on the link between the malliavin derivative operator and the...

On the link between the Malliavin derivativeoperator and the implied volatility behaviour .

Can we expect the Malliavin calculus to be useful in applications?

Elisa Alòs

Universitat Pompeu FabraBarcelona

Mathematics of Quantitative Finance, Oberwolfach 2017

Jointly with Jorge León and Josep Vives

Elisa Alòs (Universitat Pompeu Fabra) Implied Volatility and Malliavin derivatives Oberwolfach 2017 1 / 58

Contents

1 Malliavin calculus : a useful tool ?

2 A modeling problem

3 Some extensions and applications

4 Conclusions


Contents




4 Conclusions


Malliavin calculus : a useful tool ?

After a short search in the net, we can easily find sentences like :

Not to be a party-pooper, but Malliavin calculus is essentially uselessin finance. Any practical result ever obtained with Malliavin calculuscan be obtained by much simpler methods by eg differentiating thedensity of the underlying process.

So, is Malliavin calculus just a ’language’ used bymathematicians ?


Malliavin calculus : a useful tool ?

In this talk, we will take a walk through some recent applications ofMalliavin calculus.We will see examples where the Malliavin calculus techniques havebeen used to solve real problems.We will discuss about how, when and why Malliavin calculus canbecome a useful tool.


Contents




4 Conclusions


A modeling problemWe want to construct models for stock prices that reproduce theempirical properties of the implied volatility surface

Stock : Apple, Expiration : 16/4/2010. Data : courtesy of Rafael DeSantiago (IESE, Barcelona)


A modeling problem

In particular, a popular rule-of-thumb states (see Lee 2002) that skewslopes decay with maturity approximately as 1√

T .


A modeling problem

Is it possible to reproduce this short-time behaviour by usingstochastic volatility models ?

The literature, around 2005, was no so clear : It was commonlyassumed that this empirical fact could be explained by introducingjumps in the asset model. But not by introducing stochasticvolatilities. But at the same time :

Fouque, Papanicolaou, Sircar and Solna : "Maturity cycles in impliedvolatility." Finance and Stochastics 8.4 (2004) : 451-477.

Time-periodic volatility coefficients are able to reproduce theempirical skew.


A modeling problem

Can we determine the class of stochastic volatility modelsthat can reproduce this skew behaviour ?


A modeling problem

Let us try to determine this class of stochastic volatilitymodels

Our objective will be to compute the at-the-money-skew, as thederivative of the implied volatility as a function of the log-stock price.Then we will compute its short-time limit.We will do this computations not for a concrete model, but for a bigclass of stochastic volatility models that will include both the classicaldiffusion volatilities and the time-periodic volatilities introduced byFouque, Papanicolaou, Sircar and Solna.


A modeling problem : Why Malliavin calculus ?

Because of the Clark-Ocone formula, under some general hypotheses

F =∫ t

0Er (DrF )dWr ,

where D denotes the Malliavin derivative operator. Then, all therelevant information is included in the Malliavin derivative ofF .Because the future average volatility

1T − t

∫ T

tσ2

s ds

is a non-adapted process. And the anticipating Itô’s formula is anatural tool to work with anticipating processes


A modeling problem : Notations

In this talk we will consider the following model for the log-price of astock under a risk-neutral probability measure Q :

Xt = x + rt − 12

∫ t

0σ2

s ds

+∫ t

0σs(ρdWs +

√1− ρ2dBs), t ∈ [0,T ].




Xt = x + rt − 12

∫ t

0σ2

s ds

+∫ t

0σs(ρdWs +

√1− ρ2dBs), t ∈ [0,T ].

Here, x is the current log-price.




Xt = x + r t − 12

∫ t

0σ2

s ds

+∫ t

0σs(ρdWs +

√1− ρ2dBs), t ∈ [0,T ].

Here, x is the current log-price, r is the instantaneous interest rate.




Xt = x + r t − 12

∫ t

0σ2

s ds

+∫ t

0σs(ρdWs +

√1− ρ2dBs), t ∈ [0,T ].

Here, x is the current log-price, r is the instantaneous interest rate,W and B are independent standard Brownian motions.




