Malliavin Calculus and its Recent Applications in Financial...

Introduction Preliminaries Stochastic Differential Equations Evolution of Malliavin Calculus Theory Recent Studies on Malliavin Calculus

Malliavin Calculus and its RecentApplications in Financial Mathematics

Yeliz Yolcu Okur

Institute of Applied Mathematics, Financial MathematicsMiddle East Technical University, Ankara

Istanbul Analysis Seminars11 April 2014, Karaköy, Istanbul

Agenda

1 Introduction2 Preliminaries

Brownian Motion and its Properties3 Stochastic Differential Equations

Itô FormulaApplication

Black Scholes Model

4 Evolution of Malliavin Calculus TheoryStochastic Calculus of Variations: Approach I

The Derivative and Divergence OperatorsStochastic Calculus of Variations: Approach II

Wiener-Itô Chaos Expansion

5 Recent Studies on Malliavin CalculusComputation of Greeks under General StochasticVolatility Models

Observations:Stock prices move randomly because of the efficientmarket hypothesis.Absolute change in the price of a stock is not a usefulquantity. Increase of one unit in a stock worth 10 units ismuch more significant than the one with a stock worth 100units.Hence, relative change must be modeled: dS/SdS/S = (µ+ σnoise)dt (more realistic, consider someenvironmental random effects)

Random variable

(Ω,F , Ftt≥0,P)

DefinitionA real valued random variable, X , is a real valuedF-measurable function defined on Ω. i.e.

X : Ω→ R

(Ω,F ,P)→ (R,B,PX )

B = (−∞, x ], x ∈ R,

PX ((−∞, x ]) = P(ω : X (ω) ∈ (−∞, x ])= P(ω : X (ω) ≤ x) = FX (x)

Stochastic Process

DefinitionA real valued stochastic process is just a sequence of realrandom variables, i.e.

(t , ω)→ Xt (ω) : T × Ω→ R

is a stochastic process which is a parameterized collection ofreal-valued rvs Xtt∈T .

Brownian Motion and its Properties

Brownian Motion

DefinitionA Brownian is a real valued, continuous stochastic processB(t , ω) , t ≥ 0, ω ∈ Ω, with independent and stationaryincrements, i.e.,

continuity: The map t → B(t , ω) is continous.independent increments: ∀s ≤ t , B(t)− B(s) is indepedentof Fs = σ(B(u),u ≤ s).

stationary increments: ∀s ≤ t , h > 0, B(t + h)− B(s + h)have the same probability law with B(t)− B(s).

Standard Brownian Motion

A standard Brownian motion, B(t , ω), t ∈ [0,T ], ω ∈ Ω, is aBrownian motion with the following properties:

B(0) = 0 P− a.s.E[B(t)] = 0E[B(t)2] = t

Simulation of Brownian Motion

Properties of Brownian Motion

cont. everywhere but nowhere differentiable,hits any and every real value no matter how large or smallit is,It has 1/2-self similarity, i.e. B(λt) = λ1/2B(t) for λ ≥ 0.

Variation of Brownian Motion

Taylor and Duke (1972) showed that

Vp(B; a,b) <∞⇐⇒ p > 2

For p = 2, instead of considering the supremum over allpartitions, if we restrict ourselves to the sequences of partitions4n for which ||4n|| → 0, then we obtain the following result:

Theorem

Q2(B; a,b,4n)→ b − a in L2

as ||4n|| → 0. Moreover, the quadratic variation of a Brownianmotion B on the interval [a,b] is the limit of Q2(B; a,b,4n) anddenoted by < B,B >[a,b].

Stochastic Integration for Simple Processes

H(t , ω) =∑p

i=1 φi(ω)I[ti−1,ti )(t)where φ is Fti−1-measurable and bounded,0 = t0 < t1 < . . . < tp = T .

I(H)(t) :=∑

1≤i≤k

φi(B(ti)− B(ti−1)) + φk+1(B(t)− B(tk ))

for any t ∈ (tk , tk+1].

