Functional Ito calculus. hedging of path-dependent options

Transcription

1 and hedging of path-dependent options Laboratoire de Probabilités et Modèles Aléatoires CNRS - Université de Paris VI-VII and Columbia University, New York

2 Background Hans Föllmer (1979) Calcul d Itô sans probabilités, proposed a pathwise approach to Ito calculus and derived a change of variable ( Ito ) formula for paths with finite quadratic variation along a sequence of subdvisions Bruno Dupire (2009) Functional Itô calculus, proposed a notion of functional derivative which is the correct notion of sensitivity for path-dependent options. We build on these ideas to develop a non-anticipative pathwise calculus for functionals defined on cadlag paths. This leads to a non-anticipative calculus for path-dependent functionals of a semimartingale, which is (in a precise sense) a non-anticipative equivalent of the Malliavin calculus.

3 References R. Cont & D Fournié (2010) A functional extension of the Ito formula, Comptes Rendus Acad. Sci., Vol. 348, R. Cont & D Fournié (2009) and stochastic integral representation of martingales, arxiv/math.pr. R. Cont & D Fournié (2010) Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, Vol 259. R Cont (2010) Pathwise computation of hedging strategies for path-dependent derivatives, Working Paper.

4 Framework Consider a R d -valued Ito process on (Ω, B, B t, P): X (t) = t 0 t μ(u)du + σ(u).dw u 0 μ integrable, σ square integrable B t -adapted processes. Quadratic variation process [X ](t) = t 0 t σ.σ(u)du = t 0 A(u)du D([0, T ], R d ) space of cadlag functions. F t = F X t+: natural filtration / history of X C 0 ([0, T ], R d ) space of continuous paths.

5 Functional notation For a path x D([0, T ], R d ), denote by x(t) R d the value of x at t x t = x [0,t] = (x(u), 0 u t) D([0, t], R d ) the restriction of x to [0, t]. We will also denote x t the function on [0, t] given by x t (u) = x(u) u < t x t (t) = x(t ) For a process X we shall similarly denote X (t) its value and X t = (X (u), 0 u t) its path on [0, t].

6 Path dependent functionals In stochastic analysis, statistics of processes and mathematical finance, one is interested in path-dependent functionals such as (weighted) averages along a path Y (t) = t 0 f (X (t))ρ(t)dt Quadratic variation and p-variation: t/n 1 Y (t) = lim X ( k n n ) X (k 1 n ) p k=1 Exponential functionals: Y (t) = exp(x (t) [X ](t)/2 ) Functionals of quadratic variation: e.g. variance swaps and volatility derivatives ([X ](t) K) +, t 0 f (X (t))d[x ] f (t, X (t), [X ] t )

7 We are interested in developing a non-anticipative differential calculus for such functionals of a semimartingale. 1 Ito s stochastic integral allows to define T 0 YdX where Y (t) may be a path-dependent functional of X 2 The Ito formula yields a stochastic integral representation for Y (t) = f (t, X (t)) where f is a (smooth) function A key step is to obtain an Ito-type change of variable formula for path-dependent functionals.

8 Dupire s pathwise derivatives Motivated by the computation of sensitivities for path-dependent options, B. Dupire (2009) introduced a notion of pathwise derivative for functionals defined on D([0, t], R d ). Dupire proposed an Ito formula for path-dependent functionals of the type Y (t) = F t ({X (u), 0 u t} where X is an Ito process and F t : D([0, t]) R is continuous in supremum norm. This applies to functionals like F t (x) = T 0 f (x(u))ρ(u)du but excludes most other examples. We explore the mathematical foundation of Dupire s construction and extend it to cover a large class of functionals including stochastic integrals and quadratic variation.

9 Outline We define pathwise derivatives for functionals of the type Y (t) = F t ({X (u), 0 u t}, {A(u), 0 u t}) = F t (X t, A t ) where A = t σ.σ and F t : D([0, t], R d ) D([0, t], S + d ) R represents the dependence on the path of X and its quadratic variation process. Using this pathwise derivative, we derive a functional change of variable formula which extends the Ito formula in two ways: it allows for path-dependence and for dependence with respect to the quadratic variation of X. This pathwise derivative admits a closure X on the space of square integrable stochastic integrals w.r.t. X, which is shown to be a stochastic derivative i.e. an inverse of the Ito stochastic integral. We derive a (constructive) martingale representation formula and an integration by parts formula for stochastic integrals.