Xt = x + r t − 12

∫ t

0σ2

s ds

+∫ t

0σs(ρdWs +

√1− ρ2dBs), t ∈ [0,T ].

Here, x is the current log-price, r is the instantaneous interest rate,W and B are independent standard Brownian motions, ρ ∈ (−1, 1).




Xt = x + r t − 12

∫ t

0σ2

s ds

+∫ t

0σs(ρdWs +

√1− ρ2dBs), t ∈ [0,T ].

Here, x is the current log-price, r is the instantaneous interest rate,W and B are independent standard Brownian motions, ρ ∈ (−1, 1).Here, the volatility process σ is a square-integrable stochastic processwith right-continuous trajectories, adapted to the filtration generatedby W .



It is well-known the price of an European call with strike price K isgiven by

Vt = e−r(T−t)E [(eXT − K )+|Ft ],

where E is the expectation with respect to Q and F := FW ∨ FB.



BS(t, x , σ) will denote the price of an European call option under theclassical Black-Scholes model with constant volatility σ, current logstock price x , time to maturity T − t, strike price K and interest rater :

BS(t, x , σ) = exN(d+)− Ke−r(T−t)N(d−),

where N denotes the cumulative probability function of the standardnormal law and

d± := x − x∗tσ√T − t

± σ

2√T − t,

with x∗t := lnK − r(T − t).



LBS (σ) will denote the Black-Scholes differential operator, in the logvariable, with volatility σ :

LBS(σ) = ∂t + 12σ

2∂2xx + (r − 1

2σ2)∂x − r ·

It is well known that LBS(σ)BS(·, ·, σ) = 0. Moreover, we define

vt :=√

1T − t

∫ T

tσ2

s ds.


A modeling problem : Implied volatility

Let It(Xt) denote the implied volatility process, which is anF -adapted process such that

Vt = BS(t,Xt , It(Xt)).


A modeling problem : Objective

We want to computelimT→t

∂It∂Xt

(x∗t ).


A modeling problem : Malliavin calculusLet us consider a standard Browian motion W = Wt ,t ∈ [0,T ]defined in a complete probability space (Ω,F ,P) .SetH = L2 ([0,T ]) , and denote W (h) =

∫ T0 h (s) dWs the Wiener

integral of a deterministic function h ∈ H .

Let S be the set of random variables of the form

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

where n ≥ 1, f ∈ C∞b (Rn) (f and all its derivatives are bounded), andh1, .., hn ∈ H . Given a random variable F of this form, we define itsderivative as the stochastic process

DW

t F , t ∈ [0,T ]given by

DWt F =

n∑i=1

∂f∂xi

(W (h1) , ...,W (hn)) hi (t) , t ∈ [0,T ] .


A modeling problem : The Malliavin derivative

Example : If F = WT , DWt F = 1[0,T ](t).

Example : If F = W HT =

∫ T0 K (T , s)dWs , DW

t F = K (T , s)1[0,T ](t).

We see that the Malliavin derivatives of both processes arevery different


A modeling problem : Some useful properties ofthe Malliavin derivative

Notice that, for an adapted process u and for s > t

DWs ut = 0

The Malliavin derivative operator satisfies the chain rule : for anyF ∈ D1,2

W and any function f satisfying some regularity conditions

Dt f (F ) = f ′(F )DWt F

Example : If σt = σ0E(νW Ht ) (see Bayer, Friz and Gatheral (2016)),

DWr σt = σtνK (t, r).


A modeling problem : Some useful properties ofthe Malliavin derivative

In particular, the two above properties give us that, for s > t

Ds(BS(t,Xt ,K , vt

))= ∂BS

∂σ

(t,Xt ,K , vt

)DW

s vt

= ∂BS∂σ

(t,Xt ,K , vt

) 12(T − t)vt

( ∫ T

sDW

s σ2r dr

)


A modeling problem : The Skorohod integral

We denote by δW the adjoint of the derivative operator DW :

E (δW (u)F ) = E∫ T

0(DW

s F )usds

The operator δW is an extension of the Itô integral in the sense thatthe set L2

a ([0,T ]× Ω) of square integrable and adapted processes isincluded in Domδ and the operator δ restricted to L2

a ([0,T ]× Ω)coincides with the Itô stochastic integral defined in [16]). We willmake use of the notation δ (u) =

∫ T0 utdWt .

