Martingale Optimal Transport: A Nice Ride in Quantitative Finance

Transcription

1 Martingale Optimal Transport: A Nice Ride in Quantitative Finance Pierre Henry-Labordère 1 1 Global markets Quantitative Research, SOCIÉTÉ GÉNÉRALE

2 Contents Optimal transport versus Martingale optimal transport. Applications in mathematical finance: Model-independent bounds for exotic options: Numerical methods. Particle s methods for non-linear McKean SDEs: Calibration of LSVMs. Skorokhod embedding problem [see Nizar s talk]

3 Optimal Transport in Mathematics Optimal transport, first introduced by G. Monge in his work Théorie des déblais et des remblais" (1781). Has recently spread out in various mathematical domains as highlighted by the last Fields medallist C. Villani. Let us cite Analysis of non-linear (kinetic) partial differential equations arising in statistical physics such as McKean-Vlasov PDE. Mean-field limits, convergence of particle s methods. Optimal fundamental inequalities (Poincaré, (Log)-Sobolev, Talagrand...) Study of Ricci flows in differential geometry.

4 Optimal Transport in Quantitative Finance Despite these large ramifications with analysis and probability, optimal transport has not yet attracted the attention of practitioners in financial mathematics. However, various long-standing problems in quantitative finance can be tackled using the framework of optimal transport. In particular, Calibration of (hybrid) models on market smiles using particle s method. Computation of efficient model-independent bounds for exotic options. Leads to a nice modification of optimal transport [ Martingale version" of MK]

5 Optimal Transport in a Nutshell (1) Payoff c depending on two assets S 1, S 2. The distributions of S 1 and S 2 are known from Vanilla options Monge-Kantorovich 1 MK c = P i (K ) = 2 K Ci (T, K ) inf E P [c(s 1, S 2 )] P,S 1 P 1,S 2 P 2 1 S 1 P 1 means Law(S 1 ) = P 1

6 Optimal Transport in a Nutshell (2): Kantorovich duality (Linear) duality (Minimax): MK c = sup u 1 ( ),u 2 ( ) E P1 [u 1 (S 1 )] + E P2 [u 2 (S 2 )] u 1 (S 1 ) + u 2 (S 2 ) c(s 1, S 2 ), P 1 P 2 a.s The dual bound can be statically replicated by holding European options with payoffs u 1 (S 1 ) and u 2 (S 2 ) with market prices E P1 [u 1 (S 1 )] and E P2 [u 2 (S 2 )]. The intrinsic value of the portfolio u 1 (S 1 ) + u 2 (S 2 ) is lower than the payoff c(s 1, S 2 ).

7 Martingale Optimal Transport (1) Payoff c(s t1,..., S tn ) depending on one asset evaluated at t 1 <... < t n. No-arbitrage condition: S t is required to be a (local) positive martingale 2. The distribution of S ti is known from Vanilla options at t i. Primal (Lower bound): P = inf S ti P i,e P t i 1 [S ti ]=S ti 1 E P [c(s t1,..., S tn )] 2 We take zero interest rate, no dividends for the sake of simplicity. This can be easily relaxed.

8 Martingale Optimal Transport (2) Feasibility of {P : S ti P i, E P t i 1 [S ti ] = S ti 1 ]}: Convex order [Kellerer]. Convex order: P 1 P 2 if E P1 [(S t1 K ) + ] E P2 [(S t2 K ) + ]. Dual 3 : n u i (S i ) + i=1 D = inf (u i ( )) 1 i n,( i ( )) 1 i n n E Pi [u i (S i )] i=1 n i (S 1,..., S i 1 )(S i S i 1 ) c(s 1,..., S n ) i=1, P 1... P n a.s. Financial interpretation: sub-hedging strategy static portfolio of Vanillas. 3 Markov assumption: i (S 1,..., S i 1 ) = i (S i 1 )

9 Martingale version" of MK duality Theorem (Beiglböck, PHL, Penkner) Assume that P 1,..., P n are Borel probability measures on R + such that P 1... P n. Let c : R n + (, ] be a lower semi-continuous function such that c(s 1,..., S n ) K (1 + S S n ) (1) on R n + for some constant K. Then there is no duality gap, i.e. P = D MK c. Moreover, the primal value P is attained, i.e. there exists a martingale measure P with marginals (P 1,..., P n ) such that P = E P [c]. The dual supremum is in general not attained.

