Monte Carlo simulation for a doubly nonlinear problem in finance Lokman Abbas-Turki First part from a joint work with M. A. Mikou Last part from a joint work with S. Graillat UPMC, LPMA 10 June 2015 Lokman (UPMC, LPMA) Journées inaugurales 1 / 22
Plan Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Implementation issues on GPUs The difference with the references Complexity vs. accuracy Some numerical results Lokman (UPMC, LPMA) Journées inaugurales 2 / 22
Introduction Plan Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Implementation issues on GPUs The difference with the references Complexity vs. accuracy Some numerical results Lokman (UPMC, LPMA) Journées inaugurales 3 / 22
Introduction From linear/linear to linear/nonlinear Credit Valuation In a financial transaction between a party C that has to pay another party Adjustment B some amount V , the CVA value is the price of the insurance contract that covers the default of party C to pay the whole sum V . � � V + CVA t , T = ( 1 − R ) E t τ 1 t <τ ≤ T (1) R is the recovery to make if the counterparty defaults (Assume R = 0), ◮ τ is the random default time of the counterparty, ◮ T is the protection time horizon. ◮ Numerical simulation N − 1 � � � V + CVA 0 , T ≈ E t k 1 τ ∈ ( t k , t k + 1 ] , (2) k = 0 N ≤ the number of time steps used for SDEs discretization. Importance Hold sufficient amount of liquid assets to face the counterparty default. ◮ Basel III includes the calculation of the CVA (Credit Valuation ◮ Adjustment) as an important part of the prudential rules. Lokman (UPMC, LPMA) Journées inaugurales 4 / 22
Introduction From linear/linear to linear/nonlinear Various kind of t , ..., S d Simulating assets S t = ( S 1 t ) trajectories then contracts trajectories contracts to get V t as a sum: � � � � φ exp φ eui φ eud φ am V t = ie ( S t ) + ii ( S t ) + id , t ( S t ) + ia , t ( S t ) , (3) ie ii id ia where ie , ii , id and ia are the exposure indices and: φ exp explicit function, for example: φ exp ( S t k ) = S 1 t k − S 2 t k . φ eui is a path-independent European contract, φ eui ( S t k ) = E ( f eui ( S T ) | S t k ) . (4) φ eud is a path-dependent European contract, t φ eud t k ( S t k ) = E ( f eud ( S t k + 1 ) | S t k ) , (5) t k for example: f eud i = 0 ,.., k S 1 t i ∨ S 1 t k + 1 − S 2 ( S t k + 1 ) = ( max t k + 1 ) + . t k φ am is an American contract, involving an optimal stopping problem t φ am t k ( S t k ) = f ( S t k ) ∨ E ( φ am t k + 1 ( S t k + 1 ) | S t k ) (6) with f an explicit payoff that generally does not depend on the asset path. Common problem ϕ ( x ) = E ( f ( S t k + 1 ) | S t k = x ) . Lokman (UPMC, LPMA) Journées inaugurales 5 / 22
Introduction From nonlinear/linear to nonlinear/nonlinear TVA definition Total valuation adjustment ( > CVA + DVA + FVA), it covers: Both defaults: τ = τ c ∧ τ b , CVA and DVA. ◮ Funding our risk and the risk of the counterparty: Nonlinear BSDE part, ◮ FVA. � t 0 r u du where r is the S. Crépey(2012) Ignoring the external funding and denoting β t = e − risk-free short rate process, Θ satisfies the following BSDE on [ 0 , τ ∧ T ] � � τ ∧ T � � � G t β t Θ t = E β τ 1 τ< T ( V τ − R τ ) + β s g s ( V s − Θ s ) ds (7) t where G is the extension of F by the natural filtration generated by τ c and by τ b . R is the total close-out cash-flow specified thanks to CSA (Credit Support Annex) and g is the funding coefficient. TVA BSDE simulation Only for European contracts. ◮ Requires a good approximation of the exposure V . ◮ Practitioners usually use rough approximations. ◮ No trustable procedure in the general case . ◮ Lokman (UPMC, LPMA) Journées inaugurales 6 / 22