Stochastic Integration

PropositionLet H(t , ω), t ∈ [0,T ], ω ∈ Ω be a simple process. Then,

(∫ T

0 H(s)dB(s))0≤t≤T is a continuous Ft -martingale.

[(∫ T0 H(s)dB(s)

= E(∫ T

0 |H(s)|2ds)

supt≤T |∫ t

0 H(s)dB(s)|2)≤ 4E

(∫ T0 |H(s)|2ds

(H(t , ω))0≤t≤T , (Ft )t≥0 − adapted process, E(∫ T

0 H(s)2ds)<∞

Itô Process

X (t , ω), t ∈ [0,T ] is called Itô process if it can be written as

X (t) = X (0) +

0K (s)ds +

0H(s)dB(s),

whereX (0) is F0-measurable.∫ T

0 |K (s)|ds <∞ P− a.s.∫ T0 |H(s)|2ds <∞ P− a.s.

Stochastic Differential Equations

More generally, consider the following equation:

dX (t) = b(t ,X (t))dt + σ(t ,X (t))dB(t) (1)

where X (0) = x ∈ R,b : [0,T ]× R→ R,σ : [0,T ]× R→ R

Questions Come to Our Minds at First Glance

1 Can we obtain existence and uniqueness theorem for suchequations?

2 How can we solve?3 Answers:

1. b and σ satisfy Lipschitz and polynomial growthcondition and E[|x |2] <∞2. Itô formula

4 E = (Xs)0≤s≤T ,Ft −measurable continuous processess.t. E ( sups≤T |Xs|2) <∞

2 How can we solve?

3 Answers:1. b and σ satisfy Lipschitz and polynomial growthcondition and E[|x |2] <∞2. Itô formula

Itô Formula

Let X (t , ω) , t ∈ [0,T ] be an Itô process

X (t) = X (0) +

0K (s)ds +

0H(s)dB(s),

and f ∈ C1,2, then

f (t ,X (t)) = f (0,X (0)) +

∂f∂s

(s,X (s))ds +

∂f∂x

(s,X (s))dX (s)

∂f∂x2 (s,X (s))d < X ,X >s

Itô Formula

Itô Formula in Practice

f (t , x) = x2, X (t) = B(t);

B(t)2 = t + 2∫ t

0B(s)dB(s)

Application

Financial Market

Market is ideal (no constraints on consumption, notransaction costs or taxes).

Bank account process (riskless asset) S0(t),

dS0(t) = S0(t)r(t)dt ,S0(0) = 1.

1 risky asset and price process is given by S1(t),

dS1(t) = S1(t)[µdt + σdB(t)],

S1(0) > 0.

Application

Financial Market

Market is ideal (no constraints on consumption, notransaction costs or taxes).Bank account process (riskless asset) S0(t),

dS0(t) = S0(t)r(t)dt ,S0(0) = 1.

S1(0) > 0.

Application

Financial Market

Market is ideal (no constraints on consumption, notransaction costs or taxes).Bank account process (riskless asset) S0(t),

dS0(t) = S0(t)r(t)dt ,S0(0) = 1.

S1(0) > 0.

Application

Simulation of Stock Price

Malliavin calculus

Paul Malliavin1925-2010

Books on Calculus of Variations

Evolution of Malliavin Calculus Theory

DefinitionMalliavin calculus is an infinite dimensional differential calculusfor functionals of an arbitrary Gaussian process.

1 originally introduced to study the regularity of the law offunctionals of Brownian Motion:work on Wiener space (Ω = C0[0,T ] of continuousfunctions ω : [0,T ]→ R, ||ω||∞ := supt∈[0,T ] |ω(t)|)

2 Applications to finance:Representation of functionals of stochastic processesStochastic control theoryStochastic differential equationsComputation of GreeksGeneral stochastic differential games

Evolution of Malliavin Calculus Theory

DefinitionMalliavin calculus is an infinite dimensional differential calculusfor functionals of an arbitrary Gaussian process.

