Numerical Approximations of McKean Anticipative Backward Stochastic - PowerPoint PPT Presentation

Motivation A general framework for MKABSDEs Approximations and numerical analysis Numerical Approximations of McKean Anticipative Backward Stochastic Differential Equations Arising in Initial Margin Requirements A. Agarwal, S. De Marco, E. Gobet, J. G. L´ opez-Salas, F. Noubiagain, A. Zhou S´ eminaire des doctorants, CERMICS, 27 f´ evrier 2018

Motivation A general framework for MKABSDEs Approximations and numerical analysis Outline Motivation 1 A general framework for MKABSDEs 2 Approximations and numerical analysis 3

Motivation A general framework for MKABSDEs Approximations and numerical analysis Outline Motivation 1 A general framework for MKABSDEs 2 Approximations and numerical analysis 3

Motivation A general framework for MKABSDEs Approximations and numerical analysis Classical theory for option pricing and hedging Let us assume that a risky asset S follows the dynamics dS t = µ t dt + σ t dW t S t and a riskless asset S 0 , where dS 0 t = rdt S 0 t Given an European option with payoff g ( S T ), the dynamics of a self-financing hedging portfolio in S , S 0 is, using the notation θ t = µ t − r and Z t = σ t π t σ t dV t = r ( V t − π t ) dt + π t dS t = rV t dt + θ t Z t dt + Z t dW t S t V T = g ( S T ) Linear BSDE, the price of the portfolio is given by � V t = E Q � � e − r ( T − t ) g ( S T ) � F t = v ( t , S t ) � � · where W + 0 θ s ds is a Brownian motion in the probability measure Q .

Motivation A general framework for MKABSDEs Approximations and numerical analysis A first BSDE nonlinear in the sense of McKean New regulation since 2008: change in pricing and hedging rules (from linear to nonlinear BSDEs) New constraint: post collateral to Central Counterparty (CCP) to secure position CVaR variation margin: the value of the deposit depends on the CVaR of the portfolio computed over 10 days The equation satisfied by the hedging portfolio becomes (MKA) dV t = rV t dt + θ t Z t dt − R λ CVaR α, F t � V t − V ( t +∆) ∧ T � dt + Z t dW t V T = g ( S T ) Anticipative BSDE, with dependence in conditional law (MKABSDE) Correction w.r.t. the price without collateralization ? Assumption: here, no default, safe products

Motivation A general framework for MKABSDEs Approximations and numerical analysis Objectives Is (MKA) a well-posed problem? How do we compute the correction induced by the nonlinear CVaR term on the price of an European option?

Motivation A general framework for MKABSDEs Approximations and numerical analysis Outline Motivation 1 A general framework for MKABSDEs 2 Approximations and numerical analysis 3

Motivation A general framework for MKABSDEs Approximations and numerical analysis Framework We are looking for a process ( Y , Z ), such that (MKABSDE) � T � T f ( s , Y s , Z s , Λ s ( Y s : T )) ds − Z s dW s , t ∈ [0 , T ] . Y t = ξ + t t For t ∈ [0 , T ] and X ∈ H 2 0 , T ( R ) , Λ t ( X ) is a F t -measurable random variable with values in R . Λ t is a functional of the future path of ( Y s ) s ∈ [ t , T ] =: Y s : T . For s , y , z , x ∈ [0 , T ] × R × R q × R , f ( s , y , z , x ) is a F s -adapted random variable with values in R . ξ is a square integrable F T -measurable random variable.

Motivation A general framework for MKABSDEs Approximations and numerical analysis Existence and Uniqueness Theorem Under integrability Assumptions on ξ, f and Lipschitz properties on f , Λ , the BSDE (MKABSDE) has a unique solution in S 2 0 , T ( R ) × H 2 0 , T ( R q ) . Proof. Standard a priori estimates in S 2 β, T × H 2 β, T = S 2 0 , T × H 2 0 , T . For β ≥ 0 large enough, the map φ : ( U , V ) ∈ S 2 β, T × H 2 β, T → ( Y , Z ) := φ ( U , V ) ∈ S 2 β, T × H 2 β, T , where � T � T Y t = ξ + f ( s , U s , V s , Λ s ( U s : T )) ds − Z s dW s , t ∈ [0 , T ] , t t is a contraction under Assumptions ( L ) , ( I ) , ( R ), hence existence and uniqueness. Corollary BSDE (MKA) has a unique solution ( Y MK , Z MK ) in S 2 0 , T ( R ) × H 2 0 , T ( R q ) .

Motivation A general framework for MKABSDEs Approximations and numerical analysis Outline Motivation 1 A general framework for MKABSDEs 2 Approximations and numerical analysis 3

Motivation A general framework for MKABSDEs Approximations and numerical analysis A first approximation We do not know how to simulate the original CVaR BSDE Typically, ∆ = 1 week and thus ∆ ≪ 1. First approximation (MKA to nonlinear classical BSDE): in the CVaR term, at the lowest order, � t +∆ � t +∆ V t − V t +∆ ≈ Z s dW s ≈ Z t dW s = Z t ( W t +∆ − W t ) t t For C α := CVarR α ( N (0 , 1)), the CVAR term simplifies to � CVaR α, F t ( V t − V t +∆ ) ≈ C α | Z t | ( t + ∆) ∧ T − t We obtain the nonlinear BSDE � T � � V NL − rV NL − θ s Z NL + R λ C α | Z NL � = g ( S T ) + | ( s + ∆) ∧ T − s ds t s s s t � T Z NL − dW s s t

Motivation A general framework for MKABSDEs Approximations and numerical analysis A second approximation √ Let us remark that in order 0 in the parameter ∆, we have the solution of classical linear pricing framework: � T � T � � V BS − rV BS − θ s Z BS Z BS = g ( S T ) + ds − dW s t s s s t t Second Approximation (nonlinear to linear): we make a formal derivation of the √ nonlinear BSDE w.r.t the parameter ∆ to obtain a linear BSDE (Gobet-Pagliarani 2015) � T � � V L − rV L s − θ s Z L s + R λ C α | Z BS � t = g ( S T ) + | ( s + ∆) ∧ T − s ds s t � T Z L − s dW s t

Motivation A general framework for MKABSDEs Approximations and numerical analysis Numerical estimates Proposition If µ, σ are Markovian and satisfy Lipschitz properties, there exists K 1 , K 2 , K 3 > 0 , independent from ∆ , such that || V L − V BS || 2 0 , T ( R ) + || Z L − Z BS || 2 0 , T ( R q ) ≤ K 1 ∆ S 2 H 2 || V NL − V L || 2 0 , T ( R ) + || Z NL − Z L || 2 0 , T ( R q ) ≤ K 2 ∆ 2 S 2 H 2 || V MK − V NL || 2 0 , T ( R ) + || Z MK − Z NL || 2 0 , T ( R q ) ≤ K 3 ∆ 2 S 2 H 2

Motivation A general framework for MKABSDEs Approximations and numerical analysis Conclusion and future research We obtained existence and uniqueness for general MKABSDEs We do not know how to simulate the CVaR BSDE, but in this case, as ∆ ≪ 1, we approximated the CVaR BSDE with standard BSDEs We obtained estimates to control the error between the real solution and its approximation How to compute efficiently the approximations? How to simulate general MKABSDEs?

Motivation A general framework for MKABSDEs Approximations and numerical analysis Thank you! Thank you for your attention!