Chaire Risques Financiers. Responsable: Nicole El Karoui Société Générale, ENPC, UPMC, X

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1 : Responsable: Nicole El Karoui Société Générale, ENPC, UPMC, X Ecole Polytechnique Paris 29 avril 2014

2 /Sponsor SG Historique Créée en Mars 2007, pour 5 ans au sein de la FdR (2007) Equipes de Mathématiques financières de Ecole Polytechnique (X) Ecole Nationale des Ponts (ENPC) Nouvelle Chaire( ) Les mêmes + UPMC Paris VI Augmentation du budget Active participation des quants SG Lorenzo Bergomi, Julien Guyon, Pierre Henry-Labordère

3 Une chaire structurée autour du marché Produits dérivés Identification, Modélisation, Couverture robuste des "produits individuels" Méthodes numériques/monte Carlo /Approchées Risque de modèle /corrélation, dépendence Les risques agrégés au niveau du book Mesures de risques/grande dimension Simulation des événements rares Risque de crédit, risque de contrepartie, risque de liquidité et les différents indicateurs de risques associés. Risque systémique Trading haute fréquence, microstructure, carnets d ordre.

4 Une chaire très orientée autour des jeunes La formation à la recherche, soutenue par la chaire Au moins 6 thèses soutenues en 2012 Au moins quinze thèses en cours Renforcement du potentiel d encadrement par la croissance des équipes, recrutement en particulier par l association de chercheurs français ou étrangers aux équipes, garante d une diversité de compétences suffisante

5 Une activité de recherche intense Exemples de thèmes récents Autour des risques Illiquidity risk Limit order book modeling and optimal execution Credit Risk and Numerical method, Longevity Risk Long term discounting, Méthodes Numériques Approximation by asymptotic expansion Variance reduction for Monte Carlo Methods Parallelization

6 Recherche (suite) Finance statistique Analyse des processus discontinus observés à des temps aléatoires (pour la couverture en haute fréquence) Modèles de microstructure Ambiguity and model risk Robustness to uncertainty of drift, volatility, correlation... = Nonlinear models Calibration, numerical approximation, Monte Carlo Methods

7 Collaborations avec la SG Bornes Universelles de prix calibrées à surface de volatilité avec P. Henry-Labordère Nouveaux problèmes numériques Lien avec la théorie du transport optimal Littérature académique engendrée par les travaux de la chaire Approximation par développements asymptotique avec L. Bergomi, J. Guyon, et P. Henry-Labordère Risques dépendant de la trajectoire avec P. Henry-Labordère Eviter les méthodes de régressions répétées Méthodes par simulation purement forward : diffusions branchantes Equations aux Dérivées Partielles dépendant de la trajectoire

8 Examples : from Vanilla to path-dependent Gestion des risques dépendant de la trajectoire

9 Examples : from Vanilla to path-dependent Management of path-dependent risk : why should we care? After all, we are going to implement a discrete approximation! Up to increasing the state space, we can reduce to the Markov case! My objective is to convince that Path-dependent thinking is more natural Developing path-dependent tools avoids discretization errors Sometimes easier... because it fits better the nature of problems we are facing

10 Examples : from Vanilla to path-dependent EXAMPLES

11 Example 1 : Black-Scholes Pricing Examples : from Vanilla to path-dependent r : constant instantaneous interest rate Underlying security price process driven by the Black-Scholes dynamics ds t S t = rdt + σdb t (under risk-neutral measure) v(t, S t ) : timet price of a derivative defined by the payoff g(s T ) at maturity T solves : t v + rs v s σ2 s 2 2 v rv = 0, s2 v(t,.) = g

12 Sensitivities in the Black-Scholes Model Examples : from Vanilla to path-dependent Time value : Optimal hedging ratio : t v(t, S t ) t = v s (t, S t) Hedging portfolio adjustment driven by the sensitivity Γ t = 2 v s 2 These definition are related to the Markovian feature From their nature, they should have a wider definition in the path-dependent context

13 Examples : from Vanilla to path-dependent Black-Scholes Theory for Path-Dependent Derivatives Consider a derivative defined by the path-dependent ξ = ξ(s t, 0 t T ) at maturity T. Then its time t price is v t = v ( t, (S u ) u t ) = E [ e r(t t) ξ F t ] Suitable for numerical approximation by Monte Carlo Sensitivities can also be computing by bumping the corresponding parameter