1 originally introduced to study the regularity of the law offunctionals of Brownian Motion:work on Wiener space (Ω = C0[0,T ] of continuousfunctions ω : [0,T ]→ R, ||ω||∞ := supt∈[0,T ] |ω(t)|)

2 Applications to finance:Representation of functionals of stochastic processesStochastic control theoryStochastic differential equationsComputation of GreeksGeneral stochastic differential games

Stochastic Calculus of Variations: Approach I

Settings in Infinite Dimensional Framework

Let H be a separable Hilbert space (|| · ||H , 〈·, ·〉H )There exists probability space (Ω,G, µ)

M = W (h); h ∈ H := Family of centered rvs defined onthis space s.t.

h→W (h) is linear.E[W (h1)W (h2)] = 〈h1,h2〉HEach W (h) is Gaussian.

M is a closed subspace of L2(Ω) that is isometric to H.W (h1) and W (h2) are independent if and only if h1 and h2are orthogonal.The rvs eW (h),h ∈ H form a total subset of L2(Ω).

Theorem(Revuz & Yor Thm 1.2) Given a probability measure P on R,there exists a probability space (Ω,F , µ) and a sequence ofindependent rvs Xndefined on Ω s.t. Xn(P) = µ for every n.

Take an orthonormal basis en,n ≥ 1 in H. By the thm, thereis a probability space (Ω,G, µ) on which one can define asequence of independent reduced real Gaussian variables gnfor n ≥ 1. That is Ω = R⊗N, G = B⊗N and µ = µ⊗N1 , where µ1 is

the Gaussian measure on R (µ1(dx) = (2π)−1/2 exp(− |x |2

2 )dx).For each h ∈ H, the series

∑n≥1〈h,en〉Hgn converges in

L2(Ω,G, µ) to a random variable and

W (h) = limn→∞

∑n≥1

〈h,en〉Hgn

Indeed, Hn = Hn(W (h)),h ∈ H, ||h||H = 1

The Derivative Operator

There are several ways to define the Malliavin derivative forrandom vectors F : Ω→ Rn. Note that, the values of thederivative of a random variable will be H-valued.

DefinitionThe random variables of the type

F = f (W (h1), . . . , f (W (hn)),

with h1,h2, . . . ,hn ∈ H is called smooth functionals.If f ∈ C∞p (Rn), then the corresponding classes of smoothfunctionals is denoted by SIf f ∈ C∞b (Rn), then the family is denoted by SbIf f is polynomial, then the family is denoted by P

P ⊂ S, Sb ⊂ S.

Derivative Operator on Smooth Functionals

DF =n∑

∂i f (W (h1), . . . ,W (hn))hi

For a fixed h ∈ H; set

F εh = f (W (h1) + ε〈h,h1〉H , . . .W (hn) + ε〈h,hn〉H)

DhF := 〈DF ,h〉H =ddε

F εh|ε=0.

Properties

For any F ∈ SD(FG) = FDG + GDFD is closable operator; D : Lp(Ω)→ Lp(Ω; H), for anyp ≥ 1.E[〈DF ,h〉H ] = E[FW (h)]

E[G〈DF ,h〉H ] = −E[F 〈DG,h〉H ] + E[FGW (h)]

Let D1,p be the closure of the set S with respect to theseminorm for p ∈ [1,∞):

||F ||1,p :=(E[||F ||p] + E[||DF ||pH ]

Observations:D1,p is dense in Lp(Ω),

||F ||k ,p :=(E[||F ||p] +

∑kj=1E[||DjF ||pH⊗j ]

Dj,q ⊂ Dk ,p, for k ≤ j and p ≤ qD0,p = Lp(Ω)

The Divergence Operator

DefinitionThe adjoint of the derivative operator D is denoted with δ. It isan unbounded operator on L2 (Ω; H) with values in L2 (Ω) and ithas the following properties:

1 The domain of δ is the set of H valued square integrablerandom variables u ∈ L2 (Ω; H) such that,

|E [〈DF ,u〉H ]| ≤ c ‖F‖2 , (2)

for all F ∈ D1,2 and some constant c, which depends on u.2 If u ∈ Dom (δ) then δ (u) ∈ L2 (Ω) and characterized by

E [Fδ (u)] = E [〈DF ,u〉H ] , (3)

for any F ∈ D1,2.