10 Outline I: Pathwise calculus for non-anticipative functionals. II: An Ito formula for functionals of semimartingales. III: Weak derivatives and relation with Malliavin calculus. IV: Numerical computation of functional derivatives V: Functional Kolmogorov equations. Pricing equations for path-dependent options.

11 Functional representation of non-anticipative processes A process Y adapted to F t may be represented as a family of functionals Y (t,.) : Ω = D([0, T ], R d ) R with the property that Y (t,.) only depends on the path stopped at t: Y (t, ω) = Y (t, ω(. t) ) so ω [0,t] = ω [0,t] Y (t, ω) = Y (t, ω ) Denoting ω t = ω [0,t], we can thus represent Y as Y (t, ω) = F t (ω t )for some which is F t -measurable. F t : D([0, t], R d ) R

12 Non-anticipative functionals on the space of cadlag functions This motivates the following definition: Definition (Non-anticipative functional) A non-anticipative functional on the (canonical) path space Ω = D([0, T ], R d ) is a family F = (F t ) t [0,T ] where is F t -measurable. F t : D([0, t], R d ) R F = (F t ) t [0,T ] naturally induces a functional on the vector bundle t [0,T ] D([0, t], Rd ).

13 Functional representation of predictable processes An F t -predictable process Y may be represented as a family of functionals with the property We can thus represent Y as Y (t,.) : Ω = D([0, T ], R d ) R ω [0,t[ = ω [0,t[ Y (t, ω) = Y (t, ω ) Y (t, ω) = F t (ω t ) for some F t : D([0, t], R d ) R where ω t (u) = ω(u), u < t and ω t (t) = ω(t ). So: an F t -predictable Y can be represented as Y (t, ω) = F t (ω t ) for some non-anticipative functional F. Ex: integral functionals Y (t, ω) = t 0 g(ω(u))ρ(u)du

14 Functional representation of non-anticipative processes The previous examples of processes have a non-anticipative dependence in X and a predictable dependence on A since they only depend on [X ] =. 0 A(u)du. We will thus consider processes which may be represented as Y (t) = F t ({X (u), 0 u t}, {A(u), 0 u t}) = F t (X t, A t ) where the functional F t : D([0, t], R d ) D([0, t], S + d ) R represents the dependence of Y (t) on the path of X and A = t σ.σ and is predictable with respect to the 2nd variable : t, (x, v) D([0, t], R d ) D([0, t], S + d ), F t(x t, v t ) = F t (x t, v t ) F = (F t ) t [0,T ] may then be viewed as a functional on the vector bundle Υ = t [0,T ] D([0, t], Rd ) D([0, t], S + d ).

15 Horizontal extension of a path t

16 Horizontal extension of a path Let x D([0, T ] R d ), x t D([0, T ] R d ) its restriction to [0, t]. For h 0, the horizontal extension x t,h D([0, t + h], R d ) of x t to [0, t + h] is defined as x t,h (u) = x(u) u [0, t] ; x t,h (u) = x(t) u ]t, t + h]

17 d metric on Υ = t [0,T ] D([0, t], Rd ) D([0, t], S + d ) Extends the supremum norm to paths of different length. For T t = t + h t 0, (x, v) D([0, t], R d ) S t + and (x, v ) D([0, t + h], R d ) S + t+h d ( (x, v), (x, v ) ) = sup x t,h (u) x (u) u [0,t+h] + sup v t,h (u) v (u) + h u [0,t+h]

18 Continuity for non-anticipative functionals A non-anticipative functional F = (F t ) t [0,T ] is said to be continuous at fixed times if for all t [0, T [, F t : D([0, t], R d ) S t R is continuous w.r.t. the supremum norm. Definition (Left-continuous functionals) Define F l as the set of non-anticipative functionals F = (F t, t [0, T [) such that t [0, T [, ε > 0, (x, v) D([0, t], R d ) S t, η > 0, h [0, t], (x, v ) (x, v) D([0, t h], R d ) S t h, d ((x, v), (x, v )) < η F t (x, v) F t h (x, v ) < ε

