applications of large deviations in finance and ... · applications of large deviations in nance...

Applications of large deviations in finance andmathematical physics.

Martin Forde

DCU, Dept of Mathematics

23rd June 2011

Martin Forde Applications of large deviations in finance and mathematical physics.

Outline

I Motivation and definition of the large deviation principle (LDP).

I Examples - Brownian motion, Cramer’s theorem, Levy processes,Sanov’s theorem.

I The Brownian sheet.

I Saddlepoint methods; the Feynman path integral.

I The Donsker-Varadhan LDP for the occupation measure of theOrnstein Uhlenbeck process dYt = −θYtdt + dWt for θ > 0.

I Applications to stochastic volatility models - the Ornstein-Uhlenbeckand CEV-Heston models.

I Large deviations for the maximum likelihood estimator of θ.

I Application to SPDEs - Freidlin-Wentzell theory for the stochasticheat equation.


The Large deviation principle (LDP): motivation

I Suppose we have sequence of random variables (Xn) such that Xn isconcentrated around x0 as n→∞, and for sets A away from x0,P(Xn ∈ A) tends to zero exponentially rapidly in n:

limn→∞

1

nlogP(Xn ∈ A) = −I (A)

i.e. ∀δ > 0, e−n(I (A)+δ) ≤ P(Xn ∈ A) ≤ e−n(I (A)−δ)

for n = n(δ) sufficiently large, and some rate function I ≥ 0.

I Example: for standard Brownian motion (Wt), Wt → 0 a.s. ast → 0 and (by SLLN) Wt

t → 0 a.s. as t →∞, but

limt→0

t logP(Wt > x) = −1

2x2 ,

limt→∞

1

tlogP(

Wt

t> x) = −1

2x2

for x > 0.


The Large deviation principle (LDP): definition

Definition. A sequence of random variables (Xn) in a topological spaceS satisfies the LDP with non-negative lower semicontinuous rate functionI if we have the following exponential upper/lower bounds for A ∈ B(S):

− infx∈A◦

I (x) ≤ lim infn→∞

1

nlogP(Xn ∈ A)

≤ lim supn→∞

1

nlogP(Xn ∈ A) ≤ − inf

x∈AI (x) .

Definition. Xn is said to satisfy the weak LDP if

limδ→0

limn→∞

1

nlogP(Xn ∈ Bδ) = −I (x) .


Examples

I Cramer’s theorem. Let (Xi ) be an i.i.d. sequence of randomvariables with finite mean E(X1) <∞ and cumulant generatingfunction

V (p) = logE(epX1 ) .

Then Sn = 1n

∑ni=1 Xi satisfies the LDP with rate function equal to

the Fenchel-Legendre transform V ∗(x) = supp∈R{px − V (p)}. For

Brownian motion, V (p) = 12p

2, V ∗(x) = 12x

2.

I A Levy process (Xt) has i.i.d. increments, so (Xt

t ) satisfies an LDPas t →∞ with rate function V ∗(x).


Sketch proof of Cramer’s theorem

I Cramer upper bound proved using a simple Chebychev argument:

P(Sn ≥ x) = E(1{Sn≥nx}) ≤ E(e−θnx eθSn) = e−nθxenV (θ) .

We then tighten the bound by taking the inf over θ on the righthand side:

P(Sn ≥ x) ≤ e−n supθ [θx−V (θ)] = e−nV∗(x) .

I Lower bound is obtained by changing to a different measure Pθ∗(x)

under which {Sn ≥ x} is no longer a rare, large deviation event.


Sanov’s theorem

I Let (Xi ) be a sequence of n i.i.d. random variables in R withcommon probability measure µ. The sample distribution:

Ln =1

n

n∑i=1

δXi

is a random probability measure (a.k.a. the empirical measure).I Let Pn = µn ◦ (Ln)−1 denote the distribution of Ln, where µn is the

product measure. Pn is a probability measure on (P(R),B(P(R)) -

i.e. Pn ∈ P(P(R)). By SLLN, we can show that Pnw→ δµ.

I Theorem (Sanov). (Ln) satisfies an LDP in the topology of weakconvergence1 as n→∞ with rate function given by the infinitedimensional counterpart of V ∗(x):

R(ν|µ) = supp∈B(R)

[∫pdν − log

∫epdµ

],

where B(R) is the space of bounded, measurable functions on R.(see [Var10]), so P(Ln ∈ A) ≈ e−t infν∈A R(ν|µ) for µ /∈ A.

1A ⊆ P(R) is closed iff for any (µn) ∈ A with µnw→µ ∈ P(R), we have µ ∈ A.