Skorohod Integral

H = L2(T ,B, λ), where λ is the σ-finite atomless measuredefined on the set T , then, δ(u) is the Skorohod stochasticintegral of the square integrable process u ∈ L2(Ω× T );

δ(u) =

u(t)δB(t)

Stochastic Calculus of Variations: Approach II

Wiener-Itô Chaos Expansion

For the sake of simplicity assume T = [0,T ].

Definition

Let F be an FT measurable random variable in L2(Ω). Thenthere exists a unique sequence of functions, fn∞n=0, wherefn ∈ L2(T n) such that

F =∞∑

In(fn).

Here the convergence is in L2(Ω). Moreover, we have thefollowing isometry: ‖F‖2L2(Ω)

=∑∞

n=0 n!‖fn‖2L2(T n)

Malliavin Derivative

Definition

Let F ∈ L2(Ω) be FT -measurable random variable with thefollowing chaos expansion:

F =∞∑

In(fn)

Then the Malliavin derivative DtF of F at time t can be definedas the following expansion:

DtF =∞∑

nIn−1(fn(·, t)), t ∈ T

where In−1(fn(·, t)) is the (n − 1)-fold iterated Itô integral offn(t1, t2, . . . , tn−1, t).

Skorohod Integral

Definition

Let u ∈ L2(Ω× T ) and for all t ∈ T , the rv u(t) is FTmeasurable. Let it has Wiener-Itô chaos expansion be

u(t) =∞∑

In(fn(·, t)),

where fn ∈ L2(T n+1) symmetric function of first n variable.Then;

δ(u) :=

u(t)δB(t) :=∞∑

In+1(fn)

when convergent in L2(Ω).

Here, fn, n = 1,2, . . . are the symmetric functions obtained by

fn(t1, . . . , tn, t) =1

[fn(t1, . . . , tn, t) +

n∑i=1

fn(t1, · · · , ti−1, t , ti , . . . , tn, ti )

Remark

Note that

u(t)2λ(dt)]

=∞∑

n!||fn||2L2(T n+1).

Example1 F = B(T ), DtF = 12 F = B2(T ), DtF = 2B(T )

3 F =∫ T

0 g(s)dB(s), DtF = g(t) (g is a deterministicfunction)

4 F =∫ T

0 g(s)dB(s), DtF =∫ T

t Dt (g(s))dB(s) + g(t) (g is astochastic process)

5 δ(1) = B(T )

6∫ T

0 B(T ) δB(t) = B(T )2 − T

Research Areas

1 White noise generalization of the Clark-Ocone formulaunder change of measure.

2 A Malliavin calculus approach to general stochasticdifferential games with partial information.

3 SDE solutions in the space of smooth random variables.4 Analysis of volatility feedback and leverage effects using

high frequency data.5 Computation of Greeks in stochastic volatility models.6 A general formula for the Laplace transform of the hitting

times of non-Gaussian processes.

Computation of Greeks under General Stochastic Volatility Models

General Stochastic Volatility Model

(joint work with B. Yılmaz, A. Inkaya and T. Sayer)

St= rtdt + σ (t ,Vt ) dWt , (4)

dVt = u (t ,Vt ) dt + v (t ,Vt ) dZt , (5)

dZt =[ρdWt +

√1− ρ2dW 1

d〈W ,Z 〉t = ρdt ,

for t ∈ [0,T ]. Here, r represents the risk free interest rate,u (t ,Yt ) and v (t ,Yt ) ∈ C2 ([0,T ]× R).

