Non-anticipative functional calculus and applications · 2011-03-25 · Functional calculus: horizontal and vertical derivatives. A pathwise change of variable formula for functionals

Motivation: pricing and hedging of derivativesPathwise calculus for non-anticipative flows

Functional change of variable formulasFunctional Ito calculus

Martingale representation and hedging formulasExtensions

Functional equations for martingales

Non-anticipative functional calculus andapplications

Spring School on Stochastic Processes, Thuringia, March 2011

Rama Cont

Laboratoire de Probabilites et Modeles AleatoiresCNRS -Universite de Paris VI

andColumbia University, New York

Joint work with: David Fournie (Columbia University)

Rama Cont Functional Ito calculus





Outline

Motivation: path-dependent derivatives.Adapted processes as non-anticipative flows.Functional calculus: horizontal and vertical derivatives.A pathwise change of variable formula for functionals.A functional Ito formula for semimartingales.Martingale representation formula I: regular case.A universal hedging formula for path-dependent optionsWeak derivative. Integration by parts formula.Martingale representation formula II: general case.Numerical computation of hedging strategies.Martingales as solutions to functional equations.A universal pricing equation for path-dependent options.Example: Asian options.






References

Dupire, B. (2009) Functional Ito calculus, SSRN Preprint.

R. Cont & D Fournie (2010) A functional extension of the Itoformula, Comptes Rendus Acad. Sci., Vol. 348, 57–61.

R. Cont & D Fournie (2009) Functional Ito formula andstochastic integral representation of martingales,arxiv/math.PR.

R. Cont & D Fournie (2010) Change of variable formulas fornon-anticipative functionals on path space, Journal ofFunctional Analysis, Vol 259, 1043–1072.

R. Cont (2010) Martingale representation formulas for Poissonrandom measures, Working Paper.

R Cont (2010) Pathwise computation of hedging strategies forpath-dependent derivatives, Working Paper.






Motivation: modeling of derivative securities

Mathematical models of financial markets model uncertainty byrepresenting future evolution of prices in terms of a filteredscenario space (Ω,ℱ ,ℱt)t∈[0,T ]) and an ℱt−adapted price process

S : [0,T ]× Ω 7→ ℝd where

Ω represents the set of market scenarios

ℱ represents the set of market events (which we can makesense of)

ℱt represents the information revealed at date t, and

S(t, !) = (S1(t, !), ..,Sd(t, !)) represents the prices ofsecurities at date t in scenario !.

Specifying a probability measure ℙ gives a filtered probability spacebut there is often no consensus about the specification of ℙ: modeluncertainty.






Contingent claims

A contingent claim or derivative security on S is given by a(ℱT -measurable) payoff H : Ω 7→ ℝ

Call option: H(!) = (S(T , !)− K )+

Asian option: H(!) = ( 1T

∫ T0 S(t, !) dt − K )+

Barrier option: knockout call

H(!) = 1maxt∈[0,T ] S(t,!)<B(S(T , !)− K )+

Variance swap: with daily monitoring tk = kΔt

H(!) =N∑

k=1

∣ ln(S(tk+1

S(tk))∣2 − K

Option on realized variance:H(!) = (

∑Nk=1 ∣ ln(S(tk+1

S(tk ) )∣2 − K )+






Contingent claims

Some questions:

Pricing: what is the fair value F (t, !) at date t of acontingent claim with payoff H?

Hedging: can the option be hedged by trading in marketinstruments? What is the Profit/Loss (P&L) of a givenhedging strategy?

Sensitivity analysis: how do changes in market variablesand parameters affect the value of the claim?

The main issue for financial applications is to give computableanswers, not just existence/ uniqueness results.






Non path-dependent options in diffusion models

Diffusion models provide (computable) answers to all thesequestions, for non-path-dependent options, hence their popularity.If the underlying asset is given by a diffusion

dSt = �(t)S(t)dt + S(t)�(t,St)dW (t) under ℙ

then under some technical conditions on �, �(., .),

The fair value of an option with payoff H = h(S(T )) isf (t,S(t)) where f solves the valuation PDE

∂t f + r(t)x∂x f +�2(t, x)x2

2∂2xx f = r(t)f (t, x)

with terminal condition f (T , x) = h(x).






Delta hedging: The option may be hedged by aself-financing strategy where the position in the underlyingequal to the sensitivity (’delta’) of the option:

h(S(T )) = f (0, S(0)) +

∫ T

0

∂

∂xf (t,S(t)) dS(t)

+

∫ T

0[f (t, S(t))− S(t)

∂

∂xf (t, S(t))]r(t)dt ℙ− a.s.

”Robustness”: a hedging strategy (�t) computed in a(misspecified) Black-Scholes model with volatility �0 leads toa Profit/Loss given by

∫ T

0

�20(t,S(t))− �2(t, S(t))

2S(t)2e

∫ Tt r(s)ds

Γ(t)︷︸︸︷∂2xx f (t,S(t)) dt






Path-dependence:Do these results generalize to path-dependent options and/ornon-Markovian models?

Robustness to uncertainty:These results only make use of ℙ through its ’equivalenceclass’ i.e. its null sets. In actual situations there is noconsensus on the level of volatility, expected return etc. Howfar can one go without actually specifying a probabilitymeasure ℙ?






Sensitivity analysis of derivative securities

Consider a path-dependent derivative whose value at time t inmarket scenario ! ∈ D([0,T ],ℝd) is given by Ft(!) whereFt : D([0,T ],ℝ) 7→ ℝ.Practitioners often perform sensitivity analysis of derivatives by“bumping”/perturbating a variable, repricing the derivative andtaking the difference.For example, in the case of the “delta” of a path-dependent option,this amounts adding a jump/shift of size � today to the currentpath ! and recomputing the price Ft in the new path ! + �1[t,T ]

Ft(! + �1[t,T ])− Ft(!)

�






Sensitivity analysis of derivative securities

For non-path dependent options Ft(!) = f (t, !(t)) and thisdefinition is justified by the Black-Scholes delta-hedging rule:

Ft(! + �1[t,T ])− Ft(!)

�

�→0→ ∂f

∂S(t, S)

For a path-dependent option, the value at t depends on the path !on [0, t]: it is not just a function of the current price !(t) but afunctional of ! ∈ D([0, t],ℝ), where D([0, t],ℝ) is the set of(possibly discontinuous) paths which are right continuous with leftlimits.






