functional ito calculus and applications to path-dependent ......functional ito calculus and...

Functionals involving quadratic variationPathwise derivatives of functionals

Pathwise change of variable formulaIto formula for functionals of QV

Functional equations for martingales

Functional Ito calculus andapplications to path-dependent optionsPart II: Functionals involving quadratic variation

Rama Cont

Laboratoire de Probabilites et Modeles AleatoiresCNRS (Paris)

andColumbia University, New York

Joint work with: David Fournie (Columbia University)

Rama Cont Functional Ito calculus




References

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.





Functionals involving quadratic variation

Consider an ℝd -valued Ito process

X (t) =

∫ t

0�(u)du +

∫ t

0�(u).dWu

with quadratic variation process

[X ](t) =

∫ t

0

t�.�(u)du =

∫ t

0A(u)du

where A(t) = t�(t).�(t) takes values in the set S+d of positive

symmetric d × d matrices.






In stochastic analysis, statistics of processes and mathematicalfinance, one is interested in path-dependent functionals such as

Quadratic variation/ “realized variance”:

[X ](t) = limn→∞

t/n−1∑k=1

∥X (k

n)− X (

k − 1

n)∥2

Exponential functionals involving quadratic variation:Y (t) = exp(X (t)− [X ](t)/2 )Functionals of quadratic variation: e.g. variance swaps andvolatility derivatives

Y (t) = E [([X ](t)− K )+∣ℱt ],

Y (t) = E [

∫ T

0f (u,X (u))d [X ] ∣ℱt ]






Since X 7→ [X ] is not (a.s.) continuous in sup norm, the previousframework cannot be applied directly to such functionals.One idea would be to introduce weaker norms (ex: p-variationnorms/ “rough path” topology, see Lyons & Qian) and repeat theconstruction under those norms.Our approach is based on a simpler idea: introduce an auxiliaryvariable At ∈ D([0,T ],S+

d ) and view these functionals as definedon the product space D([0, t],ℝd)× D([0, t],S+

d ), equipped withthe supremum norm/ d∞ metric.





Functional representation

These examples of functionals have a non-anticipative dependencein X and a “predictable” dependence on A since they only dependon [X ] =

∫ .0 A(u)du : they may be represented as

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

where the functional Ft : D([0, t],ℝd)× D([0, t], S+d )→ ℝ

represents the dependence of Y (t) on the path of X and A = t�.�and is “predictable with respect to the 2nd variable”:

∀t, ∀(x , v) ∈ D([0, t],ℝd)×D([0, t],S+d ), Ft(xt , vt) = Ft(xt , vt−)

F = (Ft)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 ).





d∞ metric on Υ =∪

t∈[0,T ] D([0, t],ℝd )× D([0, t], S+d )

For T ≥ t ′ = t + h ≥ t ≥ 0, (x , v) ∈ D([0, t],ℝd)× S+t and

(x ′, v ′) ∈ D([0, t + h],ℝd)× S+t+h

d∞( (x , v), (x ′, v ′) ) = supu∈[0,t+h]

∣xt,h(u)− x ′(u)∣

+ supu∈[0,t+h]

∥vt,h(u)− v ′(u)∥+ h





Continuity for non-anticipative functionals

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

Ft : D([0, t],ℝd)× St → ℝ

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

Definition (Left-continuous functionals)

Define ℂ0,0l as the set of non-anticipative functionals

F = (Ft , t ∈ [0,T [) which are continuous at fixed times and

∀t ∈ [0,T [, ∀� > 0,∀(x , v) ∈ D([0, t],ℝd)× St ,∃� > 0,∀h ∈ [0, t], ∀(x ′, v ′) ∈ ∀(x , v) ∈ D([0, t − h],ℝd)× St−h,

d∞((x , v), (x ′, v ′)) < � ⇒ ∣Ft(x , v)− Ft−h(x ′, v ′)∣ < �





Boundedness-preserving functionals

We call a functional “boundedness preserving” if it is bounded oneach bounded set of paths:

Definition (Boundedness-preserving functionals)

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

∃CK ,R,t0 > 0, ∀t ≤ t0, ∀(x , v) ∈ D([0, t],K )× St ,sup

s∈[0,t]∣v(s)∣ ≤ R ⇒ ∣Ft(x , v)∣ ≤ CK ,R,t0





Horizontal derivativeVertical derivative of a functionalSpaces of regular functionalsExamples

Horizontal derivative

Definition (Horizontal derivative)

We will say that the functional F = (Ft)t∈[0,T ] on Υ([0,T ]) is

horizontally differentiable at (x , v) ∈ D([0, t],ℝd)× St if

DtF (x , v) = limh→0+

Ft+h(xt,h, vt,h)− Ft(xt , vt)

hexists

We will call (1) the horizontal derivative DtF of F at (x , v).