For the general stochastic volatility model, define

G (t ,T ) = σ (t ,Vt ) +

∂y(s,Vs) Dt Vs dWs −

∂y(s,Vs) Dt Vsσ (s,Vs) ds (6)

Proposition

If S ∈ D2,2 and V ∈ D2,2 the following equalities hold fors ≤ t ≤ v ≤ T

DtST = ST G (t ,T )

DuST := 〈DST ,u〉L2([0,T ]) = ST∫ T

0 ut G (t ,T ) dt

DuDuST = ST

(∫ T0 ut G (t ,T ) dt

+∫ T

∫ Ts usut DsG (t ,T ) dtds

DsG (t ,T ) = ∂σ

∂y (t ,Vt )DsVt +∫ T

t DsVv Dt Vv

[∂2σ∂y2 (v ,Vv ) (dWv − σ (v ,Vv ) dv)

]+∫ T

t∂σ∂y (v ,Vv ) DsDt Vv (dWv − σ (v ,Vv ) dv)

+∫ T

((∂σ∂y (v ,Vv )

)Dt Vv

(∂σ∂y (v ,Vv )

PropositionSuppose I is an open interval of R. Consider the families ofrandom variables

(F ζ)ζ∈I and

(Hζ)ζ∈I . These families are

continuously differentiable in Dom (δ) with respect to theparameter ζ ∈ I. Assume that (ut )t∈[0,T ] ∈ D1,2 satisfying

DuF ζ 6= 0 a.s. on∂ζF ζ 6= 0

, ζ ∈ I.

Furthermore assume that uHζ∂ζFζ

DuFζ is continuous with respect toζ in Dom (δ). Then we have

∂ζEQ

[Hζ f

(Fζ)]

Fζ)(Hζ∂ζFζ

DuFζδ (u)− Du

(Hζ∂ζFζ

)+ ∂ζHζ

for any function f such that f(Fζ)∈ L2 (Ω), ζ ∈ I.

Proposition

Assume that F S0 := ST ∈ Dom (δ), HS0 := e−∫ T

0 rt dt ∈ Dom (δ),(ut )t∈[0,T ] ∈ D1,2 and satisfies

DuST 6= 0 a.s. on ∂S0ST 6= 0.

Furthermore assume thatuHS0∂S0

DuSTis continuous with respect

to S0 in Dom (δ). Then, the delta under the general SV modeldefined in equations (4) and (5) is given as

∆ =e−

∫ T0 rt dt

f (ST )

1∫ T0 G (t ,T ) dt

δ (u.)−∫ T

∫ Ts usut DsG (t ,T ) dtds(∫ T

0 ut G (t ,T ) dt)2

First Variation and Malliavin Derivative

The first variation Yt := ∂∂V0

Vt of Vt is defined by:

Yt = exp(∫ t

′(s,Vs)−

(ρv′(s,Vs))2)ds +

0v′(s,Vs)[ρdWs +

√1− ρ2dW 1

for all t ∈ [0,T ]. The Malliavin derivative of the stochastic volatility process Vt at time sis given by:

DsVt = v(s,Vs)ρYt

(∫ t

′(r ,Vr ))2 1− ρ2

2dr −

sv′(r ,Vr )

√1− ρ2dW 1

Stein & Stein Model

St= rdt + VtdWt , (7)

dVt = γ(Θ− Vt )dt + κdZt , (8)

dZt =[ρdWt +

√1− ρ2dW 1

〈dWt ,dZt〉 = ρdt ,

for t ∈ [0,T ].

DsVt = κρexp(−γ(t − s)),

for s ≤ t ≤ T .

DsG(t ,T ) = DsVt −∫ T

(∂σ

∂y(u,Vu)

)(∂σ

∂y(u,Vu)

)DtVuDsVudu

= κρe−γ(t−s)

2γe−2Tγ +

). (9)

Delta of Stein & Stein Model via Malliavin Calculus

∆ =e−rT

[f (ST )

(WT∫ T

(Vt + κρeγt

(∫ Tt e−γu (dWu − Vudu)

−∫ T

∫ Ts DsG(t ,T )dtds(∫ T

(Vt + κρeγt

(∫ Tt e−γu (dWu − Vudu)

))dt)2

Stein & Stein Model for ρ = 0

CorollaryUnder the assumption that W and Z are two independentBrownian motion in equation (7), the delta of a European optionunder the Stein & Stein model is given by

∆ =e−rT

[f (ST )

(WT∫ T

0 Vtdt

)]. (10)

THANK YOU FOR YOUR ATTENTION !

E-mail: yyolcu@metu.edu.tr

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