Dupire’s functional calculus

Bruno Dupire (2009) formalized this notion and defines, for afunctional F : [0,T ]× D([0,T ],ℝ) 7→ ℝ defined on cadlag paths,

∇!Ft(!) = lim�→0

Ft(! + �1[t,T ])− Ft(!)

�

Dupire argues that this is the correct hedge ratio forpath-dependent options: if the option price F is twice differentiablein the functional sense and F ,∇!F ,∇2

!F are continuous insupremum norm, then

FT = E [FT ] +

∫ T

0∇!Ft .dSt






Summary

Dupire’s arguments apply to integral functionalsFt(!) =

∫ t0 g(!(t))dt but not to stochastic integrals or functionals

involving quadratic variation.We show that these ideas can be extended, in a mathematicallyrigorous fashion, to all square integrable functionals. To do so, wedevelop a non-anticipative pathwise calculus for suchfunctionals defined on cadlag paths.This leads to a non-anticipative calculus for path-dependentfunctionals of a semimartingale, which is (in a precise sense) a“non-anticipative” version of the Malliavin calculus.As an application, we obtain a martingale representation formulafor square-integrable martingales and a universal hedging formulafor path-dependent contingent claims.






Framework

Consider a ℝd -valued cadlag semimartingale X on a filteredprobability space (Ω,ℬ,ℬt ,ℙ)

Denote ℱt = �(ℱXt

⋁N (ℙ)) the (ℙ-completed) natural

filtration of X , assumed right-continuous: ℱt = ℱt+

D([0,T ],ℝd) space of cadlag (right continuous with leftlimits) functions.

C0([0,T ],ℝd) space of continuous paths.

Example: any cadlag Feller process verifies these assumptions.






Functional notation

For a path ! ∈ D([0,T ],ℝd), denote by

!(t) ∈ ℝd the value of ! at t

!t = !∣ [0,t] = (!(u), 0 ≤ u ≤ t) ∈ D([0, t],ℝd) therestriction of ! to [0, t].

We will also denote !t− the function on [0, t] given by

!t−(u) = !(u) u < t !t−(t) = !(t−)

For a process X we shall similarly denote

X (t) its value and

Xt = (X (u), 0 ≤ u ≤ t) its path on [0, t].






Outline

1 Functional representation of stochastic processes.

2 Pathwise functional calculus for non-anticipative flows.

3 Ito formula for functionals of semimartingales.

4 Weak functional derivatives and martingale representationformulas. Relation with Malliavin calculus.

5 Numerical computation of martingale representations

6 Functional Kolmogorov equations. Pricing equations forpath-dependent options.






Functional representation of non-anticipative processesHorizontal derivativeVertical derivative of a functionalSpaces of regular flowsExamplesObstructions to regularityNon-uniqueness of functional representation

Functional representation of non-anticipative processes

(Doob 1956): A process Y adapted to ℱt may be represented as afamily of functionals

Y (t, .) : Ω = D([0, t],ℝd) 7→ ℝ

with the property that Y (t, .) only depends on the path stopped att: Y (t, !) = Y (t, !(. ∧ t) ) so

!∣[0,t] = !′∣[0,t] ⇒ Y (t, !) = Y (t, !′)

So there exists a ℱt-measurable map Ft : D([0, t],ℝd)→ ℝ suchthat

Y (t, !) = Ft(!t) ℙ− a.s. where !t = !∣[0,t]







Non-anticipative flows on path space

This motivates the following definition:

Definition (Non-anticipative functional)

A non-anticipative flow on (D([0,T ],ℝd), (ℱt)t≥0) is a familyF = (Ft)t∈[0,T ] where

Ft : D([0, t],ℝd)→ ℝ

is ℱt-measurable.

F = (Ft)t∈[0,T ] may be viewed as a functional on the vector bundle∪t∈[0,T ] D([0, t],ℝd).







Functional representation of non-anticipative processes

Any process Y adapted to ℱt = ℱXt may be represented as

Y (t) = Ft({X (u), 0 ≤ u ≤ t}) = Ft(Xt)

where the functional Ft : D([0, t],ℝd)→ ℝ represents thedependence of Y (t) on the path of X on [0, t].F = (Ft)t∈[0,T ] may then be viewed as a functional on the vector

bundle Υ([0,T ]) =∪

t∈[0,T ] D([0, t],ℝd).







Horizontal extension of a path

0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

2

2.5

3

t







Horizontal extension of a path

Let ! ∈ D([0,T ]×Rd), !t ∈ D([0, t]×Rd) its restriction to [0, t].For h ≥ 0, the horizontal extension !t,h ∈ D([0, t + h],ℝd) of !t

to [0, t + h] is defined as

!t,h(u) = !(u) u ∈ [0, t] ; !t,h(u) = !(t) u ∈]t, t + h]







A distance between paths on different time intervals

For T ≥ t ′ = t + h ≥ t ≥ 0, ! ∈ D([0, t],ℝd) and!′ ∈ D([0, t + h],ℝd)

d∞( !, !′ ) = supu∈[0,t+h]

∣!t,h(u)− !′(u)∣+ h

defines a metric on Υ([0,T ]) =∪

t∈[0,T ] D([0, t],ℝd).

On D([0, t],ℝd), d∞ coincides with the supremum norm.







Continuity for non-anticipative flows

A non-anticipative flow F = (Ft)t∈[0,T ] is said to be continuousat fixed times if for all t ∈ [0,T [, the functional

Ft : D([0, t],ℝd)→ ℝ

is continuous w.r.t. the supremum norm.

Definition (Left-continuous flows)

Define ℂ0,0l as the set of non-anticipative flows F = (Ft , t ∈ [0,T [)

which are continuous at fixed times and

∀t ∈ [0,T [, ∀� > 0, ∀! ∈ D([0, t],ℝd),

∃� > 0, ∀h ∈ [0, t], ∀!′ ∈ D([0, t − h],ℝd),

d∞(!, !′) < � ⇒ ∣Ft(!)− Ft−h(!′)∣ < �







Boundedness-preserving flows

We call a flow “boundedness preserving” if it is bounded on eachbounded set of paths:

Definition (Boundedness-preserving flows)

Define B([0,T )) as the set of non-anticipative flows F onΥ([0,T ]) such that for every compact subset K of ℝd , everyR > 0 and t0 < T

∃CK ,R,t0 > 0, ∀t ≤ t0, ∀! ∈ D([0, t],K ),

sups∈[0,t]

∣v(s)∣ ≤ R ⇒ ∣Ft(!)∣ ≤ CK ,R,t0







Measurability and continuity

A non-anticipative flow F = (Ft) applied to X generates anℱt−adapted process

Y (t) = Ft({X (u), 0 ≤ u ≤ t}) = Ft(Xt)

Theorem

Let ! ∈ D([0,T ],ℝd). If F ∈ ℂ0,0l ,

the path t 7→ Ft(!t−) is left-continuous.