DF = (DtF )t∈[0,T ] defines a non-anticipative functional.

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






Definition (Partial vertical derivative)

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

vertically differentiable at (x , v) ∈ D([0, t]),ℝd)× D([0, t],S+d ) if

ℝd 7→ ℝe → Ft(xe

t , vt)

is differentiable at 0. Its gradient at 0 is called the verticalderivative of Ft at (x , v)

∇xFt (x , v) = (∂iFt(x , v), i = 1..d) where

∂iFt(x , v) = limh→0

Ft(xheit , v)− Ft(x , v)

h






Spaces of differentiable functionals

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 verticallyin x at all (x , v) ∈ D([0, t],ℝd)× S+

t , t < T , with

horizontal derivatives Dmt F ,m ≤ j continuous on

D([0,T ])× St for each t ∈ [0,T [

left-continuous vertical derivatives: ∀n ≤ k ,∇nxF ∈ F∞l .

Dmt F ,∇n

xF ∈ B([0,T ]).






Examples of regular functionals

Example

Y = exp(X − [X ]/2) = F (X ,A) where

Ft(xt , vt) = ex(t)− 12

∫ t0 v(u)du

F ∈ ℂ1,∞b with:

DtF (x , v) = −1

2v(t)Ft(x , v) ∇j

xFt(xt , vt) = Ft(xt , vt)






Examples of regular functionals

Example (Integrals w.r.t quadratic variation)

For g ∈ C0(ℝd), Y (t) =∫ t

0 g(X (u))d [X ](u) = Ft(Xt ,At) where

Ft(xt , vt) =

∫ t

0g(x(u))v(u)du

F ∈ ℂ1,∞b , with:

DtF (xt , vt) = g(x(t))v(t) ∇jxFt(xt , vt) = 0





Change of variable formula for functionals involving QV

Let (x , v) ∈ D([0,T ]× ℝd)× D([0,T ]× ℝn) where x has finitequadratic variation along (�n) and

supt∈[0,T ]−�n

∣x(t)− x(t−)∣+ ∣v(t)− v(t−)∣ → 0

Denote

xn(t) =

k(n)−1∑i=0

x(ti+1−)1[ti ,ti+1)(t) + x(T )1{T}(t)

vn(t) =

k(n)−1∑i=0

v(ti )1[ti ,ti+1)(t) + v(T )1{T}(t), hni = tni+1 − tni





A pathwise change of variable formula for functionals

Theorem (C. & Fournie (JFA 2010))

Let F ∈ ℂ1,2b ([0,T [) with Ft(xt , vt) = Ft(xt , vt−). Then the limit

∫ T

0

∇xFt(xt−, vt−)d�x := limn→∞

k(n)−1∑i=0

∇xFtni(x

n,Δx(tni )tni −

, vntni −

)(x(tni+1)− x(tni ))

exists and FT (xT , vT )− F0(x0, v0) =

∫ T

0

DtFt(xu−, vu−)du

+

∫ T

0

1

2tr(t∇2

xFt(xu−, vu−)d [x ]c(u))

+

∫ T

0

∇xFt(xt−, vt−)d�x

+∑

u∈]0,T ]

[Fu(xu, vu)− Fu(xu−, vu−)−∇xFu(xu−, vu−).Δx(u)]





Ito formula for functionals of quadratic variation

Theorem (Ito formula for functionals of quadratic variation)

If F ∈ ℂ1,2b ([0,T [) then for any t ∈ [0,T [,

Ft(Xt ,At)− F0(X0,A0) =

∫ t

0DuF (Xu,Au)du +∫ t

0∇xFu(Xu,Au).dX (u) +

∫ t

0

1

2tr(t∇2

xFu(Xu,Au) d [X ](u))

a.s.

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





Functional characterization of martingalesExample: a weighted variance swap

Functional equation for martingales

Consider now a semimartingale X whose characteristics areleft-continuous functionals of the path:

dX (t) = bt(Xt ,At)dt + �t(Xt ,At)dW (t)

where b, � are non-anticipative functionals on Ω with values inℝd -valued (resp. ℝd×n) whose coordinates are in ℂ0,0.Consider the topological support of the law of (X ,A) in(C0([0,T ],ℝd)× ST , ∥.∥∞):

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






A functional Kolmogorov equation for martingales

Theorem

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

only if F satisfies

DtF (xt , vt) + bt(xt , vt)∇xFt(xt , vt)

+1

2tr[∇2

xF (xt , vt)�tt�t(xt , vt)] = 0,

for (x , v) ∈supp(X ,A).