19 Boundedness-preserving functionals We call a functional boundedness preserving if it is bounded on each bounded set of paths: Definition (Boundedness-preserving functionals) Define B([0, T )) as the set of non-anticipative functionals F on Υ([0, T ]) such that for every compact subset K of R d, every R > 0 and t 0 < T C K,R,t0 > 0, t t 0, (x, v) D([0, t], K) S t, sup v(s) R F t (x, v) C K,R,t0 s [0,t]

20 Measurability and continuity A non-anticipative functional F = (F t ) applied to X generates an F t adapted process Y (t) = F t ({X (u), 0 u t}, {A(u), 0 u t}) = F t (X t, A t ) Theorem Let (x, v) D([0, T ], R d ) D([0, T ], S + d ). If F F l, the path t F t (x t, v t ) is left-continuous. Y (t) = F t (X t, A t ) defines an optional process. If A is continuous, Y (t) = F t (X t, A t ) is a predictable process.

21 Horizontal derivative Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Definition (Horizontal derivative) We will say that the functional F = (F t ) t [0,T ] on Υ([0, T ]) is horizontally differentiable at (x, v) D([0, t], R d ) S t if D t F (x, v) = lim h 0 + F t+h (x t,h, v t,h ) F t (x t, v t ) h exists We will call (1) the horizontal derivative D t F of F at (x, v). DF = (D t F ) t [0,T ] defines a non-anticipative functional. If F t (x, v) = f (t, x(t)) with f C 1,1 ([0, T ] R d ) then D t F (x, v) = t f (t, x(t)).

22 Vertical perturbation of a path Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Figure: For e R d, the vertical perturbation xt e obtained by shifting the endpoint: xt e (u) = x(u) for u < t and xt e (t) = x(t) + e. of x t is the cadlag path

23 Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Definition (Dupire 2009) A non-anticipative functional F = (F t ) t [0,T [ is said to be vertically differentiable at (x, v) D([0, t]), R d ) D([0, t], S + d ) if R d R e F t (x e t, v t ) is differentiable at 0. Its gradient at 0 is called the vertical derivative of F t at (x, v) x F t (x, v) = ( i F t (x, v), i = 1..d) where F t (xt hei, v) F t (x, v) i F t (x, v) = lim h 0 h

24 Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Vertical derivative of a non-anticipative functional x F t (x, v).e is simply a directional (Gateaux) derivative in the direction of the indicator function 1 {t} e. Note that to compute x F t (x, v) we need to compute F outside C 0 : even if x C 0, x h t / C 0. x F t (x, v) is local in the sense that it is computed for t fixed and involves perturbating the endpoint of paths ending at t.

25 Spaces of differentiable functionals Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Definition (Spaces of differentiable functionals) For j, k 1 define C j,k b ([0, T ]) as the set of functionals F F r which are differentiable j times horizontally and k times vertically at all (x, v) D([0, t], R d ) S t +, t < T, with horizontal derivatives D m t F, m j continuous on D([0, T ]) S t for each t [0, T [ left-continuous vertical derivatives: n k, n xf F l. D m t F, n xf B([0, T ]). We can have F C 1,1 b ([0, T ]) while F t not Fréchet differentiable for any t [0, T ].

26 Examples of regular functionals Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Example Y = exp(x [X ]/2) = F (X, A) where F t (x t, v t ) = e x(t) 1 2 t 0 v(u)du F C 1, b with: D t F (x, v) = 1 2 v(t)f t(x, v) j xf t (x t, v t ) = F t (x t, v t )

27 Examples of regular functionals Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Example (Cylindrical functionals) For g C 0 (R d ), F t (x t, v t ) = [x(t) x(t n )] 1 t tn g(x(t 1 ), x(t 2 )..., x(t n )) is in C 1,2 b