Introduction From nonlinear/linear to nonlinear/nonlinear WOLOG: V = P For one American contract P t is the American derivative exposition. ◮ Let τ ∗ ∈ [ 0 , T ] be the optimal stopping time associated to P t . ◮ Theorem Θ = ˜ Θ J + ( 1 − J ) ξ τ , where J t = 1 { τ> t } and the pre-default TVA ˜ Θ satisfies the following BSDE on [ 0 , τ ∗ ] �� τ ∗ � β t ˜ ˜ ˜ g s ( P s − ˜ g t ( P t − ˜ Θ t ) = g t ( P t − ˜ Θ t ) + γ t ˜ Θ t = E t β s ˜ Θ s ) ds , ˜ ξ t . (8) t � t β t = e − ˜ 0 ( γ u + r u ) du . ◮ ˜ 1 ξ t := E t ( 1 { τ> t } ) E t ( ξ t 1 { τ> t } ) with ξ t := P t − E ( R 1 { t =¯ τ } |G t ) . ◮ Extension Possible to extend (8) on a portfolio of various American options. Lokman (UPMC, LPMA) Journées inaugurales 7 / 22
Simulation algorithms Plan Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Implementation issues on GPUs The difference with the references Complexity vs. accuracy Some numerical results Lokman (UPMC, LPMA) Journées inaugurales 8 / 22
Simulation algorithms Without funding � � N − 1 � constraints h = T P + CVA 0 , T = k + 1 1 τ ∈ ( kh , ( k + 1 ) h ] , N . E k = 0 With funding Θ k = E k (Θ k + 1 + hg ( k + 1 , P k + 1 , Θ k + 1 )) , Θ N = 0 . constraints � � The exposure P k ( x ) = E Φ k ,τ k ( S τ k ) | S k = x with τ N = N , (9) ∀ k ∈ { N − 1 , ..., 0 } , τ k = k 1 A k + τ k + 1 1 A c k , � � � �� � where A k = Φ k + 1 , k ( S k ) > E P k + 1 ( S k + 1 ) � S k . The conditional expectation involved in A k is approximated using a regression on a basis of monomial functions where K k is its cardinal. Lokman (UPMC, LPMA) Journées inaugurales 9 / 22
Simulation algorithms Without funding N − 1 � � � h = T constraints P + CVA 0 , T = E k + 1 1 τ ∈ ( kh , ( k + 1 ) h ] , N . k = 0 With funding Θ k = E k (Θ k + 1 + hg ( k + 1 , P k + 1 , Θ k + 1 )) , Θ N = 0 . constraints An example of a two stage simulation with M 0 = 2, M 6 = 8 and M 8 = 4 Lokman (UPMC, LPMA) Journées inaugurales 10 / 22
Simulation algorithms CVA 0 , T N − 1 M 0 � � � � 1 � F 2 P 1 ( S i � 1 ) , ..., � P k + 1 ( S i approximation CVA 0 , T = k + 1 ) (10) k + 1 M 0 k = 0 i = 1 With � � � F 1 k + 1 ( x 1 , ..., x k + 1 ) = E 1 τ ∈ ( kh , ( k + 1 ) h ] | P 1 = x 1 , ..., P k + 1 = x k + 1 , (11) F 2 k + 1 ( x 1 , ..., x k + 1 ) = ( x k + 1 ) + F 1 k + 1 ( x 1 , ..., x k + 1 ) . Θ k approximation For k = 1 , ..., N − 1 � � M 0 � � � 1 k + 1 ) + 1 Θ k ( x )= t ψ ( x )Ψ − 1 � ψ ( S j Θ k + 1 ( S j � k + 1 , � Θ k + 1 ( S j k + 1 ) , � P k + 1 ( S j k ) N g k + 1 ) k M 0 j = 1 (12) � � � � M 0 � 1 1 ) + 1 and � � Θ 1 ( S j � 1 , � Θ 1 ( S j 1 ) , � P 1 ( S j Θ N ( x ) = 0 , Θ 0 ( S 0 ) = 1 ) . N g M 0 j = 1 M 0 � 1 ψ ( � k ) t ψ ( � with: { S i } i ∈{ 1 ,..., M 0 } and { � S i S i S i } i ∈{ 1 ,..., M 0 } are two Where Ψ k = T k ) M 0 i = 0 independent simulations of the underlying asset S , ψ is a basis of monomial functions where K is its cardinal and T is an operator that must satisfy some desired properties. Lokman (UPMC, LPMA) Journées inaugurales 11 / 22
Simulation algorithms Without funding constraints Theorem �� � 2 � � � �� N 2 � � 1 ) , ..., � F 2 P 1 ( S i P k + 1 ( S i CVA 0 , T − CVA 0 , T ≤ k ∈{ 0 ,..., N − 1 } Var max k + 1 ) E k + 1 M 0 N � � �� 2 � 1 V j ( S i j ) f ′′ j ( P j ( S i j )) F 3 j ( P 1 ( S i 1 ) , ..., P j ( S i + j )) E 4 NM 2 j j = 1 2 N − 1 � N � 1 V j ( S i j ) f ′′ j ( P j ( S i k + 1 ( P 1 ( S i 1 ) , ..., P j ( S i j ) , S i F 4 + E j )) j ) 4 NM 2 j j = 1 k = j � � � � � � 2 � N � N 1 N V j ( S i j ( P 1 ( S i 1 ) , ..., P j ( S i j )) | P j ( S i j ) F 1 ( N − j + 1 ) 2 O + E j ) = 0 ϕ j ( 0 ) + N 4 M 2 M 4 j j j = 1 j = 1 �� M j � �� � Where ϕ j is the density of P j ( S i j ) , V j ( x ) = Var P j ( x ) − P j ( x ) . If ϕ j ( 0 ) is big then take M j ∼ √ M 0 , otherwise M j ∼ √ M 0 / N . In both Good choices cases, N must be small when compared to √ M 0 . When American options are involved, make sure that K 3 j / M j is small enough. For example √ M 0 � M j = N − j N − 1 M 1 with either M 1 = or M 1 = M 0 . (13) N Lokman (UPMC, LPMA) Journées inaugurales 12 / 22
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