By solving the variational problem on the previous slide, we can showthat the rate function simplifies to

R(ν|µ) =

∫ ∞−∞

(logdν

dµ) dν =

∫ ∞−∞

dν

dµ(log

dν

dµ) dµ if (ν � µ) ,

∞ otherwise


The Brownian sheet

I Let (Zt) be the Brownian sheet, i.e. the centred Gaussian process on[0, 1]2 with zero mean and covariance structure

E(ZtZs) = (s1 ∧ t1)(s2 ∧ t2)

where t = (t1, t2), s = (s1, s2). Z is a “two-parameter” Brownianmotion.

I Then√εZ satisfies the LDP on C0([0, 1]2) with rate function

I (f ) =

1

2

∫[0,1]2

(∂2f

∂s∂t)2 dsdt if

∂2f

∂s∂t∈ L2 ,

+∞ otherwise .


Saddlepoint approximations

The LDP gives crude exponential bounds. For a Levy process (Xt) withdensity pt(x), we can sharpen these bounds using saddlepoint methods,proved using contour integration:

I Large-time estimate

pt(xt) ∼ e−t(p∗x−V (p∗))√2πtV ′′(p∗)

=e−tV

∗(x)√2πtV ′′(p∗)

(t →∞)

(see F-Lopez,Forde&Jacquier[FLFJ11]).

I Tail estimate

pt(x) ∼ e−p∗( x

t )x+tV (p∗( xt ))√

2πtV ′′(p∗( xt ))

(x →∞).

(see F-Lopez,Forde[FLF11]). p∗ = p∗(x) is the unique solution tothe saddlepoint equation V ′(p∗) = x .

I Similar saddlepoint estimates can be obtained for the well knownHeston stochastic volatility model for large-time[FJ09],[FJM10],small-time[FJL10] and tail regimes (see Friz et al.[FGGS10]).


Saddlepoint methods in infinite dimensions - the Feynmanpath integral

I Consider the Feynman path integral for a wavefunction ψ(x , t):

ψ(x , t) = (2πi)−n2

∫γ:γt=x

ei} [ 1

2 m∫ t

0γ2dτ −

∫ t0V (γτ )dτ ] ψ(γ0, 0)Dγ

= (2πi)−n2

∫He−

i2

m}∫ t

0γ2dτdµ(γ)

for x ∈ Rn, with ψ(x , 0) = ei} f (y)χ(y).

I The first line is the formal expression for the path integral which wedefine rigorously via the Fresnel integral in the second line overH = {γ ∈ C [0, t] : γ ∈ L2[0, t], γt = x} for V , ψ(., 0) ∈ F(H).

I f (γ) = e−i}∫ t

0V (γτ )dτψ(γ0, 0) is the Fourier transform:

f (γ) =

∫He(γ,γ1)dµ(γ1)

of µ ∈M(H) with B.V., where (γ, γ1) = m}∫ 1

0γγ2dτ , [AHM08]2.

I ψ(x , t) satisfies the Schrodinger eq: i}∂ψ∂t = − }2

2m∆ψ + V (x)ψ .2Feynman integral can also be defined via analytic continuation of Wiener measure.


Letting }→ 0 - the semi-classical expansion

I The integrand ei}St = e

i} [ 1

2 m∫ t

0γ2dτ −

∫ t0V (γτ )dτ ] is an

infinite-dimensional oscillatory integral. If we let }→ 0, we tendtowards classical everyday Newtonian mechanics and the integralbecomes highly oscillatory, so we expect the main contribution tocome from the classical path γ∗ which make St stationary (inanalogy with the finite-dimensional method of stationary phase).

I From this we can compute the semi-classical expansion:

ψ(x , t) ∼ (2πi)−n2

1√det(...)

ei} [ 1

2 m∫ t

0(γ∗)2dτ −

∫ t0V (γ∗τ )dτ ] χ(y)

as }→ 0 (see [AHM08]).

I The stationary path γ∗ is just the classical path mγ = −∇Vfollowed by a particle moving under the potential V (x), which goesfrom y to x in time t with initial momentum f ′(y) (IF there is aunique non-degenerate stationary path γ∗ with this property).


The Donsker-Varadhan LDP for the occupation measure ofthe Ornstein Uhlenbeck process

Let dYt = −θYtdt + dWt be an OU process for θ > 0. Let

µt(A) =1

t

∫ t

0

1A(Ys)ds

denote the proportion of time that Y spends in A, for A ∈ B(R). Foreach t > 0 and ω, µt(ω, .) ∈ P(R). Then from [DV76] (or [Str84]) µt(.)satisfies the LDP as t →∞ in the topology of weak convergence, with agood3, convex, lower semicontinuous rate function given by:

IB(µ) = − infu∈D+

∫ ∞−∞

Lu

udµ

where L = −θy ddy + 1

2d2

dy2 is the infinitesimal generator for Y and D+ isthe set of u in the domain D of L with u > ε for some ε > 0.