28 Examples of regular functionals Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Example (Integrals w.r.t quadratic variation) For g C 0 (R d ), Y (t) = t 0 g(x (u))d[x ](u) = F t(x t, A t ) where F C 1, b, with: F t (x t, v t ) = t 0 g(x(u))v(u)du D t F (x t, v t ) = g(x(t))v(t) j xf t (x t, v t ) = 0

29 Obstructions to regularity Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Example (Jump of x at the current time) F t (x t, v t ) = x(t) x(t ) has regular pathwise derivatives: But F / F r D t F (x t, v t ) = 0 x F t (x t, v t ) = 1 F l. Example (Jump of x at a fixed time) F t (x t, v t ) = 1 t t0 (x(t 0 ) x(t 0 )) F F has horizontal and vertical derivatives at any order, but x F t (x t, v t ) = 1 t=t0 fails to be left (or right) continuous.

30 Obstructions to regularity Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Example (Maximum) F t (x t, v t ) = sup s t x(s) F F but is not vertically differentiable on {(x t, v t ) D([0, t], R d ) S t, x(t) = sup x(s)}. s t

31 Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Non-uniqueness of functional representation Take d = 1. F 0 (x t, v t ) F 1 t2 n (x t, v t ) = (lim n i=0 = t 0 v(u)du x( i+1 2 n ) x( i 2 n ) 2 ) 1 i+1 limn i t2n (x( 2 n ) x( i 2 n ))2 < F 2 (x t, v t ) = lim i + 1 x( n 2 n ) x( i 2 n ) 2 Δx(s) 2 }{{} s<t 1 V 2(x)< 1 J2(x)< V 2(x) }{{} J 2(x) Then F 0 t (X t, A t ) = F 1 t (X t, A t ) = F 2 t (X t, A t ) = [X ](t) Yet F 0 C 1,2 b but F 1, F 2 / F r.

32 Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Non-uniqueness of functional representation F 1, F 2 C 1,1 coincide on continuous paths t < T, (x, v) C 0 ([0, t], R d ) D([0, t], S + d ), F 1 t (x, v) = F 2 t (x, v) then P( t [0, T ], F 1 (X t, A t ) = F 2 (X t, A t ) ) = 1 Yet, x F depends on the values of F computed at discontinuous paths...

33 Horizontal derivative Vertical derivative of a functional Spaces of regular functionals Examples Obstructions to regularity Non-uniqueness of functional representation Derivatives of functionals defined on continuous paths Theorem If F 1, F 2 C 1,1 coincide on continuous paths t < T, (x, v) C 0 ([0, t], R d ) D([0, t], S + d ), Ft 1 (x, v) = Ft 2 (x, v) then their pathwise derivatives also coincide: t < T, (x, v) C 0 ([0, t], R d ) D([0, t], S + d ), x Ft 1 (x, v) = x Ft 2 (x, v), D t Ft 1 (x, v) = D t Ft 2 (x, v)

34 Quadratic variation for cadlag paths Föllmer (1979): f D([0, T ], R) is said to have finite quadratic variation along a subdivision π n = (t0 n <..tk(n) n = T ) if the measures: ξ n = k(n) 1 i=0 (f (t n i+1) f (t n i )) 2 δ t n i where δ t is the Dirac measure at t, converge vaguely to a Radon measure ξ on [0, T ] such that [f ](t) = ξ([0, t]) = [f ] c (t) + (Δf (s)) 2 0<s t where [f ] c is the continuous part of [f ]. [f ] is called the quadratic variation of f along the sequence (π n ).

35 Change of variable formula for cadlag paths Let (x, v) D([0, T ] R d ) D([0, T ] R n ) where x has finite quadratic variation along (π n ) and Denote v n (t) = sup x(t) x(t ) + v(t) v(t ) 0 t [0,T ] π n k(n) 1 i=0 x n (t) = k(n) 1 i=0 x(t i+1 )1 [ti,t i+1 )(t) + x(t )1 {T } (t) v(t i )1 [ti,t i+1 )(t) + v(t )1 {T } (t), h n i = t n i+1 t n i