Theorem (Uniqueness of solutions)

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

b of the functional equation (1), verifying∀(x , v) ∈ C0([0,T ])× ST ,

DtF (xt , vt)+ bt(xt , vt)∇xFt(xt , vt)

+ 12tr[∇

2xF (xt , vt)�t

t�t(xt , vt)] = 0

FT (x , v) = h(x , v), E [supt∈[0,T ] ∣Ft(Xt ,At)∣] <∞

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

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

∀(x , v) ∈ supp(X ,A), ∀t ∈ [0,T ], F 1t (xt , vt) = F 2

t (xt , vt).






Pricing equation for volatility derivatives

Theorem (Pricing equation for volatility derivatives)

Let ∃F ∈ ℂ1,2b , Ft(Xt ,At) = E [H∣ℱt ] then F is the unique

solution of the pricing equation

DtF (xt , vt) + bt(xt , vt)∇xFt(xt , vt)

+1

2tr[∇2

xF (xt , vt)�tt�t(xt , vt)] = 0,

for (x , v) ∈supp(X ,A).






A diffusion example

Consider a scalar diffusion

dX (t) = b(t,X (t))dt + �(t,X (t))dW (t) X (0) = x0

defined as the solution ℙx0 of the martingale problem onD([0,T ],ℝd) for

Lt f =1

2�2(t, x)∂2

x f (t, x) + b(t, x)∂x f (t, x)

where b and � ≥ a > 0 are continuous and bounded functions.By the Stroock-Varadhan support theorem, the topological supportof (X ,A) under ℙx0 is

{(x , (�2(t, x(t)))t∈[0,T ]) ∣x ∈ C0(ℝd , [0,T ]), x(0) = x0}.






Weighted variance swap

A weighted variance swap with weight functiong ∈ Cb([0,T ]× ℝd), we are interested in computing

Y (t) = E [

∫ T

0g(t,X (t))d [X ](t)∣ℱt ]

If Y = F (X ,A) with F ∈ ℂ1,2b ([0,T ]) then F is X -harmonic and

solves the functional Kolmogorov equation.Taking conditional expectations and using the Markov property ofX :

Ft(xt , vt) =

∫ t

0g(u, x(u))v(u)du + f (t, x(t))

Elementary computation show that F ∈ ℂ1,2b iff f is a smooth C 1,2

function.







Ft(xt , vt) =

∫ t

0g(u, x(u))v(u)du + f (t, x(t))

solves the functional Kolmogorov eq. iff f solves

1

2�2(t, x)∂2

x f (t, x) + b(t, x)∂x f (t, x) + ∂t f (t, x) = −g(t, x)�2(t, x)

with terminal condition f (T , x) = 0

so that Y (T ) = FT (XT ,AT ).This PDE has a unique C 1,2 solution with exponential growth. Fthen defines a unique ℂ1,2

b ([0,T ]) functional on supp(X,A) and theprice is given by

Y (t) =

∫ t

0g(u,X (u))d [X ](u) + f (t,X (t))







Applying now the Ito formula to f (t,X (t)) we obtain that thehedging strategy is given by

�(t) =∂f

∂x(t, x)

where f is the unique solution of the PDE with source term:

1

2�2(t, x)∂2

x f (t, x) + b(t, x)∂x f (t, x) + ∂t f (t, x) = −g(t, x)�2(t, x)

with terminal condition f (T , x) = 0







Note that:

Note that the new definition of �/ horizontal derivativerequires extending both the path of X and A by a constant:

�(t) = DtF (Xt ,At) = limh→0

Ft+h(Xt,h,At,h)− Ft(Xt ,At)

h

and [Xt,h] ∕= At,h. In particular this is NOT equal to thedefinition in Dupire (2009) where only X is extended and [X ]is computed along the extended path.

It is this � which verifies the � − Γ tradeoff, not ∂t f (t,X (t)),which is not the rate of time decay of the position.






References

R. Cont & D Fournie (2010) A functional extension of the Itoformula, Comptes Rendus Acad. Sci. Paris, Ser. I , Vol. 348,51-67, Jan 2010.

R. Cont & D Fournie (2010) Functional Ito formula andstochastic integral representation of martingales,arxiv.org/math.PR.

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

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


functional ito calculus and applications to path-dependent ......functional ito calculus and...

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