Y (t) = Ft(Xt) defines a predictable process.







Definition (Horizontal derivative)

We say that a non-anticipative flow F = (Ft)t∈[0,T ] is horizontally

differentiable at ! ∈ D([0, t],ℝd) if

DtF (!) = limh→0+

Ft+h(!t,h)− Ft(!)

hexists

We will call DtF (!) the horizontal derivative DtF of F at !.

By right continuity of (ℱt), DtF is ℱt-measurable:DF = (DtF )t∈[0,T ] defines a non-anticipative flow.

If Ft(!) = f (t, !(t)) with f ∈ C 1,1([0,T ]× ℝd) thenDtF (!) = ∂t f (t, !(t)).







Vertical perturbation of a path

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

2.5

3

Figure: For e ∈ ℝd , the vertical perturbation !et of !t is the cadlag path

obtained by shifting the endpoint:!et (u) = !(u) for u < t and !e

t (t) = !(t) + e.







Definition

A non-anticipative flow F = (Ft)t∈[0,T [ is said to be vertically

differentiable at ! ∈ D([0, t]),ℝd) if

ℝd 7→ ℝe → Ft(!t + e1t)

is differentiable at 0. Its gradient at 0 is called the verticalderivative of Ft at !:

∇!Ft (!) = (∂iFt(!), i = 1..d) where

∂iFt(!) = limh→0

Ft(!heit )− Ft(!)

h







Vertical derivative of a flow

∇!Ft (!).e is simply a directional derivative in the directionof the indicator function 1{t}e.

Note that to compute ∇!Ft (!) we need to compute Foutside C0: even if ! ∈ C0, !h

t /∈ C0.

∇!Ft (!) is ’local’ in the sense that it is computed for t fixedand involves perturbing the endpoint of paths ending at t.

∇!F = (∇!Ft)t∈[0,T ] is a non-anticipative flow.

This derivative was introduce by B. Dupire to compute thesensitivity of an option price to its underlying.







Spaces of differentiable flows

Definition (Spaces of differentiable functionals)

For j , k ≥ 1 define ℂj ,kb ([0,T ]) as the set of functionals F ∈ ℂ0,0

r

which are differentiable j times horizontally and k times verticallyat all ! ∈ D([0, t],ℝd), t < T , with

horizontal derivatives Dmt F ,m ≤ j continuous on D([0,T ]) for

each t ∈ [0,T [

left-continuous vertical derivatives: ∀n ≤ k ,∇n!F ∈ ℂ0,0

l .

Dmt F ,∇n

!F ∈ B([0,T ]).

We can have F ∈ ℂ1,1b ([0,T ]) while Ft not Frechet differentiable

for any t ∈ [0,T ].







Examples of regular functionals

Example (Cylindrical functionals)

For g ∈ C 0(ℝd), h ∈ C k(ℝd) with h(0) = 0. Then

Ft(!) = h (!(t)− !(tn−)) 1t≥tn g(!(t1−), !(t2−)..., !(tn−))

is in ℂ1,kb and

DtF (!) = 0, and ∀j = 1..k,

∇j!Ft(!) = h(j) (!(t)− !(tn−)) 1t≥tng (!(t1−), !(t2−)..., !(tn−))







Examples of regular functionals

Example (Integral functionals)

For g ∈ C0(ℝd), Y (t) =∫ t

0 g(X (u))�(u)du = Ft(Xt) where

Ft(!) =

∫ t

0g(!(u))�(u)du

F ∈ ℂ1,∞b , with:

DtF (!) = g(!(t))�(t) ∇j!Ft(!) = 0







Obstructions to regularity

Example (Jump of x at the current time)

Ft(!) = !(t)− !(t−) has regular pathwise derivatives:

DtF (!) = 0 ∇!Ft(!) = 1

But F /∈ ℂ0,0r ∪ ℂ0,0

l .

Example (Jump of x at a fixed time)

Ft(!) = 1t≥t0(!(t0)− !(t0−))F ∈ ℂ0,0 has horizontal and vertical derivatives at any order, but∇!Ft(!) = 1t=t0 fails to be left (or right) continuous.







Obstructions to regularity

Example (Maximum)

Ft(!) = sups≤t !(s)F ∈ ℂ0,0 but is not vertically differentiable on

{! ∈ D([0, t],ℝd), !(t) = sups≤t

!(s)}.







Non-uniqueness of functional representation

Given a process Y (say, the value of a path dependent optionIf F 1,F 2 ∈ ℂ1,1 coincide on continuous paths

∀t < T , ∀! ∈ C0([0, t],ℝd),

F 1t (!) = F 2

t (!)

thenℙ(∀t ∈ [0,T ],F 1(Xt) = F 2(Xt) ) = 1

Yet, ∇!F depends on the values of F computed at discontinuouspaths...







Derivatives of flows defined on continuous paths

Theorem

If F 1,F 2 ∈ ℂ1,1 coincide on continuous paths

∀t < T , ∀! ∈ C0([0, t],ℝd),F 1t (!t) = F 2

t (!t)

then their pathwise derivatives also coincide:

∀t < T , ∀! ∈ C0([0, t],ℝd),

∇!F 1t (!t) = ∇!F 2

t (!t), DtF1t (!t) = DtF

2t (!t)






Paths of finite quadratic variationA pathwise change of variable formula for functionalsIto-Follmer integral

Quadratic variation for cadlag paths [Follmer (1979)]

x ∈ D([0,T ],ℝ) has finite quadratic variation along the sequence

of partitions �n = (0 = tn0 < ..tk(n)n = T ) if the discrete measures

�n =

k(n)−1∑i=0

(x(tni+1)− x(tni ))2�tni

converge weakly to some Radon measure � on [0,T ]:

∀f ∈ C 0b (ℝ),

k(n)−1∑i=0

(x(tni+1)− x(tni ))2f (tni )n→∞→

∫ T

0f d�

In particular

k(n)−1∑i=0

(x(tni+1)− x(tni ))2 n→∞→ [x ](t) := �([0, t])