36 for functionals Theorem (C. & Fournié (2009)) For any F C 1,2 ([0, T [), the Föllmer integral, defined as T 0 b k(n) 1 x F t (x t, v t )d π x := lim x F t n n i (x n,δx(tn i ) ti n, vt n i )(x(ti+1) n x(t n n i )) i=0 exists and F T (x T, v T ) F 0 (x 0, v 0 ) = T tr (t 2 xf t (x u, v u )d[x] c (u) ) + + u ]0,T ] T 0 T 0 D t F t (x u, v u )du x F t (x t, v t )d π x [F u (x u, v u ) F u (x u, v u ) x F u (x u, v u ).Δx(u)]

37 This pathwise formula implies a functional change of variable formula for semimartingales: Theorem () Let F C 1,2 b ([0, T [). For any t [0, T [, F t (X t, A t ) F 0 (X 0, A 0 ) = t 0 x F u (X u, A u ).dx (u) + t 0 t 0 D u F (X u, A u )du tr (t 2 xf u (X u, A u ) d[x ](u) ) a.s. In particular, Y (t) = F t (X t, A t ) is a semimartingale.

38 If F t (X t, A t ) = f (t, X (t)) where f C 1,2 ([0, T ] R d ) this reduces to the standard Ito formula. Y = F (X ) depends on F and its derivatives only via their values on continuous paths: Y can be reconstructed from the second-order jet of F on Υ c = t [0,T ] C 0([0, t], R d ) D([0, T ], S + d ) Υ.

39 Sketch of proof Consider first a cadlag piecewise constant process: n X (t) = 1 [tk,t k+1 [(t)φ k φ k F tk measurable k=1 Each path of X is a sequence of horizontal and vertical moves: X tk+1 = (X tk,h k ) φ k+1 φ k h k = t k+1 t k F tk+1 (X tk+1, A tk+1 ) F tk (X tk, A tk ) = F tk+1 (X tk+1, A tk+1 ) F tk+1 (X tk+1, A tk,h k )+ F tk+1 (X tk+1, A tk,h k ) F tk+1 (X tk,h k, A tk,h k )+ F tk+1 (X tk,h k, A tk,h k ) F tk (X tk, A tk ) vertical move horizontal move

40 Sketch of proof Horizontal step: fundamental theorem of calculus for φ(h) = F tk +h(x tk,h, A tk,h) F tk+1 (X tk,h k, A tk,h k ) F tk (X tk, A tk ) = φ(h k ) φ(0) = tk+1 t k D t F (X t, A t )dt frozen {}}{ Vertical step: apply Ito formula to ψ(u) = F tk+1 (Xt u k,h k, A tk,h k ) F tk+1 (X tk+1, A tk,h k ) F tk+1 (X tk,h k, A tk,h k ) = ψ(x (t k+1 ) X (t k )) ψ(0) = tk+1 t k x F t (X t, A tk,h k ).dx tr( 2 xf t (X t, A tk,h k )d[x ])

41 Sketch of proof General case: approximate X by a sequence of simple predictable processes n X with n X (0) = X (0): F T ( n X T ) F 0 (X 0 ) = T 0 T D t F ( n X t )dt T 0 0 X F ( n X t ).dx tr[ t 2 xf ( n X t ) A(t)] dt The C 1,2 b assumption on F implies that all derivatives involved in the expression are left continuous in d metric, which allows to control their convergence as n using dominated convergence + the dominated convergence theorem for stochastic integrals.

42 Definition (Vertical derivative of a process) Define C 1,2 b (X ) the set of processes Y which admit a representation in C 1,2 b : C 1,2 b (X ) = {Y, F C1,2 b ([0, T ]), Y (t) = F t(x t, A t ) a.s.} If det(a) > 0 a.s. then for Y C 1,2 b (X ), the predictable process: X Y (t) = x F t (X t, A t ) is uniquely defined up to an evanescent set, independently of the choice of F C 1,2 b. We call X Y the vertical derivative of Y with respect to X.

43 Vertical derivative for Brownian functionals In particular when X is a standard Brownian motion, A = I d : Definition Let W be a standard d-dimensional Brownian motion. For any Y C 1,2 b (W ) with representation Y (t) = F t(w t, t), the predictable process W Y (t) = x F t (W t, t) is uniquely defined up to an evanescent set, independently of the choice of the representation F C 1,2 b.