3good means that the level set {x : I (x) ≤ α} is compact.Martin Forde Applications of large deviations in finance and mathematical physics.

Simplifying the rate function

I We can simplify IB to the following:

IB(µ) =1

2

∫ ∞−∞|∂y√

(dµ

dµ∞)(y)|2 µ∞(dy)

for µ� µ∞, where µ∞(y) = ( θπ )12 e−θy

2

is the unique stationary

distribution for Y , i.e. N(0, 1/(2θ)). dµdµ∞

is the Radon-Nikodym

derivative. If µ is not absolutely cts wrt µ∞, then IB(µ) =∞.

I P(R) can be made into a (non-compact) metric space using theProkhorov metric.

I IB(µ) clearly attains its minimum value of zero at µ = µ∞, and wecan show that µ∞ is the unique minimizer of IB(µ)


An uncorrelated Stochastic volatility model

I Consider a stochastic volatility model for a log stock price processXt = log St : {

dXt = − 12σ

2(Yt)dt + σ(Yt)dW1t ,

dYt = −θYtdt + dW 2t

(1)

for θ > 0, where f (y) = σ2(y) is a continuous non-decreasingfunction with 0 < fmin ≤ f (y) ≤ fmax and d〈W1,W2〉 = 0 withx0 = 0.

I The distribution of Xt , conditional on At = 1t

∫ t

0σ2(Ys)ds, is

N(− 12Att,Att).


Using the contraction principle

I Let F (µ) =∫∞−∞ f (y)µ(dy) for µ ∈ P(R). Then we can re-write At

as

At = F (µt) =

∫ ∞−∞

f (y)µt(dy) =1

t

∫ t

0

f (Ys)ds .

I F : P(R) 7→ [fmin, fmax] is a bounded, continuous functional 4,

because if µnw→µ then

∫f (y)µn(dy)→

∫f (y)µ(dy), because

f ∈ Cb.

I Thus, by the contraction principle from large deviations theory, At

also satisfies the LDP, with rate function

If (a) = infµ∈P(R) : F (µ)=a

IB(µ) , a ∈ [fmin, fmax] . (2)

4in the topology of weak convergence.Martin Forde Applications of large deviations in finance and mathematical physics.

A joint LDP for (Xt/t,At)

Proposition [Forde11a]. (Xt/t,At) satisfies a LDP on R× [fmin, fmax] ast →∞ with rate function

I (x , a) = aV ∗(x

a) + If (a)

where V ∗(x) = 12 (x + 1

2 )2.

Sketch proof. Let Zt = Xt/t. We first note that

(Zt ,At)d= ( 1

tWtAt − 12At ,At). Conditioning on At , formally we have

P(|Zt − x | < δ, |At − a| < δ) ≈ (.)× e−aV∗( y

a )/t e−If (a)/t

as t →∞, where aV ∗( xa ) is the rate function of Wta − 1

2a, for a fixed.This argument can be made rigorous.


I Corollary [Forde11a]. (Xt/t) satisfies the LDP as t →∞ with agood rate function given by

I (x) = infa∈[fmin,fmax]

{(x + 1

2a)2

2a+ If (a)

}≤

(x + 12 σ

2)2

2σ2(3)

Proof The LDP with a good rate function just follows from thecontraction principle.

I This can be applied to price call options with value E(eXt − K )+.

I We can relax the assumption that σ is bounded to a sublineargrowth condition σ(y) ≤ A(1 + |y |p), A > 0, p ∈ (0, 1); in this casewe take the infimum over all a ∈ (0,∞) in (3) (see [Forde11b]).

I The LDP can be also be extended to a Levy process or a CEVprocess evaluated the OU time-change

∫ t

0f (Ys)ds.


For the case of sublinear growth, the following lemma is the keyobservation:

Lemma. If IB(µ) ≤ α and k ∈ (0, 1), we have∫ ∞−∞

y2µ(dy) ≤ α + k

2k(1− k).

Proof. If we consider the test function u = eky2

inIB(µ) = − infu∈D+

∫∞−∞

Luu dµ, then −Lu

u (y) = k [2(1− k)y2 − 1] . Fromthis we obtain

α ≥ IB(µ) = − infu∈D+

∫ ∞−∞

Lu

udµ = sup

u∈D+

−∫ ∞−∞

Lu

udµ

≥∫ ∞∞

k [2(1− k)y2 − 1]µ(dy)

= 2k(1− k)

∫ ∞−∞

y2µ(dy)− k .