Quadratic variation for cadlag paths

Denote Q([0,T ], �) the set of x ∈ D([0,T ],ℝd) with finitequadratic variation with respect to the partition � = (�n)n≥1.For x ∈ Q([0,T ], �), the increasing function

k(n)−1∑i=0

(x(tni+1)− x(tni ))2 n→∞→ [x ](t) := �([0, t])

is called the quadratic variation of x along the partition (�n) andhas Lebesgue decomposition

[x ](t) = [x ]c(t) +∑

0<s≤t(Δx(s))2

where [x ]c is the continuous part of [x ].Rama Cont Functional Ito calculus






Quadratic variation for cadlag paths

Similarly x = (x1, ..., xd) ∈ D([0,T ],ℝd) is said to have finite

quadratic variation along �n = (0 = tn0 < ..tk(n)n = T ) if x i , x i + x j

have finite quadratic variation along �n for i , j = 1..d . Then

k(n)−1∑i=0

t(x(tni+1)− x(tni )).(x(tni+1)− x(tni ))n→∞→ [x ](t) ∈ Sym+(d × d)

is an increasing function with values in Sym+(d × d) and[x ]ij = ([x i + x j ]− [x i ]− [x j ])/2.







Processes with finite quadratic variation

For a stochastic process Z : [0,T ]× Ω 7→ ℝd , define

Ω�(Z ) = {! ∈ Ω, Z (., !) ∈ Q([0,T ], �)}.

Wiener process: if W is a Wiener process under ℙ, then forany sequence of partitions � with ∣�n∣ → 0, the paths of Wlie in Q([0,T ], �) with probability 1:

ℙ(Ω�(W )) = 1 and ∀! ∈ Ω�(W ), [W (., !)](t) = t.








Fractional Brownian motion: if W H is fractional Brownianmotion with Hurst index H ∈ (0.5, 1) then for any sequence ofpartitions � with ∣�n∣ → 0, the paths of W H lie inQ([0,T ], �) with probability 1:

ℙ(Ω�(W H)) = 1 and ∀! ∈ Ω�(W H), [W (., !)](t) = 0.

(See e.g. Zahle 1998)

Levy process: if L is a Levy process with triplet (b,A, �) thenfor any sequence of partitions � with ∣�n∣ → 0, ℙ(Ω�(L)) = 1and

∀! ∈ Ω�(L), [L(., !)](t) = tA +∑

s∈[0,t]

∣X (s, !)− X (s−, !)∣2.








Theorem (Follmer (1981))

Let S be a semimartingale on (Ω,ℱ ,ℙ,F = (ℱt)t≥0), T > 0.There exists a sequence of (random) partitions � = (�n)n≥1 of[0,T ], composed of F-stopping times, with ∣�n∣ → 0 a.s., suchthat the paths of S lie in Q([0,T ], �) with probability 1:

ℙ({! ∈ Ω, S(., !) ∈ Q([0,T ], �(!))}) = 1.

Proof: take the dyadic partition tkn = kT/2n, k = 0..2n. Since∑(S(tnk+1)− S(tnk ))2 → [S ]T in probability, there exists a

subsequence �n such that∑

�n(S(ti )− S(ti+1))2 → [S ]T a.s.







Change of variable for functionals of cadlag paths

Let ! ∈ D([0,T ]× ℝd) where ! has finite quadratic variationalong (�n) and

supt∈[0,T ]−�n

∣!(t)− !(t−)∣ → 0

Denote hni = tni+1 − tni ,

!n(t) =

k(n)−1∑i=0

!(ti+1−)1[ti ,ti+1)(t) + !(T )1{T}(t)

Consider the ’Riemann sum’:

k(n)−1∑i=0

∇!Ftni(!

n,�!(tni )tni −

)(!(tni+1)− !(tni )) �!(tni ) = !(tni+1)− !(tni )







A pathwise change of variable formula for functionals

Theorem (R.C. & D Fournie (Jour. Func. Analysis 2010))

For any F ∈ ℂ1,2b ([0,T [), ! ∈ Q([0,T ], �) the Follmer integral

∫ T

0

∇!Ft(!t−)d�! := limn→∞

k(n)−1∑i=0

∇!Ftni(!

n,�!(tni )tni −

)(!(tni+1)− !(tni ))

exists and FT (!T )− F0(!0) =

∫ T

0

DtFt(!u−)du

+

∫ T

0

1

2tr(t∇2

!Ft(!u−)d [!]c(u))

+

∫ T

0

∇!Ft(!t−)d�!

+∑

u∈]0,T ]

[Fu(!u)− Fu(!u−)−∇!Fu(!u−).Δ!(u)]







Sketch of proof: continuous case

! ∈ C0([0,T ]) is (uniformly) continuous on [0,T ] so

�n = sup0≤i≤k(n)−1,u∈[tni ,t

ni+1){∣!(u)− !(tni )∣+ ∣tni+1 − tni ∣} →n→∞ 0

Consider the decomposition:

Ftni+1(!n

tni+1−)− Ftni

(!ntni −

) = Ftni+1(!n

tni+1−)− Ftni

(!ntni

)

+ Ftni(!n

tni)− Ftni

(!ntni −

)

First term= (hni )− (0) where (u) = Ftni +u(!n

tni ,u).

Since F ∈ ℂ1,2([0,T ]), is right-differentiable so

Ftni+1(!n

tni ,hni)− Ftni

(!ntni

) =

∫ tni+1−tni

0Dtni +uF (!n

tni ,u)du







Sketch of proof: continuous case

The second term= �(�!ni )− �(0), where: �(u) = Ftni

(!n,utni −

). Since

F ∈ ℂ1,2([0,T ]), � is C 2 and:

�′(u) = ∇!Ftni(!n,u

tni −) �′′(u) = ∇2

!Ftni(!n,u

tni −)

Second order Taylor expansion of � at u = 0:

Ftni(!n

tni)− Ftni

(!ntni −

) = ∇!Ftni(!n

tni −)�!n

i

+1

2tr(∇2!Ftni

(!ntni −

) t�!ni �!

ni

)+ rni

where �!ni = !(tni+1)− !(tni ) and

rni ≤ K ∣�!ni ∣2 sup

t∈B(!(tni ),�n)∣∇2

!Ftni(!

n,!(t)−!(tni )tni −

)−∇2!Ftni

(!ntni −

)∣







So finally we have the decomposition

Ftni+1(!n

tni+1−)− Ftni

(!ntni −

)

=

∫ tni+1

tni

DuF (!ntni ,u−t

ni)du +∇!Ftni

(!ntni −

)�!ni

+1

2tr(∇2!Ftni

(!ntni −

) t�!ni �!

ni

)+ rni

As n→∞:∫ tni+1

tniDuF (!n

tni ,u−tni)du →

∫ T0 DuF (!u)du by dominated

convergence.rni ≤ �ni ∣�!n

i ∣2 where C ≥ ∣�ni ∣ → 0 so∑

i rni → 0.