44 Consider now the case where X (t) = t 0 σ(t).dw (t) is a Brownian martingale. Consider an F T -measurable functional H = H(X (t), t [0, T ]) = H(X T ) with E[ H 2 ] < and define the martingale Y (t) = E[H F t ]. Theorem If Y C 1,2 b (X ) then Y (T ) = E[Y (T )] + T 0 X Y (t)dx (t) = E[H] + T 0 X Y (t)σ(t)dw (t) This is a non-anticipative version of Clark s formula (under weaker assumptions).

45 A hedging formula for path-dependent options Consider now a (discounted) asset price process S(t) = t 0 σ(t).dw (t) assumed to be a square-integrable martingale under the pricing measure P. Let H = H(S(t), t [0, T ]) with E[ H 2 ] < be a path-dependent payoff. The price at date t is then Y (t) = E[H F t ]. Theorem (Hedging formula) If Y C 1,2 b (S) then H = E[H] + T 0 SY (t)ds(t) P a.s. The hedging strategy for H is given by the vertical derivative of the option price with respect to S:

46 A hedging formula for path-dependent options So the hedging strategy for H may be computed pathwise as φ(t) = X Y (t, X t (ω)) = lim h 0 Y (t, X h t (ω)) Y (t, X t (ω)) h where Y (t, X t (ω)) is the option price at date t in the scenario ω. Y (t, X h t (ω)) is the option price at date t in the scenario obtained from ω by moving up the current price ( bumping the price) by h. So, the usual bump and recompute sensitivity actually corresponds to.. the hedge ratio!

47 Pathwise computation of hedge ratios Consider for example the case where X is a (component of a ) multivariate diffusion. Then we can use a numerical scheme (ex: Euler scheme) to simulate X. Let n X be the solution of a n-step Euler scheme and Ŷn a Monte Carlo estimator of Y obtained using n X. Compute the Monte Carlo estimator Ŷn(t, n X h t (ω)) Bump the endpoint by h. Recompute the Monte Carlo estimator Ŷn(t, n X h t (ω)) (with the same simulated paths) Approximate the hedging strategy by ˆφ n (t, ω) := Ŷn(t, n X h t (ω)) Ŷn(t, n X h t (ω)) h

48 Numerical simulation of hedge ratios ˆφ n (t, ω) Ŷn(t, n X h t (ω)) Ŷn(t, n X h t (ω)) h For a general C 1,2 b (S) path-dependent claim, with a few regularity assumptions 1/2 > ε > 0, n 1/2 ε ˆφ n (t) φ(t) 0 P a.s. This rate is attained for h = cn 1/4+ε/2 By exploiting the structure further (Asian options, lookback options,...) one can greatly improve this rate.

49 An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative A non-anticipative integration by parts formula I 2 (X ) = {. 0 φdx, φ F t adapted, E[ T 0 φ(t) 2 d[x ](t)] < } Theorem Let Y C 1,2 b (X ) be a (P, (F t))-martingale with Y (0) = 0 and φ an F t adapted process with E[ T 0 φ(t) 2 d[x ](t)] <. Then E ( Y (T ) T 0 ) φdx = E ( T ) X Y.φd[X ] This allows to extend the functional Ito formula to the closure of (X ) I2 (X ) wrt to the norm C 1,2 b E Y (T ) 2 = E[ T 0 0 X Y (t) 2 d[x ](t) ]

50 Martingale Sobolev space An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative Definition (Martingale Sobolev space) Define W 1,2 (X ) as the closure in I 2 (X ) of D(X ) = C 1,2 b (X ) I2 (X ). Lemma { X Y, Y D(X )} is dense in L 2 (X ) and W 1,2 (X ) = {. 0 φdx, E T 0 φ 2 d[x ] < }. So W 1,2 (X )=all square-integrable integrals with respect to X.

51 Weak derivative An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative Theorem (Weak derivative on W 1,2 (X )) The vertical derivative X : D(X ) L 2 (X ) is closable on W 1,2 (X ). Its closure defines a bijective isometry X : W 1,2 (X ) L 2 (X ) T 0 φ.dx φ characterized by the following integration by parts formula: for Y W 1,2 (X ), X Y is the unique element of L 2 (X ) such that Z D(X ), [ T E[Y (T )Z(T )] = E 0 ] X Y (t) X Z(t)d[X ](t).