The CEV model

I The CEV model is defined by the SDE

dSt = δSβt dWt (4)

with β ∈ (0, 1), δ > 0 and S = 0 absorbing so (St) is a martingale.

I The transition density is

p(t,S0,S) =S−2β− 3

2 S12

0

δ2|β|texp(−S−2β

0 + S−2β

2δ2β2t)Iν(

S−β0 S−β

δ2β2t) (S > 0),

(5)where β = β − 1, ν = 1

2|β |, and Iν(.) is the modified Bessel function

of the first kind (see [DavLin01]).

I Proposition. Let γ = 1/|β|. Then using (5) we can show that(St/t

γ) satisfies the LDP on [0,∞) as t →∞ with continuous ratefunction

ICEV(K ) =K 2|β|

2δ2β2(K ≥ 0) .


The CEV-Heston model

I Combining the CEV model with a CIR time-change, we can definethe uncorrelated CEV-Heston model, governed by the following SDEs{

dSt = Sβt√YtdW

1t ,

dYt = κ(θ − Yt)dt + σ√YtdW

2t

with dW 1t dW

2t = 0, Y0 = y0 > 0.

I Conditioning on∫ t

0Ysds, we can write St = X∫ t

0Ysds

, where X is

now just the standard CEV process dXt = δXβt dWt with δ = 1.

I By a similar argument to that used for the OU model, we have:Proposition. (St/t

γ) satisfies the LDP on [0,∞) as t →∞ with agood rate function given by

ICEVH(K ) = infa∈(0,∞)

[aICEV(K

aγ) + ICIR(a)] ≤ θICEV(

K

θγ) (K ≥ 0) ,

where ICIR(a) is the rate function of At = 1t

∫ t

0Ysds, and the

infimum of I is attained uniquely at K = 0, where I (K ) = 0.


Call options

I We can show that call options have the same large-time behaviour

limt→∞

1

tlogE(St − Ktγ)+ = ICEVH(K ) .

I For the large-time, fixed-strike regime, we can show that

S0 − E(St − K )+ = cK (θt)−γ2 (1 + o(1)) (t →∞)

where c = 1Γ(1+ γ

2 ) [ 12 (

S−2β0

δ2β2 )]γ2 .


The large-maturity smile for the CEV-Heston model

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.060

0.065

0.070

0.075

Figure: Here we have plotted the implied volatility for the CEV-Heston modelin the large-time, large-strike regime for t = 30 years using Corollary 7.1 inGao&Lee[GL11], with δ = 1, β = .7,S0 = 1 and κ = 1.15, θ = .04, σ = 0.2.Working in the large-time, large-strike parameterizaton allows us to see theslope and the convexity effect.


The Maximum likelihood estimator of θ for the OU process

I Let θ0 denote the true value of θ.

I Let PTθ be the measure induced on (C[0,T ],B(C[0,T ])) by the

solution of dYt = −θYtdt + dWt . Then, from Girsanov’s theorem,we have the likelihood ratio

L(θ) =dPT

θ

dPT0

= e−∫ T

0θYtdYt− 1

2

∫ T0θ2Y 2

t dt (6)

(note that PT0 is just the Wiener measure).

I Taking the log of L(θ), differentiating wrt θ and setting to zero, weobtain the classical maximum likelihood estimator for θ:

θT = −∫ T

0YtdYt∫ t

0Y 2t dt

,

(see [Kut04]) and θT is a consistent estimator of θ0 (i.e. θT → θ0 inprobability as T →∞).


Large deviations for θT

It can be shown (see [FLP99]) that θT satisfies the LDP with good ratefunction

J(θ) =

{1

4θ (θ − θ0)2 (θ ≥ 13θ0) ,

−2θ + θ0 (θ < 13θ0) .

0.02 0.04 0.06 0.08

0.01

0.02

0.03

0.04

Figure: Here we have plotted J(θ) for θ = .04.Martin Forde Applications of large deviations in finance and mathematical physics.

SPDEs

I Consider the stochastic heat equation with small-noise:

∂tuεt,x =

1

2∂xxu

εt,x +

√ε Wt,x (7)

on [0,T ]× [0, 1], with Dirichlet boundary condition uε0,. ∈ C 2α,

0 ≤ α < 14 and uεt,0 = uεt,1 = 1. W is space-time white noise, which

is a Gaussian random set function such that WA ∼ N(0,Leb(A)) forA ∈ B([0,T ]× [0, 1]) and E(WAWB) = Leb(A ∩ B).