The expectation of the Skorohod integral is zero.


A modeling problem : The Skorohod integral

limn→∞

n−1∑i=1

g(iTn ,XiT/n,YiT/n

)(W(i+1)T/n −WiT/n)

=∫ T

0g(s,Xs ,Ys)dWs +

∫ T

0(DW

s )− (g(s,Xs ,Ys)) ds

=∫ T

0g(s,Xs ,Ys)dWs +

∫ T

0∂y(s,Xs ,Ys)

(∫ T

sDW

s σ2r dr

)ds,

where

(DWs )− (g(s,Xs ,Ys)) := limr→s(DW

s )− (g(r ,Xr ,Yr ))


The anticipating Itô’s formulaWe define Yt =

∫ Tt σ2

r dr . Then

f (T ,XT ,YT )= f (t,Xt ,Yt)

+∫ T

t

∂f∂s (s,Xs ,Ys)ds

+∫ T

t

∂f∂x (s,Xs ,Ys)

((r − σ2

s2

)ds + σs

(ρdWs +

√1− ρ2dBs

))

+12

∫ T

t

∂2f∂x2 (s,Xs ,Ys)σ2

s ds

+∫ T

t

∂f∂y (s,Xs ,Ys)dYs

+ρ∫ T

0(DW

s )−(∂f∂x (s,Xs ,Ys)

)σsds


A modeling problem : The anticipating Itô’sformulaThat is

f (T ,XT ,YT )

= f (t,Xt ,Yt) +∫ T

t

∂f∂s (s,Xs ,Ys)ds

+∫ T

t

∂f∂x (s,Xs ,Ys)((r − σ2

s )ds + σs(ρdWs +√1− ρ2dBs))

+12

∫ T

t

∂2f∂x2 (s,Xs ,Ys)σ2

s ds

−∫ T

t

∂f∂y (s,Xs ,Ys)σ2

s

+ρ∫ T

0

( ∂2f∂x∂y (s,Xs ,Ys)

)Λsσsds,

where Λs :=∫ T

s DWs σ2

r dr .Elisa Alòs (Universitat Pompeu Fabra) Implied Volatility and Malliavin derivatives Oberwolfach 2017 31 / 58

A modeling problem : computing the derivative ofthe implied volatility

How can we compute the derivative of the implied volatility ?

Take into account that

∂Vt

∂Xt= ∂xBS(t,Xt , It(Xt)) + ∂σBS(t,Xt , It(Xt)) ∂It

∂Xt(Xt).

Then, if we find a way to compute the derivative of Vt , it seems wecan compute the derivative of the implied volatility.


A modeling problem : computing the derivative ofthe implied volatility

Previous results (see for example Renault and Touzi (1996)) give usthat, when ρ = 0

∂It∂Xt

(x∗t ) = 0.

So the skew effect is given by the correlation. So we need tounderstand the effect of the correlation on option prices, and then onthe implied volatility.To this end, we will prove a formula that decomposes option prices asthe sum of the price when ρ = 0, plus a term due by correlation.


A modeling problem : An extension of the Hull andWhite formula

TheoremUnder some regularity conditions in the sense of Malliavin calculus,we have the following decomposition formula

Vt = E (BS(t,Xt , vt)|Ft)

+ ρ

2E(∫ T

te−r(s−t)∂xG(s,Xs , vs)σsΛsds|Ft

)

where G = (∂2xx − ∂x )BS

Notice that this expression is an exact decomposition.


A modeling problem : proof of the decompositionformula

Applying Itô’s formula to the process

t → e−rtBS(t,Xt , vt)

and taking conditional expectations we get



E(e−rTBS(T ,XT , vT )

∣∣∣Ft)

= E(e−rtBS(t,Xt , , vt)

∣∣∣Ft)

+∫ T

tLBS(vs)BS(s,Xs , vs)ds

+12E

(∫ T

te−rs∂σBS(s,Xs , vs) v 2

s − σ2s

vs(T − s)ds∣∣∣Ft

)

+E(∫ T

te−rs∂2

xσBS(s,Xs , vs) σsρΛWs

2vs(T − s)ds∣∣∣Ft

)

+12E

(∫ T

te−rs

(∂2

xx − ∂x)BS(s,Xs , vs)

(σ2

s − v 2s

)ds∣∣∣Ft

)



Now, taking into account that

LBS(vs)BS(s,Xs , vs) = 0

and

∂2xσBS(s,Xs , vs) 1

vs(T − s)=(∂3

xxx − ∂2xx

)BS(s,Xs , vs)

the result follows.