52 Computation of the weak derivative An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative For D(X ) = C 1,2 b (X ) I2 (X ), the weak derivative may be computed pathwise For Y W 1,2 (X ), X Y = lim n X Y n where Y n D(X ) is an approximating sequence with E Y n (T ) Y (T ) 2 n 0 An example of such an approximation is given by a Monte Carlo estimator Ŷ n (computed for example from an Euler scheme for X ). X Y (t, X t (ω)) = lim lim Ŷ n (t, Xt h (ω)) Ŷ n (t, X t (ω)) n h 0 h In practice one may compute instead ˆ X Y (t, X t (ω)) = Ŷ n (t, X h(n) t (ω)) Ŷ n (t, X t (ω)) h(n)

53 Relation with Malliavin derivative An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative Consider the case where X = W. Then for Y W 1,2 (W ) T Y (T ) = E[Y (T )] + W Y (t)dw (t) 0 If H = Y (T ) is Malliavin-differentiable e.g. H = Y (T ) D 1,1 then the Clark-Haussmann-Ocone formula implies T Y (T ) = E[Y (T )] + p E[D t H F t ]dw (t) 0 where D is the Malliavin derivative.

54 Relation with Malliavin derivative An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative Theorem (Intertwining formula) Let Y be a (P, (F t ) t [0,T ] ) martingale. If Y C 1,2 (W ) and Y (T ) = H D 1,2 then E[D t H F t ] = ( W Y )(t) dt dp a.e. i.e. the conditional expectation operator intertwines W and D: E[D t H F t ] = W (E[H F t ]) dt dp a.e.

55 Relation with Malliavin derivative An integration by parts formula Martingale Sobolev space Weak derivative Relation with Malliavin derivative The following diagram is commutative, in the sense of dt dp almost everywhere equality: W 1,2 (W ) W L 2 (W ) (E[. F t]) t [0,T ] (E[. F t]) t [0,T ] D 1,2 D L 2 ([0, T ] Ω) Note however that X may be constructed for any Ito process X and its construction does not involve Gaussian properties of X.

56 Functional equation for martingales Functional characterization of martingales Example: a weighted variance swap Consider now a semimartingale X whose characteristics are left-continuous functionals: dx (t) = b t (X t, A t )dt + σ t (X t, A t )dw (t) where b, σ are non-anticipative functionals on Ω with values in R d -valued (resp. R d n ) whose coordinates are in F l. Consider the topological support of the law of (X, A) in (C 0 ([0, T ], R d ) S T,. ): supp(x, A) = {(x, v), any neighborhood V of (x, v), P((X, A) V ) > 0}

57 Functional characterization of martingales Example: a weighted variance swap A functional Kolmogorov equation for martingales Theorem Let F C 1,2 b. Then Y (t) = F t(x t, A t ) is a local martingale if and only if F satisfies D t F (x t, v t ) + b t (x t, v t ) x F t (x t, v t ) tr[ 2 xf (x t, v t )σ t t σ t (x t, v t )] = 0, for (x, v) supp(x, A). We call such functionals X harmonic functionals.

58 Functional characterization of martingales Example: a weighted variance swap Structure equation for Brownian martingales In particular when X = W is a d-dimensional Wiener process, we obtain a characterization of regular Brownian local martingales: Theorem Let F C 1,2 b. Then Y (t) = F t(w t ) is a local martingale on [0, T ] if and only if t [0, T ], (x, v) C 0 ([0, T ], R d ), D t F (x t ) tr ( 2 xf (x t ) ) = 0.