A diagonal lemma

Lemma (R.C. and D Fournie, 2010)

Let (�n)n≥1 be a sequence of Radon measures on [0,T ] convergingweakly to a Radon measure � with no atoms, and (fn)n≥1, f beleft-continuous functions on [0,T ] with

∀t ∈ [0,T ], limn

fn(t) = f (t) ∀t ∈ [0,T ], ∥fn(t)∥ ≤ K

then

∫ t

sfnd�n

n→∞→∫ t

sfd�

So the sum of third terms converges to∫ T

0

1

2tr(t∇2

!Ft(!u).d [!](u))

=

∫ T

0

1

2tr(t∇2

!Ft(!u).d [!](u))







The Follmer integral

Since all other terms converge, we conclude that the limit

∫ T

0∇!Ft(!t−)d�! := lim

n→∞

k(n)−1∑i=0

∇!Ftni(!

n,Δ!(tni )tni −

)(!(tni+1)−!(tni ))

exists pathwise.Such integrals were defined in [Follmer 1981] for integrands of theform f ′(X (t)) where f ∈ C 2(ℝd).The construction depends a priori on the sequence � of partitions.But, for semimartingales and, more generally, Dirichlet processes,one obtains a limit object which is a.s. independent of the choiceof �.






Functional Ito formula: continuous caseFunctional Ito formula: cadlag caseFunctional Ito formula: proofFunctional of Dirichlet processes

Ito formula for functionals of semimartingales

Applied to a semimartingale, these results lead to a functionalextension of the Ito formula:

Theorem (Functional Ito formula (R.C.& Fournie, 2009))

Let X be a continuous semimartingale and F ∈ ℂ1,2b ([0,T [). For

any t ∈ [0,T [,

Ft(Xt)− F0(X0) =

∫ t

0DuF (Xu)du +∫ t

0∇!Fu(Xu).dX (u) +

∫ t

0

1

2tr(t∇2

!Fu(Xu) d [X ](u))

a.s.

In particular, Y (t) = Ft(Xt) is a semimartingale.







Let X be a cadlag semimartingale and denote for t > 0,Xt−(u) = X (u)1[0,t) + X (t−)1t .

Theorem (Functionals of cadlag semimartingales)

For any F ∈ ℂ1,2b ([0,T [), t ∈ [0,T [,

Ft(Xt)− F0(X0) =

∫ t

0∇!Fu(Xu).dX (u) +∫ t

0DuF (Xu)du +

∫ t

0

1

2tr(t∇2

!Fu(Xu) d [X ](u))

+∑

u∈]0,T ]

[Fu(Xu)− Fu(Xu−)−∇!Fu(Xu−).ΔX (u)]a.s.

In particular, Y (t) = Ft(Xt) is a semimartingale.







Functional Ito formula

If Ft(Xt) = f (t,X (t)) where f ∈ C 1,2([0,T ]× ℝd) thisreduces to the standard Ito formula.

For F ∈ ℂ1,2, Y (t) = Ft(Xt) can be reconstructed from the‘second-order jet’ (F ,DF ,∇!F ,∇2

!F ) of F along the path ofX .

If X has continuous paths then Y = F (X ) depends on F andits derivatives only via their values on continuous paths: Ycan be reconstructed from the second-order jet of F onΥc =

∪t∈[0,T ] C0([0, t],ℝd) ⊂ Υ.







Sketch of proof

Consider first a cadlag, “simple predictable” process:

X (t) =n∑

k=1

1[tk ,tk+1[(t)�k �k ℱtk −measurable

Each path of X is a sequence of horizontal and vertical moves:

Xtk+1= (Xtk ,hk )�k+1−�k hk = tk+1 − tk

Ftk+1(Xtk+1

)− Ftk (Xtk ) =

Ftk+1(X tk+1

)− Ftk+1(X tk ,hk )+ vertical move

Ftk+1(Xtk ,hk )− Ftk (Xtk ) horizontal move







Sketch of proof

Horizontal step: fundamental theorem of calculus for�(h) = Ftk+h(Xtk ,h)

Ftk+1(Xtk ,hk )− Ftk (Xtk )

= �(hk)− �(0) =

∫ tk+1

tk

DtF (Xt)dt

Vertical step: apply Ito formula to (u) = Ftk+1(X u

tk ,hk)

Ftk+1(Xtk+1

)− Ftk+1(Xtk ,hk ) = (X (tk+1)− X (tk))− (0)

=

∫ tk+1

tk

∇!Ftk+1(Xt).dX +

1

2tr(∇2

!Ftk+1(Xt)d [X ])







Sketch of proof

General case: approximate X by a sequence of simple predictableprocesses nX with nX (0) = X (0):

FT (nXT )− F0(X0) =

∫ T

0DtF (nXt)dt +

∫ T

0∇!F (nXt).dX

+1

2

∫ T

0tr[t∇2

!F (nXt)] d [X ]

The ℂ1,2b assumption on F implies that all derivatives involved in

the expression are left continuous in d∞ metric, which allows tocontrol their convergence as n→∞ using dominated convergence+ the dominated convergence theorem for stochastic integrals.







Similar formulas are obtained for a ’Dirichlet process’(=semimartingale + process with zero quadratic variation).Example:

Theorem (Functional change of variable formula for fBm)

Let W H be a fractional Brownian motion with H ∈ (0.5, 1). Thenfor any F ∈ ℂ1,2

b ([0,T [), t ∈ [0,T ]

Ft(W Ht )− F0(0) =

∫ t

0DuF (W H

u )du +

∫ t

0∇!Fu(W H

u ).dW H(u) a.s.

where∫ t

0 ∇!Fu(W Hu ).dW H(u) is the pathwise integral

∫ T

0∇!Ft(!t−)d! := lim

n→∞

k(n)−1∑i=0

∇!Ftni(W H

tni)(W H(tni+1)−W H(tni ))







Definition (Vertical derivative of a process)

Define C1,2b (X ) the set of processes Y which admit a

representation in ℂ1,2b :

C1,2b (X ) = {Y , ∃F ∈ ℂ1,2

b ([0,T ]), Y (t) = Ft(Xt) a.s.}

If det(A) > 0 a.s. then for Y ∈ C1,2b (X ), the predictable process:

∇XY (t) = ∇!Ft(Xt)

is uniquely defined up to an evanescent set, independently of thechoice of F ∈ ℂ1,2

b . We call ∇XY the vertical derivative of Y withrespect to X .