A modeling problem : An expression for thederivative of the implied volatility

Let It(Xt) denote the implied volatility process, which is anF -adapted process such that

Vt = BS(t,Xt , It(Xt)).


A modeling problem : An expression for thederivative of the implied volatilityPropositionUnder some general assumptions

∂It∂Xt

(x∗t ) =E (∫ T

t (∂xF (s,Xs , vs)− 12F (s,Xs , vs))ds|Ft)

∂σBS(t, x∗t , It(x∗t ))

∣∣∣∣∣Xt=x∗

t

,

whereF (s,Xs , vs) = ρ

2e−r(s−t)∂xG(s,Xs , vs)σsΛs

This is the expression from where we will compute theshort-time limit of the at-the-money derivative



To prove this result, notice that

Vt = BS(t,Xt , It(Xt))

and

Vt = E (BS(t,Xt , vt)|Ft) + E(∫ T

tF (s,Xs , vs)ds|Ft

),



Then we have

∂Vt

∂Xt= ∂xBS(t,Xt , It(Xt)) + ∂σBS(t,Xt , It(Xt)) ∂It

∂Xt(Xt).

and

∂Vt

∂Xt= E (∂xBS(t,Xt , vt)|Ft) + E

( ∫ T

t∂xF (s,Xs , vs)ds|Ft

).

Then the result follows after some algebra.


A modeling problem : Short-time limit behavior

We consider that the volatility is a lower-bounded process, regularenough (in the Malliavin calculus sense) and that There exists aconstant δ > −1

2 such that, for all 0 < t < s < r < T ,

E((

DWs σr

)2∣∣∣∣Ft

)≤ C (r − s)2δ ,

E((

DWθ DW

s σr)2∣∣∣∣Ft

)≤ C (r − s)2δ (r − θ)2δ .


A modeling problem : Short-time limit behaviorPropositionAssume that the previous hypotheses hold. Then :

∂σBS(t, x∗t , It(x∗t )) ∂It∂Xt

(x∗t )

= ρ

2E (L(t, x∗t , vt)∫ T

tΛsds|Ft) + O(T − t)(1+2δ)∧1

as T → t, where

L(t, x∗t , vt) = (∂2xx −

12∂x )G(t, x∗t , vt)

Now, we only need to take limits !



TheoremSuppose that the previous hypotheses (jointly with some continuityproperties on the volatility process) holld. Then1) Assume that δ is nonnegative and that there exists aFt-measurable random variable D+

t σt such that, for every t > 0,

sups,r∈[t,T ]∣∣∣E ((DW

s σr − D+t σt

)∣∣∣Ft)∣∣∣→ 0,

a.s. as T → t. Then

limT→t

∂It∂Xt

(x∗t ) = − ρ

2σtD+

t σt .



Theorem2) Assume that δ is negative and that there exists a Ft-measurable

random variable Lδ,+t σt such that, for every t > 0,

1(T − t)2+δ

∫ T

t

∫ T

sE(DW

s σr∣∣∣Ft

)drds − Lδ,+t σt → 0,

a.s. as T → t. Then

limT→t

(T − t)−δ ∂It∂Xt

(x∗t ) = − ρ

σtLδ,+t σt .

We have solved the puzzle ! ! !


A modeling problem : Examples

ExampleAssume σ = f (Y ), where f ∈ C1

b(R) and Y is the solution of astochastic differential equation :

dYr = a (r ,Yr ) dr + b (r ,Yr ) dWr ,

for some real functions a, b ∈ C1b(R).

limT→t

∂It∂Xt

(x∗t ) = − ρ

2σt(f ′(Yt)b (t,Yt)),

that agrees with previous limit results.



b(R) and Y is a process of the form

Yr = m + (Yt −m) e−α(r−t) + c√2α∫ r

te−α(r−s)dW H

s ,

where W Hs :=

∫ s0 (s − u)H− 1

2dWu.