59 Functional characterization of martingales Example: a weighted variance swap Theorem (Uniqueness of solutions) Let h be a continuous functional on (C 0 ([0, T ]) S T,. ). Any solution F C 1,2 b of the functional equation (1), verifying (x, v) C 0 ([0, T ]) S T, D t F (x t, v t )+ b t (x t, v t ) x F t (x t, v t ) tr[ 2 xf (x t, v t )σ t t σ t (x t, v t )] = 0 F T (x, v) = h(x, v), E[sup t [0,T ] F t (X t, A t ) ] < is uniquely defined on the topological support supp(x, A) of (X, A): if F 1, F 2 C 1,2 ([0, T ]) are two solutions then b (x, v) supp(x, A), t [0, T ], F 1 t (x t, v t ) = F 2 t (x t, v t ).

60 A universal pricing equation Functional characterization of martingales Example: a weighted variance swap Theorem (Pricing equation for path-dependent options) Let F C 1,2 b, F t(x t, A t ) = E[H F t ] then F is the unique solution of the pricing equation for (x, v) supp(x, A). D t F (x t, v t ) + b t (x t, v t ) x F t (x t, v t ) tr[ 2 xf (x t, v t )σ t t σ t (x t, v t )] = 0, This equations implies all known PDEs for path-dependent options: barrier, Asian, lookback,...but also leads to new pricing equations for other examples.

61 A diffusion example Functional characterization of martingales Example: a weighted variance swap Consider a scalar diffusion dx (t) = b(t, X (t))dt + σ(t, X (t))dw (t) X (0) = x 0 defined as the solution P x 0 D([0, T ], R d ) for of the martingale problem on L t f = 1 2 σ2 (t, x) 2 x f (t, x) + b(t, x) x f (t, x) where b and σ a > 0 are continuous and bounded functions. By the Stroock-Varadhan support theorem, the topological support of (X, A) under P x 0 is {(x, (σ 2 (t, x(t))) t [0,T ] ) x C 0 (R d, [0, T ]), x(0) = x 0 }.

62 Weighted variance swap Functional characterization of martingales Example: a weighted variance swap A weighted variance swap with weight function g C b ([0, T ] R d ), we are interested in computing T Y (t) = E[ g(t, X (t))d[x ](t) F t ] 0 If Y = F (X, A) with F C 1,2 b ([0, T ]) then F is X -harmonic and solves the functional Kolmogorov equation. Taking conditional expectations and using the Markov property of X : F t (x t, v t ) = t 0 g(u, x(u))v(u)du + f (t, x(t))

63 An example Functional characterization of martingales Example: a weighted variance swap F t (x t, v t ) = t 0 g(u, x(u))v(u)du + f (t, x(t)) solves the functional Kolmogorov eq. iff f solves 1 2 σ2 (t, x) 2 x f (t, x) + b(t, x) x f (t, x) + t f (t, x) = g(t, x)σ 2 (t, x) with terminal condition f (T, x) = 0 so that Y (T ) = F T (X T, A T ). On the other hand, the (unique) C 1,2 solution of this PDE defines a unique C 1,2 ([0, T ]) functional on supp(x,a). b

64 Weighted variance swap Functional characterization of martingales Example: a weighted variance swap Applying now the Ito formula to f (t, X (t)) we obtain that the hedging strategy is given by φ(t) = f (t, x) x where f solves the PDE with source term: 1 2 σ2 (t, x) 2 x f (t, x) + b(t, x) x f (t, x) + t f (t, x) = g(t, x)σ 2 (t, x) with terminal condition f (T, x) = 0

65 Extensions and applications Functional characterization of martingales Example: a weighted variance swap The result can be extended to discontinuous functionals of cadlag processes i.e. Y and X can both have jumps. The result can be localized using stopping times: important for applying to functionals involving stopped processes/ exit times. Pathwise maximum principle for non-markovian control problems. Infinite-dimensional extensions. θ Γ tradeoff for path-dependent derivatives.

66 References Functional characterization of martingales Example: a weighted variance swap R. Cont & D Fournié (2010) A functional extension of the Ito formula, Comptes Rendus Acad. Sci. Paris, Ser. I, Vol. 348, 51-67, Jan R. Cont & D Fournié (2010) and stochastic integral representation of martingales, arxiv.org/math.pr. R. Cont & D Fournié (2010) Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, Vol 259. R Cont (2010) Pathwise computation of hedging strategies for path-dependent derivatives, Working Paper.