10 MK c versus MK c MK c > MK c = tight bounds. MK c MK c inf P,S1 P 1,S 2 P 2 E P [c(s 1, S 2 )] inf P,S1 P 1,S 2 P 2 E P [c(s,e[s 2 S 1 ]=S 1, S 2 )] 1 sup u1,u E P1 [u 2 1 (S 1 )] + E P2 [u 2 (S 2 )] sup u1,u 2, EP1 [u 1 (S 1 )] + E P2 [u 2 (S 2 )] u 1 (S 1 ) + u 2 (S 2 ) c(s 1, S 2 ) u 1 (S 1 ) + u 2 (S 2 ) + (S 1 )(S 2 S 1 ) c(s 1, S 2 ) sup u E P2 [u(s 2 )] + E P1 [u c (S 1 )] 4 sup u E P2 [u(s 2 )] + E P1 [(c(s 1, ) u( )) conv (S 1 )] 5 Important results in optimal transport are derived for the quadratic cost c(s 1, S 2 ) = S 2 S 1 2 [see Brenier s Theorem]. In the Martingale version, the quadratic cost is degenerate: E P [(S 2 S 1 ) 2 ] = E P2 [S 2 2 ] EP1 [S 2 1 ] P mart. S i P i = Important results in MK need to be rewritten for MK! 4 u c (S 1 ) inf S2 c(s 1, S 2 ) u(s 2 ) 5 f conv : largest convex function smaller than or equal to f

11 Optimal Transport on the real line We note F 1 the cumulative distribution associated to P 1. Let c(s 1, S 2 ) = c(s 2 S 1 ) be a C 1 strictly concave. Proposition The upper bound is given by [Fréchet copula] MK c = 1 0 c(f (u), F2 (u))du The (optimal) upper bound is reached for û 2 (y) = y 0 c (F 1 1 F 2(z), z)dz û 1 (x) = c(x, F 1 2 F 1(x)) û 2 (F 1 2 F 1(x))

12 Brenier s theorem Let c(s 1, S 2 ) = c(s 2 S 1 ) be a C 1 strictly convex. Theorem (Brenier) There exists a unique optimal transference plan for the MK c transportation problem and it has the form P (S 1, S 2 ) = δ (S 2 T (S 1 )) P 1 (S 1 ), T # P 1 = P 2 and T (x) = x c 1 ( ψ) for some c-concave function ψ. The optimal lower bound is given by MK c = 0 c(x, T (x))p 1 (x)dx On the real line, T (x) = F 1 2 F 1(x): monotone rearrangement map.

13 Martingale version of Brenier s theorem (1) [Hobson-Neuberger], [Beiglböck-Juillet] Let c(s 1, S 2 ) = c(s 2 S 1 ) be a C 1 function such that c is strictly concave. Suppose P 1 P 2. Theorem (Beiglböck-Juillet) There exists a unique optimal transference plan for MK c : ( P T 2 (S 1 ) S 1 (S 1, S 2 ) = δ(s 2 T 1 (S 1 )) T 2 (S 1 ) T 1 (S 1 ) ) S 1 T 1 (S 1 ) +δ(s 2 T 2 (S 1 )) P 1 (S 1 ) T 2 (S 1 ) T 1 (S 1 ) The optimal upper bound is given by 0 (T 2 (x) x) c(x, T 1 (x)) + (x T 1 (x)) c(x, T 2 (x)) P 1 (x)dx T 2 (x) T 1 (x)

14 Explicit characterization of T 1, T 2 [PHL] The maps (T 1, T 2 ) are solutions of the equations (T 1 (x) x T 2 (x), T 1, T 2 C 1 functions) c 2 (T 1 1 (x), x) c 2 (T 1 (x), x) = 2 P 2 (x) = T 2T 1 (x) T T 2 T 1 (x) x 1 (x) P 1 (T T (x) 1 T 1 (x) 2 Semi-static superreplication: c 1 (y, T 2 (y)) c 1 (y, T 1 (y)) dy T 2 (y) T 1 (y) (x)) T 1 (x) + T 1 (x) T 2 1 T 1 (x) 2 1 x T 1 T 1 (x) 2 P 1 (T 1 2 (x)) T 1 (x) 2 1 du 2 (x) T = c 2 (T 1 (x) 1 c 1 (y, T 2 (y)) c 1 (y, T 1 (y)) (y), x) dy 1 dx 0 T 2 (y) T 1 (y) u 1 (x) = (c(x, T 1(x)) u 2 (T 1 (x))) (x T 2 (x)) (c(x, T 2 (x)) u 2 (T 2 (x))) (x T 1 (x)) T 1 (x) T 2 (x) (x) = (c(x, T 1(x)) u 2 (T 1 (x))) (c(x, T 2 (x)) u 2 (T 2 (x))) T 1 (x) T 2 (x)