Vertical derivative for Brownian functionals

In particular when X is a standard Brownian motion, A = Id :

Definition

Let W be a standard d-dimensional Brownian motion. For anyY ∈ C1,2

b (W ) with representation Y (t) = Ft(Wt , t), thepredictable process

∇W Y (t) = ∇!Ft(Wt , t)

is uniquely defined up to an evanescent set, independently of thechoice of the representation F ∈ ℂ1,2

b .






Martingale representation formulaMartingale representation formulaA hedging formula for path-dependent optionsPathwise computation of hedge ratios

Martingale representation theorem

Consider now the case where X (t) =∫ t

0 �.dW where � isℱt-adapted with

det(�(t)) > 0a.s, E

(∫ T

0�2(t)dt

)<∞

X is then a square integrable martingale, with the predictablerepresentation property: for any ℱT -measurable variableH = H(X (t), t ∈ [0,T ]) = H(XT ) with E [∣H∣2] <∞ there exists apredictable process � with

H = E [H] +∫ T

0 �(t)dX (t)

This theorem is not constructive: various methods have beenproposed for computing � (Clark, Haussmann, Ocone,Jacod-Meleard-Protter, Picard, Fitzsimmons) using variousassumptions on H. Rama Cont Functional Ito calculus






Martingale representation formula

Theorem

Consider an ℱT -measurable functionalH = H(X (t), t ∈ [0,T ]) = H(XT ) with E [∣H∣2] <∞ and definethe martingale Y (t) = E [H∣ℱt ]. If Y ∈ C1,2

b (X ) then

Y (T ) = E [Y (T )] +∫ T

0 ∇XY (t)dX (t)

= E [H] +∫ T

0 ∇XY (t)�(t)dW (t)

This is a non-anticipative version of Clark’s formula.







A hedging formula for path-dependent options

Consider now a (discounted) asset price processS(t) =

∫ t0 �(t).dW (t) with det(�(t)) > 0a.s and

Eℚ(∫ T

0 �2(t)dt)<∞. Let H = H(S(t), t ∈ [0,T ]) with

E [∣H∣2] <∞ be a path-dependent payoff. The price at date t isthen Y (t) = E [H∣ℱt ].

Theorem (Hedging formula)

If Y ∈ C1,2b (S) then

H = Eℚ[H] +∫ T

0 ∇SY (t)dS(t) ℚ− a.s.

The hedging strategy for H is given by the vertical derivative ofthe option price with respect to S .







A hedging formula for path-dependent options

So the hedging strategy for H may be computed pathwise as

�(t) = ∇XY (t,Xt(!)) = limh→0

Y (t,X ht (!))− Y (t,Xt(!))

h

where

Y (t,Xt(!)) is the option price at date t in the scenario !.

Y (t,X ht (!)) is the option price at date t in the scenario

obtained from ! by moving up the current price (“bumping”the price) by h.

So, the usual “bump and recompute” sensitivity actually gives..the hedge ratio!







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 nX be the solution of a n-step Euler scheme and Yn a MonteCarlo estimator of Y obtained using nX .

Compute the Monte Carlo estimator Yn(t, nXt(!))

Bump the endpoint by h.

Recompute the Monte Carlo estimator Yn(t, nX ht (!)) (with

the same simulated paths)

Approximate the hedging strategy by

�n(t, !) :=Yn(t, nX h

t (!))− Yn(t, nXt(!))

h







Numerical simulation of hedge ratios

�n(t, !) ≃ Yn(t, nX ht (!))− Yn(t, nXt(!))

h

For a general C1,2b (S) path-dependent claim, with a few regularity

assumptions

∀1/2 > � > 0, n1/2−�∣�n(t)− �(t)∣ → 0 ℙ− a.s.

This rate is attained for h = cn−1/4+�/2

By exploiting the structure further (Asian options, lookbackoptions,...) one can greatly improve this rate.






An integration by parts formulaExtension to square-integrable martingalesWeak derivativeMartingale representation: general caseComputation of the weak derivativeRelation with Malliavin derivative

A non-anticipative integration by parts formula

ℐ2(X ) = {∫ .

0 �dX , � ℱt−adapted, E [∫ T

0 ∥�(t)∥2d [X ](t)] <∞}

Theorem

Let Y ∈ C1,2b (X ) be a (ℙ, (ℱt))-martingale with Y (0) = 0 and �

an ℱt−adapted process with E [∫ T

0 ∥�(t)∥2d [X ](t)] <∞. Then

E

(Y (T )

∫ T

0�dX

)= E

(∫ T

0∇XY .�d [X ]

)This allows to extend the functional Ito formula to the closure ofC1,2b (X ) ∩ ℐ2(X ) wrt to the norm

E∥Y (T )∥2 = E [

∫ T

0∥∇XY (t)∥2d [X ](t) ]







Regular functionals are dense among square-integrablefunctionals

ℒ2(X ) = {� ℱt − adapted, E [∫ T

0 ∥�(t)∥2d [X ](t)] <∞}ℐ2(X ) = {

∫ .0 �dX , � ∈ ℒ2(X )}

Lemma

Let D(X ) = C1,2b (X ) ∩ ℐ2(X ) be the set of processes which are

regular functionals of X . {∇XY ,Y ∈ D(X )} is dense in ℒ2(X )and the closure in ℐ2(X ) of D(X ) is the set of all square-integrablestochastic integrals with respect to X :

W1,2(X ) = {∫ .

0�dX , E

∫ T

0∥�∥2d [X ] <∞}.







Theorem (Weak derivative on W1,2(X ))

The vertical derivative ∇X : D(X ) 7→ ℒ2(X ) is closable onW1,2(X ). Its closure defines a bijective isometry

∇X : W1,2(X ) 7→ ℒ2(X )∫ .

