Theorem 3 Under the assumptions of Theorem 2, � � �� M � 2 p t − I M C j ≤ max 1 ≤ j ≤ M � | I M t | 2 p sup j = 1 w j T 0 ≤ t ≤ T ��� M � 2 p � � 2 p �� M � p �� M j = 1 w 2 × j = 1 w j | β j − β | + + j = 1 w j | δ j − δ | j Mohamed Sbai (UPE-CERMICS) Coupling Index and Stocks Paris, 10-13 Jan. 2011 11 / 37