15 Examples Spread option (S 2 S 1 ) + [Fréchet]: MK c = 0 (T (x) x) + P 1 (x)dx, T (x) = F 1 2 F 1(x) Forward-start options [Hobson-Neuberger] (S t2 S t1 ) + : MK 2 = 0 (T 2 (x) x) (x T 1 (x)) P 1 (x)dx T 2 (x) T 1 (x) Variance swap c(s t2, S t1 ) = ln 2 S t 2 S t1 0 (T 2 (x) x) ln 2 T 1(x) x [PHL]: + (x T 1 (x)) ln 2 T 2(x) x T 2 (x) T 1 (x) P 1 (x)dx

16 Optimal Transport and Hamilton-Jacobi (1) Here c(s 1, S 2 ) := c(s 2 S 1 ), c is strictly concave. Theorem (see Villani, Topics in Optimal Transport, AMS) MK c = sup E P1 [u(0, S 1 )] + E P2 [u(1, S 2 )] where the supremum is taken over all continuous viscosity solutions u to the following HJ equation: t u(t, x) + c ( u) = 0, c (p) := sup{pq c(q)} q Proof uses Hopf-Lax s formula: u(0, x) = inf c(y x) u(1, y) y Guess: Martingale optimal transport = HJB. See Nizar s talk: Generalization of Mikani-Thiellen approach.

17 Hopf-Lax s formula: Reminder 1 Dynamic programming: 1 u(t, x) = sup u(1, x + ζ(s)ds) ζ t 2 Maximization over ζ: ζ is a constant q. 1 t c( ζ(s))ds u(t, x) = sup u(1, x + q(1 t)) c(q)(1 t) q 3 Set y = x + q(1 t). Get the Hopf-Lax solution: u(t, x) = sup y u(1, y) c( y x )(1 t) 1 t 4 For t = 0, u(0, ) is the c-transform of u(1, ): u(0, x) = inf c(y x) u(1, y) y

18 Time-continuous limit Robust super-hedging price of a payoff given vanilla options (S ti µ i, µ(λ) := E µ [λ]): T U n µ (ξ) := inf{u 0 :, λ : U n λ i (S ti ) i=1 0 s ds s n µ i (λ i ) ξ, P Mart.} i=1 Measures are singular: Quasi-sure analysis (see Nizar s talk)

19 Duality in continuous-time Theorem (Galichon, PHL, Touzi) Let ξ UC(Ω S0 ) be such that ξ + L 1 (P) for all P Mart.. Then, for all µ := (µ i ) i M(R + ) in convex order: U µ n (ξ) = inf sup λ i Λ µ UC P Mart. { n µ i (λ i ) + E P[ ξ i=1 n λ i (S ti ) ]}. i=1 Robust version of [Kramkov, Schachermayer] duality. If we can apply formally a min-max duality, U µ n (ξ) = sup E P [ξ] P Mart., S ti µ i Martingale optimal transport problem. Give models calibrated to vanilla options.

20 Models calibrated to Vanillas: Some examples Local volatility model [Dupire]: df t = σ loc (t, f t )dw t σ loc (t, f ) 2 = 2 tc(t, f ) f 2C(t, f ) Local stochastic volatility models: Equivalent to df t = σ(t, f t )a t dw t σ loc (t, f ) 2 = σ(t, f ) 2 E[at 2 f t = f ] df t = σ loc (t, f ) a t dw t E[at 2 f t] = Non-linear McKean SDEs for which optimal transport shows up again! [see Tanaka s approach for Boltzmann equation]

21 McKean SDEs Definition dx t = b(t, X t, P t )dt + σ(t, X t, P t ) dw t with W t a d-dimensional Brownian motion and P t = Law(X t ). Example: McKean-Vlasov SDEs: b ( ) b i (t, x, P t ) i=1,...,n = σ {σ i j (t, x, P t)} i=1,...,n;j=1,...d = b i (t, x, y)p(t, y X 0 )dy σ i j (t, x, y)p(t, y X 0)dy

22 Existence result Theorem (Sznitman) Let b : R + R n P 2 (R n ) R n and σ : R + R n P 2 R n d be Lipschitz continuous functions for the sum of canonical metric on R n and the MK metric d on the set P 2 of probability measures with finite second order moments. Then the non-linear SDE dx t = b(t, X t, P t ) + σ(t, X t, P t )dw t, X 0 R n where P s denotes the probability distribution of X s admits an unique solution such that E(sup t T X t p ) < for all p 2. Open problem: Existence of LSVMs? Proof: fixed point.