I Wt,x := W[0,t]×[0,x] is the previously defined Brownian sheet.

I We can give a rigorous meaning to (7) by writing the solution in theintegrated form

uεt,x =

∫ 1

0

Gt(x , y)u0(y)dy +√ε

∫ t

0

∫ 1

0

Gt−s(x , y)W (ds, dy)

where the stochastic integral on the right is defined in a similar wayto the classical Ito integral, and Gt(x , y) is the usual Green kernelfor the non-stochastic heat eq ∂tu = 1

2∂xxu with the same Dirichletboundary condition (see Pardoux[Par93]).


Large deviations for the stochastic heat equation

I The skeleton of h = h(t, x) in the Cameron-Martin space for W isgiven by

Z ht,x =

∫ 1

0

Gt(x , y)u0(y)dy +√ε

∫ t

0

∫ 1

0

Gt−s(x , y)∂2h

∂t∂x(s, y)dsdy .

I By a generalized contraction principle, uε satisfies the LDP onχ = Cα,0([0,T ]× [0, 1]) with rate function

S(f ) =

{inf{I (h) : Z h = f } , f ∈ Im(Z )+∞ (otherwise)

(8)

(see [CM07]), where I (h) = 12

∫[0,1]2 ( ∂

2h∂s∂t )2 dsdt is the previously

defined rate function for the Brownian sheet.

I We can also compute a small-noise LDP for the alternative way ofapproaching SPDEs as a Hilbert-space valued SDE driven by aHilbert-spaced valued Brownian motion (see [DPZ92]).


References

Albeverio,S., Hoegh-Krohn, R. and S.Mazzuchi, “Mathematicaltheory of Feynman path integrals. An introduction”, Second edition.Lecture Notes in Mathematics, 523. Springer Verlag, Berlin, 2008.

Chenal, F. and A.Millet, “Uniform large deviations for parabolicSPDEs and applications”, Stochastic Processes and theirApplications, 72, 161-186, 2007.

Davydov, D. and V. Linetsky, “The Valuation and Hedging of Barrierand Lookback Options under the CEV Process,” ManagementScience, 47, 949-965, 2001.

Da Prato, G. and J.Zabczyk, ‘Stochastic equations in infinitedimensions”, Cambridge University Press, 1992, XVIII.

Donsker, M.D. and S.R.S Varadhan, “Asymptotic evaluation ofMarkov process expectations for large time, III”, Comm. Pure Appl.Math., 29, pp.389-461 (1976).

P.Friz, S.Gerhold, A.Gulisashvili, S.Sturm, “On refined volatilitysmile expansion in the Heston model”, forthcoming in QuantitativeFinance.


References

Figueroa-Lopez, J., M.Forde and A.Jacquier, “The large-time smileand skew for exponential Levy models”, (2011), submitted.

Forde, M., A.Jacquier and A.Mijatovic (2010), “Asymptoticformulae for implied volatility in the Heston model”, Proc. R. Soc.A, 466, 3593-3620, 2010.

Forde, M., “Large-time asymptotics for an uncorrelated stochasticvolatility model”, (2011), to appear in Statistics and ProbabilityLetters.

Forde, M., “Large-time asymptotics for general stochastic volatilityand time-changed Levy models”, (2011), submitted.

Forde, M., Forde, M. and A.Jacquier, “The large maturity smile forthe Heston model” (2009), forthcoming in Finance and Stochastics.

M.Forde, A.Jacquier and R.Lee , “The small-time smile and termstructure of implied volatility for the Heston model”, (2010).

Figueroa-Lopez, J. and M.Forde, “Sharp tail estimates forexponential Levy models”, (2011).


Florens-Landais, D. and H.Pham, “Large deviations in estimation ofan Ornstein Uhlenbeck model”, J.Appl.Prob.,36, 60-77 (1999).

Gao, K., and R. Lee, “Asymptotics of Implied Volatility in ExtremeRegimes”, 2011, preprint.

Kutoyants Yu. A., “Statistical Inference for Ergodic DiffusionProcesses”, Springer Series in Statistics, London, 2004, 496p.

Pardoux, E., “Stochastic Partial Differential Equations, a review”,Bull. Sc. Math., 117, 29-47, 1993.

Stroock, D.W. “An introduction to the theory of large deviations”,Springer-Verlag, Berlin, (1984).

Varadhan, S.R.S., “Large deviations” lecture notes at NYU, Spring2010.

Walsh, J., “An introduction to stochastic partial differential

equations”, in Ecole d’ete de Probabilite de Saint Flour XIV, LNM,1180, 265-439, Springer, 1986.


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