0�.dX 7→ �

characterized by the following integration by parts formula: forY ∈ W1,2(X ), ∇XY is the unique element of ℒ2(X ) such that

∀Z ∈ D(X ), E [Y (T )Z (T )] = E

[∫ T

0∇XY (t)∇XZ (t)d [X ](t)

].

In particular, ∇X is the adjoint of the Ito stochastic integral

IX : ℒ2(X ) 7→ W1,2(X )

� 7→∫ .

0�.dX

in the following sense:

∀� ∈ ℒ2(X ), ∀Y ∈ W1,2(X ), < Y , IX (�) >W1,2(X )=< ∇XY , � >ℒ2(X )

i .e. E [Y (T )

∫ T

0�.dX ] = E [

∫ T

0∇XY �.d [X ] ]







A general martingale representation formula

X (t) =∫ t

0 �.dW � ℱt-adapted, det(�(t)) > 0 a.s,

E (∫ T

0 �2(t)dt) <∞.

Theorem

For any square-integrable ℱt-martingale Y ,(i) ∃F n ∈ ℂ1,2

b , E∥F nT (XT )− Y (T )∥2→n→∞ 0.

(ii) The vertical derivative ∇XY of Y with respect to X ,

∇XY (t) := limn→∞

∇!F nt (Xt)

is uniquely determined outside an evanescent set, independently ofthe approximating sequence (F n)n≥1.

(iii) Y (T ) = E [H] +∫ T

0 ∇XY (t)dX (t) ℙ− a.s.







General hedging formula for path-dependent options

This allows to extend the hedging formula to all square integrableclaims:

Theorem (Hedging formula)

Let H = H(S(t), t ∈ [0,T ]) with E [∣H∣2] <∞ be a(ℱT -measurable) path-dependent payoff. Define price at date t byY (t) = Eℚ[H∣ℱt ]. Then

H = Eℚ[H] +∫ T

0 ∇SY (t)dS(t) ℚ− a.s.

The hedging strategy for H is given by the (weak) verticalderivative of the option price with respect to S .







Computation of the weak derivative

For D(X ) = C1,2b (X ) ∩ ℐ2(X ), the weak derivative may be

computed pathwise: for Y ∈ W1,2(X ), ∇XY = limn∇XY n whereY n ∈ D(X ) is an approximating sequence with

E∥Y n(T )− Y (T )∥2 n→∞→ 0

An example of Y n is the Euler scheme for X .

∇XY (t,Xt(!)) = limn→∞

limh→0

Y n(t,X ht (!))− Y n(t,Xt(!))

h

In practice one may compute the estimator

∇XY (t,Xt(!)) =Y n(t,X

h(n)t (!))− Y n(t,Xt(!))

h(n)

where h(n) ∼ cn−� is chosen according to the “smoothness” of Y .Rama Cont Functional Ito calculus






Malliavin derivative

Consider the case where X = W . For a functionalF : C0([0,T ],ℝd)→ ℝ which possess a derivative DℋF in thedirection of absolutely continuous functions

∀h ∈ ℋ1([0,T ],ℝd), < DℋF (!), h >L2= lim�→0

F (! + �h)− F (!)

�

where ℋ1([0,T ],ℝd) = {f ∈ L2([0,T ],ℝd), f (0) = 0,∇f ∈L2([0,T ],ℝd)}, one can define the Malliavin derivative DF of F as

DtF (!) =∂

∂tDℋF (!)

DF may be defined forF ∈ D1,2([0,T ]) = {G ,DℋG ∈ ℋ1([0,T ],ℝd)}.In general DtF is an anticipative functional: DF ∈ L2([0,T ]× Ω)







Clark-Haussman-Ocone formula

Note that the Malliavin derivative maps a ℱT -measurablefunctional into an (anticipative) process

D : D1,2([0,T ]) 7→ L2([0,T ]× Ω)

F → (DtF )t∈[0,T ]

An important by-product of Malliavin calculus is theClark-Haussmann-Ocone formula: for F ∈ D1,2([0,T]),

F = E [F ] +

∫ T

0

pE [DtF ∣ℱt ]dW (t)

See e.g. Watanabe (1987) Nualart (2009)







Relation between vertical derivative and Malliavinderivative

Consider the case where X = W . Then for Y ∈ W1,2(W )

Y (T ) = E [Y (T )] +

∫ T

0∇W Y (t)dW (t)

If H = Y (T ) is Malliavin-differentiable e.g. H = Y (T ) ∈ D1,2

then the Clark-Haussmann-Ocone formula implies

Y (T ) = E [Y (T )] +

∫ T

0

pE [DtH∣ℱt ]dW (t)







Intertwining formula.

Theorem (Intertwining formula)

Let Y be a (ℙ, (ℱt)t∈[0,T ]) square-integrable martingale withY (T ) = H ∈ D1,2. Then

E [DtH∣ℱt ] = (∇W Y )(t) dt × dℙ− a.e.

i.e. the conditional expectation operator intertwines ∇W and D:

E [DtH∣ℱt ] = ∇W (E [H∣ℱt ]) dt × dℙ− a.e.







Relation with Malliavin derivative

In the Markovian case, the Clark-Ocone formula has beenexpressed [Nualart, Peng & Pardoux] in terms of the ‘diagonal’Malliavin derivative Dt [Y (t)].In this case one may show similarly that ∇W Y is a version ofDt [Y (t)]







Relation with Malliavin derivative

The following diagram is commutative, in the sense of dt × dℙalmost everywhere equality:

W1,2(W )∇W→ ℒ2(W )

↑(E [.∣ℱt ])t∈[0,T ] ↑(E [.∣ℱt ])t∈[0,T ]

D1,2 D→ L2([0,T ]× Ω)

Note however that ∇X may be constructed for any Ito process Xand its construction does not involve Gaussian properties of X .






Functional characterization of martingalesMartingale functionals of a Levy processUniversal pricing equation

Functional equation for martingales

Consider now a semimartingale X solution of a functional SDE:

dX (t) = bt(Xt)dt + �t(Xt)dW (t)

where b, � are non-anticipative flows on Ω with values inℝd -valued (resp. ℝd) whose coordinates are in ℂ0,0

l and areLipschitz-continuous w.r.t sup-norm.Consider the topological support of the law of X in(C0([0,T ],ℝd), ∥.∥∞):

supp(X ) = {!, any neighborhood V of !,ℙ(X ∈ V ) > 0}







A functional Kolmogorov equation for martingales

Theorem

Let F ∈ ℂ1,2b . Then Y (t) = Ft(Xt) is a local martingale if and

only if F satisfies

DtF (!) + bt(!)∇!Ft(!) +1

2tr[∇2

!F (!)�tt�t(!)] = 0,

for ! ∈supp(X ).