23 Monte-Carlo simulation: interacting particle system Replace P t by its empirical measure: Let Xt 1,... X t N be i.i.e. with law P t : P N t = 1 N N i=1 δ Xt i. Note that P N t is a random probability measure. N interacting bosons (i.e. symmetric): N dft i = ft i σ loc (t, ft i ) j=1 δ(ln f j t ln ft i ) N j=1 (aj t )2 δ(ln f j tdw i t ln ft i)ai t Needs to be replaced δ( ) by a regularizing kernel. Propagation of chaos for McKean-Vlasov SDEs: If at t = 0, X i,n 0 are independent particles then as N, for any fixed t > 0, the X i,n t are asymptotically independent and their empirical measure P N t converges in distribution towards the true measure P t.

24 Algorithm [Guyon-PHL] 1 Initialize k = 0 and set σ(t, f ) = σ Dup(0,f ) a 0 t [k, (k + 1) ]. for all 2 Simulate the N processes {f i t, ai t } i=1,...,n from t = k to (k + 1) using a discretization scheme such as Euler. 3 Compute the local volatility σ((k + 1), f ) on a space-grid f [fk min, f k max] using σ(t, f ) = σ Dup (t, f ) N j=1 (a(j) t ) 2 δ t,n (f (j) t f ) N j=1 δ t,n (f (j) t f ) Set σ(t, f ) σ((k + 1), f ) for all t [(k + 1), (k + 2) ]. 4 k := k + 1. Iterate step 2 and 3 up to the maturity date T. Convergence issue: prove the propagation of chaos for LSVMs?

25 Local Bergomi model DAX market smiles (30-May-11): Fit of the market smile for T = 4Y 2^10 particles 2^12 particles 2^13 particles Mkt No calibration Approx Strike

26 Local stochastic volatility model: Existence under question The existence of LSV models for a given market smile is not at all obvious although this seems to be a common belief in the quant community. Checked our algorithm with a volatility-of-volatility σ = 350%. Our algorithm converge with N = 2 13 particles but the market smile is not properly calibrated: Fit of the market smile for T = 4Y VolVol = 350% 2^13 particles 40 2^15 particles Mkt Approx Strike

27 Digression: Talagrand-like inequality Talagrand inequality: Relative entropy: T(λ) : P 1 W 2 (P 1, P 2 ) 2 2 λ H(P1 P 2 ) H(P P 0 ) = E P [ln dp ], P is absolutely continuous w.r.t. P0 dp0 = +, otherwise Villani-Otto, Ledoux-al: LSI(λ) T(λ) [Proof: dual expression for the Talagrand inequality + contraction of HJ ] A similar dual expression appears in mathematical finance (Martingale) Weighted Monte-Carlo.

28 (Martingale) Weighted Monte-Carlo [Avellaneda-al], [PHL] Consider instruments c a, a = 1,..., N, with bid/ask market prices c a /c a : c a E P [c a ] c a M(P 1,..., P n c 1,..., c N ): the set of all martingale measures P on (R d +) n with prescribed marginals {P i } i=1,...,n and satisfying (2). Primal: P λ sup{e P [c] : P M(P 1,..., P n c 1,..., c N ), H(P, P 0 ) λ} Some particular limits: P = MK c P 0 = inf{h(p, P 0 ) : P M(P 1,..., P n c 1,..., c N )}

29 (Martingale) Weighted Monte-Carlo [PHL] Dual: D λ inf (u i ( )) 1 i n,( i ( )) 1 i n,λ a R +,Λ a R +,ζ R + n E Pi [u i ] + i=1 +ζ N ( ) Λa c a Λ a c a a=1 (λ + ln E P0 [e ζ 1 (c N a=1(λ a Λ a )c n a i=1 u i ) n i=1 i (S i S i 1 )) ]

30 (Martingale) Weighted Monte-Carlo [PHL] Theorem There is no duality gap D λ = P λ. The supremum is attained by the optimal measure P given by dp dp 0 = where e (ζ ) 1 (c N a=1(λ a Λ a)c a n i=1 u i n i=1 i (S i S i 1 )) E P0 [e (ζ ) 1 (c N a=1(λ a Λ a)c n a i=1 u i n i=1 i (S i S i 1 )) ] ) ((ui ( )) 1 i n, ( i ( )) 1 i n, Λ a, Λ a, ζ infimum in D λ. achieves the