14 Example 2 : American Options Pricing Examples : from Vanilla to path-dependent Consider an American option defined by the payoff g(s t ) upon exercise at time t T The no-arbitrage price of this derivative in the Black-Scholes model satisfies { min t v rs v s 1 2 σ2 s 2 2 v } s 2 + rv, v g = 0 Similar to European derivatives, optimal hedging strategy given by (t, S t ) = v s Hedging portfolio adjustment driven by Γ(t, S t ) = 2 v s 2 (t, S t)

15 Path-Dependent American Options Examples : from Vanilla to path-dependent ξ t = ξ ( t, (S u ) u t ) : payoff upon exercising at time t Then, path-dependent American option price given by the probabilistic representation v t = v ( t, (S u ) u t ) = sup τ T [t,t ] E [ e r(τ t) ξ τ F t ] where T [t,t ] the collection of stopping times valued in [t, T ] What about numerical approximation?

16 Examples : from Vanilla to path-dependent Numerical approximation by repeated backward regressions Bermudian approximation : let 0 = t0 n <... < tn n = T, and vt n = = sup E [ e r(τ t) ] ξ Ft τ τ T {t n} i i n Snell envelope representation : v n 0 = Y 0, where : Y t n n = ξ t n n and Y t n i = max { ξ t n i, E [ Y t n i+1 Fti ]} Implementable scheme by numerical approximation of the conditional expectation (Carrière 84, Longstaff & Schwartz 01,...) But unfeasible in path-dependent case!

17 Examples : from Vanilla to path-dependent Example 3 : Counterparty Value Adjustment (CVA) In the Black-Scholes models, the typical valuation PDE is : t v + rs v s σ2 s 2 2 v s 2 rv Rv + = 0, v(t,.) = g This is a semilinear PDE (t, S t ) = v s (t, S t) and Γ(t, S t ) = 2 v (t, S s 2 t ) have the same interpretation as in previous examples Any characterization in the path-dependent case? Numerical scheme in the path-dependent case?

18 Examples : from Vanilla to path-dependent Backward Stochastic Differential Equation Representation The CVA price process {v t, t [0, T ]} solves the BSDE : T ( v t = ξ + rvu + Rv u + ) T du Z u db u t t Discrete-time approximation (0 = t0 n <... < tn n = T ) : vt n n = ξ and v t n i n = E [ vt n i+1 n + ( ti+1 n ti n )( rvt n + Rv + ) ] i+1 ti+1 n F t n i i.e. repeated backward regressions as in the case of American options But unfeasible in the path-dependent case!

19 Example 4 : Uncertain Volatility Models Examples : from Vanilla to path-dependent Now, suppose that the volatility parameter is only known to lie in between two bounds 0 σ σ : ds t S t = rdt + σ t db t, σ σ t σ The the superhedging cost of a European derivative defined by payoff g(s T ) at time T satisfies t v rs v s + rv 1 2 s2 max σ σ σ σ2 2 v s 2 = 0, v(t,.) = g Avellaneda, Lévy & Parás 95, Lyons 95 (t, S t ) = v s (t, S t) and Γ(t, S t ) = 2 v (t, S s 2 t ) again have the same interpretation as in previous examples

20 Examples : from Vanilla to path-dependent 2BSDE Representation and Numerical Approximation The price process {v t, t [0, T ]} solves the 2BSDE... Suppose σ σ 0, r = 0 (for simplicity), and rewrite the PDE t v 1 2 σ2 0s 2 2 v ( 2 s 2 v ) s2 G s 2 = 0 where G(γ) := 1 2 max σ σ σ(σ 2 σ 2 0 )γ Let S 0 be defined by ds 0 t = S 0 t σ 0 db t, then v n t n n v n (t n i, s) = E [ v n (t n i+1, S 0 t n i+1 ) S 0 t n i+1 = s] = g, and ( +s 2 G E [v n (ti+1, n St 0 ni+1 ) B t n 2 i+1 t ]) S 0 ( t) 2 t n = s i = repeated backward regressions But unfeasible in the path-dependent case!

21 Example 5 : Portfolio Optimization Examples : from Vanilla to path-dependent The underlying asset s dynamics under historical measure is ds t S t = µdt + σdb t r = 0 for simplicity. Portfolio optimization with final liability g(s T ) : v(t, x, s) = sup E [ U ( XT g(s T ) ) (X, S) t = (x, s) ] where the wealth process is given by T XT = x + u ds u t

22 Examples : from Vanilla to path-dependent HJB Equation for the Portfolio Optimization Problem By the dynamic programming approach, the value function v(t, x, s) satisfies : 0 = t v µs v s 1 2 σ2 s 2 2 v s 2 { sup sµ v x + s2 σ 2 2 v x s s 2 σ 2 2 v } x 2 v(t, x, s) = g(s) Suppose the liability is ξ path-dependent = path-dependent HJB equation Any implementable numerical method in path-dependent case?

