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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´
• pez-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

1

Motivation

2

A general framework for MKABSDEs

3

Approximations and numerical analysis

Motivation A general framework for MKABSDEs Approximations and numerical analysis

Outline

1

Motivation

2

A general framework for MKABSDEs

3

Approximations and numerical analysis

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 dSt St = µtdt + σtdWt and a riskless asset S0, where dS0

t

S0

t

= rdt Given an European option with payoff g(ST ), the dynamics of a self-financing hedging portfolio in S, S0 is, using the notation θt = µt−r

σt

and Zt = σtπt dVt =r (Vt − πt) dt + πt St dSt = rVtdt + θtZtdt + ZtdWt VT =g(ST ) Linear BSDE, the price of the portfolio is given by Vt = EQ e−r(T−t)g(ST )

• Ft
• = v(t, St)

where W + ·

0 θsds 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) dVt =rVtdt + θtZtdt−RλCVaRα,Ft

• Vt − V(t+∆)∧T
• dt + ZtdWt

VT =g(ST ) 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

1

Motivation

2

A general framework for MKABSDEs

3

Approximations and numerical analysis

Motivation A general framework for MKABSDEs Approximations and numerical analysis

Framework

We are looking for a process (Y , Z), such that (MKABSDE) Yt = ξ + T

t

f (s, Ys, Zs, Λs (Ys:T )) ds − T

t

ZsdWs, t ∈ [0, T]. For t ∈ [0, T] and X ∈ H2

0,T (R) , Λt (X) is a Ft-measurable random variable with

values in R. Λt is a functional of the future path of (Ys)s∈[t,T] =: Ys:T . For s, y, z, x ∈ [0, T] × R × Rq × R, f (s, y, z, x) is a Fs-adapted random variable with values in R. ξ is a square integrable FT -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 S2

0,T (R) × H2 0,T (Rq).

Proof. Standard a priori estimates in S2

β,T × H2 β,T = S2 0,T × H2 0,T . For β ≥ 0 large enough,

the map φ : (U, V ) ∈ S2

β,T × H2 β,T → (Y , Z) := φ (U, V ) ∈ S2 β,T × H2 β,T ,where

Yt = ξ + T

t

f (s, Us, Vs, Λs (Us:T )) ds − T

t

ZsdWs, t ∈ [0, 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 S2

0,T (R) × H2 0,T (Rq).

Motivation A general framework for MKABSDEs Approximations and numerical analysis

Outline

1

Motivation

2

A general framework for MKABSDEs

3

Approximations and numerical analysis

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, Vt − Vt+∆ ≈ t+∆

t

ZsdWs ≈ t+∆

t

ZtdWs = Zt(Wt+∆ − Wt) For Cα := CVarRα(N(0, 1)), the CVAR term simplifies to CVaRα,Ft (Vt − Vt+∆) ≈ Cα|Zt|

• (t + ∆) ∧ T − t

We obtain the nonlinear BSDE V NL

t

=g (ST ) + T

t

• −rV NL

s

− θsZ NL

s

+ RλCα|Z NL

s

|

• (s + ∆) ∧ T − s
• ds

− T

t

Z NL

s

dWs

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: V BS

t

= g (ST ) + T

t

• −rV BS

s

− θsZ BS

s

• ds −

T

t

Z BS

s

dWs 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) V L

t =g (ST ) +

T

t

• −rV L

s − θsZ L s + RλCα|Z BS s

|

• (s + ∆) ∧ T − s
• ds

− T

t

Z L

s dWs

Motivation A general framework for MKABSDEs Approximations and numerical analysis

Numerical estimates

Proposition If µ, σ are Markovian and satisfy Lipschitz properties, there exists K1, K2, K3 > 0, independent from ∆, such that ||V L − V BS||2

S2

0,T (R) + ||Z L − Z BS||2

H2

0,T (Rq) ≤K1∆

||V NL − V L||2

S2

0,T (R) + ||Z NL − Z L||2

H2

0,T (Rq) ≤K2∆2

||V MK − V NL||2

S2

0,T (R) + ||Z MK − Z NL||2

H2

0,T (Rq) ≤K3∆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!