31 P : Semi-Infinite Linear Programming Approach Dual: inf (u i ( )) 1 i n,( i ( )) 2 i n,λ a 0,Λ a 0 subject to the constraints n n u i + i=1 i=2 i (S i S i 1 ) + a n i=1 E Pi [u i ] + a ( Λa Λ a ) ca c ( Λa c a Λ a c a ) Deltas i are decomposed over a finite-dimensional basis: i (S 0,, S i 1 ) = [ i ] b e b (S 0,, S i 1 ) b Similarly, European options with payoffs u i are decomposed over a finite set of call options: u i (S i ) = [u i ] b (S i K b ) +

32 P : Semi-Infinite Linear Programming Approach Leads to a semi-infinite linear program: U = min x R n c x A(S)x B(S) S (R + ) d Our algorithm will produce an upper bound D basis D = P

33 Dealing with constraints: Cutting-plane method Let G (R + ) d, G < be a given initial grid and (ɛ k ) a sequence of non-negative numbers converging to 0. Let TOL > 0 be a suitable convergence tolerance and set k = 0. 1 Solve the relaxed finite-dimensional LP: optimal solution x = x U min x R n c x A(S)x B(S) S G 2 Determine the constraint violation: δ = min S R d + A(S)x B(S) 3 If δ > TOL then stop. Otherwise add the constraints A(S)x B(S) < δ + ɛ k 4 Go to step 1.

34 Algorithm for the risk-neutral WMC: calibration and pricing 1 Simulate Monte-Carlo paths under the measure P 0. 2 Solve the non-linear programming problem (2) using for example a gradient-based optimization routine (Note that this problem is strictly convex and admits an unique solution. 3 The exotic option price with payoff c is given by D λ. Note that the optimal measure P as given by Equation (2) can be used to value any exotic options depending on (S 1,, S n ).

35 Pricing variance swap on an illiquid stock (1) Assumption: Diffusion VS = 2 T E[ln S T ]. Input: finite set of strikes (with K = 0). Dual: n min ν + ω i C(K i ) ; ν + (ω i ) i=1,...,n,ν i=1 n ω i (S K i ) + 2 T ln S, S R Input: Smile DAX 5/07/2011 T = 1.5Y, static replication: i=1 Strike range Lower Upper Mid [ ] [ ] [ ] [ ]

36 Illiquid Fx smile Input: Smile 1 & 2, ATM smile 3, call spread on 3. Dual: min ν + (ω j i ) j=1,2;i=1,...,n,ν,ω 3, ν + 2 n j=1 i=1 2 j=1 i=1 n ω j i Cj (K i ) + ω 3 C 3 (S0 3 ) + CS3 ω j i (Sj K j i )+ + ω 3 (S 2 S 3 0 S1 ) + + ( (S S 3 0 S1 ) + (S S 3 0 S1 ) +) (S 2 KS 1 ) + Fact: constraints are piecewise linear w.r.t. S 1, S 2 : Extremal points: prob. with a discrete support.

37 Illiquid Fx smile (1) σ ATM = 27 Call Spread: σ(0.95) = 25.5, σ(1.05) = Illiquid Fx Smile Axis Title Smile1 Smile2 Smile Axis Title

38 ( ) + Cliquet: S2 S 1 K Eurostock implied volatilities(2-feb-2010). t 1 = 1 year and t 2 = 1.5 years. Parameters for the Bergomi model: σ = 2.0, θ = 22.65%, k 1 = 4, k 2 = 0.125, ρ = 34.55%, ρ SX = 76.84%, ρ SX = 86.40% Upper LV Bergomi+LV Lower Bergomi

39 Asian option with monthly returns 1Y (1) Input: DAX 5/09/2011. Parameters for the Bergomi model: θ = 25%, k 1 = 8, k 2 = 0.3, ρ = 0%, ρ SX = 80%, ρ SX = 48%. LV 8.36% Bergomi % Bergomi+LV 8.71% Upper/Lower 9.42%/5.47% Minimal entropy martingale 8.32% WMC 8.84%/7.51% 6 calibrated on VS

40 Some references (1) Beiglböck, M., PHL and Penkner, F. : Model independent Bounds for Option Prices: A Mass Transport Approach. arxiv: , submitted (2011). Galichon, A., PHL, Touzi, N. : A stochastic control approach to no-arbitrage bounds given marginals, with an application to Lookback options, submitted. Guyon, J., Henry-Labordère, P. : Being particular about calibration, Risk magazine (2012). Guyon, J., Henry-Labordère, P. : Non-linear Methods in Quantitative Finance, CRC Chapman-Hall, to appear in PHL : Automated Option Pricing: Numerical methods, submitted (2012).

41 Some references (2) Villani, C. : Topics in Optimal Transportation, Graduate Studies in Mathematics (58), AMS(2003). Villani, C. : Limite de Champ Moyen, Cours DEA (2002), cvillani/cours/moyen.pdf