23 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs NUMERICAL APPROXIMATION OF PATH-DEPENDENT PDEs

24 Intuition from the linear case Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs The equation : t v v βv = 0, 2 x 2 v(t,.) = g has two possible probabilistic representations : v(t, x) = E P [ 0 e β(t t) g(b T ) B t = x ] v(t, x) = E P 0[ g(bt )1I {T <τ} B t = x ], where τ is an independent exponential random variable with mean β Both representations are valid in the path-dependent case

25 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs A first class of Nonlinear Path-Dependent PDEs Let a k 0 with n k=0 a k = 1, and consider the equation : t v + 1 ( n ) 2 2 ωωv + β a k v k v = 0, v T = ξ k=0 Define the branching Brownian motion : Start from one particle driven by a Brownian motion T1 t independent exponential distribution with parameter β if T1 t < T, the first particle dies out and is replaced by k independent particles with probability a k NT t : Number of particles at time T, Z i. : path of particle i Then (Watanabe, McKean) [ Nt T ] v t = E ξ(z. i ) i=1

26 Path-Dependent KPP equation (Kolmogorov-Petrovsky-Piskounov) Let n k=0 a k = 1, and consider the equation : k=0 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs t v + 1 ( n ) 2 2 ωωv + β a k v k v = 0, v T = ξ Define the branching Brownian motion with probabilities ( a k ) 0 k n. Then v t = E [ Nt T ] ( 1) ε i ξ(z i i=1 Possible extension to include random drift and random diffusion For an analytic nonlinearity R(v) = i=0 a kv k, approximation by substitution R n (v) := n i=0 a kv k to R(v). )

27 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs Exact Simulation of Path-Dependent Diffusions I Let b : [0, T ] Ω R, (Lipschitz, say) and consider the equation t v + 1 ( ) 2 2 ωωv + β b ω v v = 0, v(t,.) = ξ There are two possible representations of the solution : v t = E [ e β(t t) ξ(x t,x. ) ] with dx s = βb(s, X. )ds + db s (requires discrete-time approximation) or by Brownian motion branching at independent Exponential(β) times...

28 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs Branching Brownian motion for Exact Simulation (θ 0 i ) i 1 : iid Expo(β) r.v. T t i := T (t + θ θ0 i ) := max{i : T t i < T } : Number of defaults on [t, T ] N t T Then, an alternative representation of the solution of the previous PDE is : [ v t = E ξ(b t,x. ) N t T i=1 ( b ( T t i, B. t,x )B T t i+1 B T t i Ti+1 t T i t does not require any discrete-time approximation... )]

29 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs Branching Brownian motion representation of semilinear PDEs For k = 0,..., n, let b k : [0, T ] Ω R be such that k b k = 1, and consider the equation t v + 1 ( n ) 2 2 ωωv + β b k ( ω v) k v = 0, v T = ξ k=0 = At any default time (Exponential(β) distribution), k particles are created with probability b k... Can be extended to a polynomial in (v, ω v) For an analytic function of (v, ω v), we may use a polynomial approximation

30 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs Exact Simulation of Path-Dependent Diffusions II Let σ : [0, T ] Ω R, (Lipschitz, say) and consider the equation t v + 1 ( ) 2 2 ωωv + β σ ωωv 2 v = 0, v T = ξ There are two possible representations of the solution : v t = E [ e β(t t) ξ(x t,x. ) ] with dx s = ( 1 + βσ(s, X. ) ) db s (requires discrete-time approximation) or by Brownian motion branching at independent Exponential(β) times...

31 Path-Dependent KPP Equations Fully Nonlinear Path-Dependent PDEs Branching Brownian motion for Exact Simulation II (θ 0 i ) i 1 : iid Expo(β) r.v. T t i := T (t + θ θ0 i ) := max{i : T t i < T } : Number of defaults on [t, T ] N t T Then, an alternative representation of the solution of the previous PDE is : [ v t = E ξ(b t,x. ) N t T i=1 ( σ ( T t i, B. t,x )(B T t i+1 B T t i ) 2 (T t i+1 T t (T t i+1 T t i )2 Polynomial nonlinearity in 2 ωωv can be addressed as for v and ω v... i ) )]