We call such functionals X−harmonic functionals.







Structure equation for Brownian martingales

In particular when X = W is a d-dimensional Wiener process, weobtain a characterization of ‘regular’ Brownian local martingales:

Theorem

Let F ∈ ℂ1,2b . Then Y (t) = Ft(Wt) is a local martingale on [0,T ]

if and only if

∀t ∈ [0,T ], ! ∈ C0([0,T ],ℝd),

DtF (!t) +1

2tr(∇2!F (!t)

)= 0.







Theorem (Uniqueness of solutions)

Let h be a continuous functional on (C0([0,T ]), ∥.∥∞). Anysolution F ∈ ℂ1,2

b of the functional equation (1), verifying∀! ∈ C0([0,T ]),

DtF (!) + bt(!)∇!Ft(!) +1

2tr[∇2

!F (!)�tt�t(!)] = 0

FT (!) = h(!), E [ supt∈[0,T ]

∣Ft(Xt)∣] <∞

is uniquely defined on the topological support supp(X ) of (X ): ifF 1,F 2 ∈ ℂ1,2

b ([0,T ]) are two solutions then

∀! ∈ supp(X ), ∀t ∈ [0,T ], F 1t (!) = F 2

t (!).







Functional equation for martingales

Consider now a ℝd -valued Levy process L with characteristic triplet(b,A, �)Denote supp(L) the topological support of the law of L in(D([0,T ],ℝd), ∥.∥∞):

supp(L) = {!, any neighborhood V of !, ℙ(L ∈ V ) > 0}

The topological support may be characterized in terms of (b,A, �)see e.g. Th Simon (2000)







Harmonic functionals of a Levy process

Theorem (Harmonic functionals of a Levy process)

Let F ∈ ℂ1,2b . Then Y (t) = Ft(Lt) is a local martingale if and

only if F satisfies the following functional equation:∫ℝd

(Ft(!t− + z1{t})− Ft(!t−)− 1∣z∣≤1 z .∇!Ft(Xt−)

)�(dz) +

DtF (!t) + b.∇!Ft(!t) +1

2tr[∇2

!F (!t).A] = 0

for ! ∈supp(X ).







A universal pricing equation

Consider now a (discounted) asset price with dynamicsS(t) =

∫ t0 �(u, Su).dW (u) under a pricing measure ℚ.

Theorem (Pricing equation for path-dependent options)

Consider a path-dependent payoff H(S(t), t ∈ [0,T ]). If

∃F ∈ ℂ1,2b , Ft(St) = Eℚ[H∣ℱt ]

then F is the unique solution of the pricing equation

DtF (!) +1

2∇2!Ft(!)�t(!)2 = 0,

for ! ∈supp(X ) with the terminal condition

FT (!) = H(!)Rama Cont Functional Ito calculus






Universal pricing equation

DtF (!) +1

2∇2!Ft(!)�t(!)2 = 0,

∀! ∈ supp(X ), FT (!) = H(!)

This equations implies all known PDEs for path-dependent options:barrier, Asian, lookback,...but also leads to new pricing equations.This equation shows the � − Γ tradeoff still holds forpath-dependent derivatives provided � and Γ are properly definedin terms of horizontal and vertical derivatives.







Example: Asian options

Consider an option with payoff H(!) = h(∫ T

0 !(s)ds) whereh : ℝ→ ℝ is measurable and has linear growth.

Ft(!t) = f (t,

∫ t

0!(s)ds, !(t))

is a ℂ1,2 solution of the functional pricing equation if and only iff ∈ C 1,1,2([0,T [×]0,∞[×]0,∞[) is continuous on[0,T ]×]0,∞[×]0,∞[ and solves

∂t f (t, a, x) + r(t)x∂x f (t, a, x) + x∂af (t, a, x) +1

2x2�2(t, x)∂xx f (t, a, x) = r(t)f (t, a, x)

with terminal condition: f (T , a, x) = h(a)







Theta-Gamma tradeoff for path-dependent options

In particular this equation extends the Black-Scholes Theta-Gammatradeoff relation for call options to path-dependent options:

DtF (!)︸︷︷︸�(t)

+r(t)∇!Ft(!) +1

2∇2!Ft(!)︸︷︷︸

Γ(t)

�t(!)2 = 0,

where the time sensitivity (�) is computed as a horizontalderivative andthe Gamma is computed as the second vertical derivative of theprice







Summary

The definition of the vertical derivative restores the relationbetween the first-order sensitivity to the underlying and itshedging strategy:∇SY (t) = sensitivity of price to underlying = hedgingportfolio,just like for European options in the Black-Scholes model.

The horizontal derivative is the correct definition of timedecay of a path-dependent options: �(t) = DtF (t, St). Inparticular it verifies the � − Γ tradeoff relation, just like forEuropean options in the Black-Scholes model.

The � − Γ tradeoff relation gives a unified derivation of allknown pricing PDEs for path-dependent options: Asian,barrier, weighted variance swap,...







Extensions and applications

The result can be extended to functionals depending onquadratic variation and to discontinuous processes: Y and Xcan both have jumps.

The result can be localized using stopping times: importantfor applying to functionals involving stopped processes/ exittimes.

Pathwise maximum principle for non-Markovian controlproblems.

� − Γ tradeoff for path-dependent derivatives.







References

H Follmer (1981) Calcul d’Ito sans probabilites, Seminaire deProbabilites (1979).B Dupire (2009) Functional Ito calculus, Working Paper.R. Cont & D Fournie (2009) Functional Ito calculus andstochastic integral representation of martingales, arxiv.R. Cont & D Fournie (2010) A functional extension of the Itoformula, Comptes Rendus Acad. Sci., Vol. 348, 51-67.R. Cont & D Fournie (2010) Change of variable formulas fornon-anticipative functionals on path space, Journal ofFunctional Analysis, Vol 259, 1043–1072.R. Cont (2010) A martingale representation formula forPoisson random measures.R Cont (2010) Pathwise computation of hedging strategies forpath-dependent derivatives, Working Paper.


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Non-anticipative functional calculus and applications · 2011-03-25 · Functional calculus: horizontal and vertical derivatives. A pathwise change of variable formula for functionals