Yr = m + (Yt −m) e−α(r−t) + c√2α∫ r

te−α(r−s)dW H

s ,

where W Hs :=

∫ s0 (s − u)H− 1

2dWu.

Case H > 1/2. In this case∫ r

te−α(r−s)dW H

s

=(H − 1

2

) ∫ r

0

(∫ r

s11[t,r ](u)e−α(r−u)(u − s)H− 3

2du)dWs .




Yr = m + (Yt −m) e−α(r−t) + c√2α∫ r

te−α(r−s)dW H

s ,

where W Hs :=

∫ s0 (s − u)H− 1

2dWu.

Case H > 1/2. In this case∫ r

te−α(r−s)dW H

s

=(H − 1

2

) ∫ r

0

(∫ r

s11[t,r ](u)e−α(r−u)(u − s)H− 3

2du)dWs .

andlimT→t

∂It∂Xt

(x∗t ) = 0.




Yr = m + (Yt −m) e−α(r−t) + c√2α∫ r

te−α(r−s)dW H

s ,

where W Hs :=

∫ s0 (s − u)H− 1

2dWu.

Case H < 1/2. In this case∫ r

te−α(r−s)dW H

s

=(12 − H

) ∫ r

0

(∫ r

s

[11[t,r ](u)e−α(r−u) − 11[t,r ](s)e−α(r−s)

]×(u − s)H− 3

2du)dWs

+∫ r

te−α(r−s)(r − s)H− 1

2dWs .




Yr = m + (Yt −m) e−α(r−t) + c√2α∫ r

te−α(r−s)dW H

s ,

where W Hs :=

∫ s0 (s − u)H− 1

2dWu.

Case H < 1/2. In this case∫ r

te−α(r−s)dW H

s

=(12 − H

) ∫ r

0

(∫ r

s

[11[t,r ](u)e−α(r−u) − 11[t,r ](s)e−α(r−s)

]×(u − s)H− 3

2du)dWs +

∫ r

te−α(r−s)(r − s)H− 1

2dWs .

andlimT→t

(T − t) 12−H ∂It

∂Xt(x∗t ) = −c

√2α ρσt

f ′ (Yt) .Elisa Alòs (Universitat Pompeu Fabra) Implied Volatility and Malliavin derivatives Oberwolfach 2017 51 / 58

ExampleIf σt = σ0E(νW H

t ) (see Bayer, Friz and Gatheral (2016)) we get

DWr σt = νσtK (t, r)

and then, if K (t, r) ∼ (t − r)H−1/2, (H) holds with δ = H − 1/2.Moreover,

limT→t

(T − t) 12−H ∂It

∂Xt(x∗t ) = − ρν

(H + 1/2)(H + 3/2)


Contents




4 Conclusions


Some extensions and applications : The secondderivative

We can study also the at-the-money short-time limit of the secondderivative. The first step is to study this derivative in the uncorrelatedcase. Then by using similar techniques, we can prove that, undersome regularity conditions

limT→t

∂2I∂k2 (t, k∗t ) =

( 112−

724ρ

2) (D+

t σ2t )2

σ5t

+ ρ2

6σ3t

((D+

t

)2σ2

t

),

which also agrees with previous results. Moreover, in the case ofnegative δ,

∂2It∂k2 (k∗t ) ∼ (T − t)2δ


Contents




4 Conclusions


Conclusions

We have seen that (in the context of stochastic volatility models) theproperties of the implied volatility (and VIX, VVIX, etc.) can be’translated’ in terms of the Malliavin derivative of the volatilityprocess. Then, if we want to construct a model satisfying someproperties :

a) We will be able to see easily if there exist a model satisfying thisset or properties. Moreover, if the answer is yes :

b) We will be able to construct this model.

We have a powerful tool in modeling problems !


Open problems

Out-of-the-money implied volatilities, long-time behaviour, etc. etc.


Many thanks !


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