Large deviations and asymptotic methods in finance · 2015. 12. 10. · Large-time asymptotics: the Donsker-Varadhan LDP for the occupation measure of the Ornstein-Uhlenbeck process

Large deviations and asymptoticmethods in finance

Martin FordeKing’s College London

Feb 2014

Martin Forde King’s College London Feb 2014 Large deviations and asymptotic methods in finance

Contents

I The Large deviation principle; outline proof of theDonsker-Varadhan LDP for occupation measures; applications tostochastic volatility models.

I Small-time asymptotics - the heat kernel expansion fromdifferential geometry; application to small-time asymptotics forstochastic volatility models.

I Tail asymptotics - sharp density estimates for the SABR modelfor ρ 6= 0, extending earlier work of Gulisashvili[GS10] and Friz et al[BFL08].

I Large deviations in portfolio optimization under proportionaltransaction costs.

I The Brownian random bridge; small-time asymptotics usingsaddlepoint methods, and application in information-based assetpricing.

I SPDEs - large deviations for the stochastic Burgers equationfrom fluid dynamics.


The Large deviation principle (LDP)

I Suppose we have a sequence of random variables (Xn) such that Xn

is concentrated 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) (1)

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), we havelimt→0 t logP(Wt > x) = − 1

2x2, limt→∞

1t logP(Wt

t > x) = − 12x

2

for x > 0, so here I (x) = 12x

2 in both cases.

I Definition. A sequence of random variables (Xn) in a topologicalspace S satisfies the LDP with a LSC rate function I ≥ 0 if we havethe 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) .


Large-time asymptotics: the Donsker-Varadhan LDP forthe occupation measure of the Ornstein-Uhlenbeck process

I Let dYt = −αYtdt + dWt be an OU process for α > 0, and let

µt(A) =1

t

∫ t

0

1A(Ys)ds

denote the proportion of time that Y spends in A, for A ∈ B(R).µt ∈ P(R) is a random probability measure on R.

I Then from [DV76], µt satisfies the LDP as t →∞ in the topology ofweak convergence1, with a non-negative, convex, LSC rate function

Iα(µ) = − infu∈D+

∫ ∞−∞

Luu

dµ

where L is the infinitesimal generator for Y and D+ is the set of uin the domain D of L with u ≥ ε > 0 for some ε > 0.

I µtw→µ∞ as t →∞, where µ∞(y) = (απ )

12 e−αy

2

is the uniquestationary distribution for Y , and intuitively for A ∈ B(P(R)) withµ∞ /∈ A, P(µt ∈ A) ≈ e−t infµ∈A Iα(µ) as t →∞.

1Generated by Uφ,x,δ = {µ ∈ P(R) : |∫φdµ − x | < δ, φ ∈ Cb(R), δ > 0, x ∈ R.


Proof of the large deviation upper bound for compact sets

I Let u ∈ D and ψ(y , t) = Ey (u(Yt)e−

∫ t0Luu (Yt)dt . Then from the

Feynman-Kac formula ψ(y , t) is the unique solution to

∂tψ = Lψ − Luu

(y)ψ

and ψ(y , t) = u(y) is the solution. Using that u ≥ ε, we see that

u(y) = Ey (u(Yt)e−

∫ t0Luu (Yt)dt) ≥ εEy (e−

∫ t0Luu (Yt)dt) .

I Then for C ∈ B(P(R)), we have

u(y)

ε≥ Ey (1µt∈Ce

−∫ t

0Luu (Yt)dt) = Ey (1µt∈C e−t

∫ Luu (y)µt(dy))

≥ Ey (1µt∈C e−t supµ∈C∫ Lu

u (y)µ(dy)) .

I Re-arranging, we obtain

Py (µt ∈ C ) ≤ u(y)

εet supµ∈C

∫ Luu (y)µ(dy))

⇒ lim supt→∞

1

tlogPy (µt ∈ C ) ≤ inf

u∈D+supµ∈C

∫Luu

(y)µ(dy) .


I If C is compact, then we can interchange the inf and the sup toobtain

lim supt→∞

1

tlogPy (µt ∈ C ) ≤ sup

µ∈Cinf

u∈D+

∫Luu

(y)µ(dy)

= − infµ∈C− inf

u∈D+

∫Luu

(y)µ(dy)

= − infµ∈C

Iα(µ) .

I (see [DV75],[Var84]). For the OU process, it can be shown that therate function simplifies to

Iα(µ) =1

2

∫ ∞−∞

ψ′(y)2 µ∞(dy)

for µ� µ∞, and dµdµ∞

= ψ2. If µ is not absolutely cts wrt µ∞,

Iα(µ) =∞.

I If µ = µ∞, then clearly ψ′ = 0 and Iα(µ∞) = 0, and we can showthat µ∞ is the unique minimizer of Iα(µ) (see [FK13]).


Application to stochastic volatility models

I Now consider a stochastic volatility model{dSt = Stσ(Yt)dW

1t ,

dYt = −αYtdt + dW 2t

and dW 1t dW

2t = ρdt. Then under a mild sublinear growth condition

on σ(.), we can use the D-V LDP to show that 1t log St satisfies the

LDP as t →∞ with rate function:

I (x) = infµ∈P(R)

[(x −M(µ))2

2ρ2F (µ)+ Iα(µ)]

(see Forde&Kumar[FK13]), where ρ =√

1− ρ2,b(y) = αyσ(y)− 1

2σ′(y), F (µ) =

∫σ2(y)µ(dy),

G (µ) =∫b(y)µ(dy), M(µ) = − 1

2F (µ) + ρG (µ).

I Use the Ritz method to compute I (x) numerically. Can also dealwith a more general process dYt = β(Yt)dt + α(Yt)dW

2t if β has

mean reverting behaviour for |y | � 1, and we can incorporatestochastic interest rates with an independent CIR short rateprocess (important for t � 1).


Small-time asymptotics: the heat kernel expansion

I Consider a general uncorrelated stochastic volatility model for a logstock price Xt :{

dXt = − 12Y

2t dt + YtdW

1t ,

dYt = µ(Yt)dt + α(Yt)dW2t

(2)

I Under suitable regularity conditions on µ, α, the transition densityfor (Xt ,Yt) satisfies

pt(x, y) ∼ 1

2πtu0(x, y) e−

12 d(x,y)2/t + A(x,y) (t → 0)

where d(x, y) = infγ∈C [0,1]:γ(0)=x,γ(1)=y

∫ 1

0

√(∑

i,j gijdγidt

dγjdt ) dt is

the shortest distance from x = (x0, y0) to y = (x1, y1) under themetric ds2 =

∑ij gijdx

idx j , where gij = a−1ij is the inverse of the

diffusion matrix for the model (2), and

u0(x, y) = |g |(x)−12 det(−∂

2φ(x, y)

∂xi∂yj) , A(x, y) =

∫ 1

0

∑ij

gijAi dγ∗j

dsds

and |g | = | det gij |, φ(x, y) = 12d(x, y)2,

Ai = bi − 12

∑j

1√|g |∂j(√|g | g ij) where b = (− 1

2y2, µ(y)).


I By integrating in the y variable and again using saddlepointmethods, we can integrate the heat kernel expansion to obtain asmall-time expansion for call options and implied volatility, seeArmstrong,Forde&Zhang[AFZ13] for details.

I Our proof also requires use of the Davies upper bound for the heatkernel on a Riemmanian manifold with curvature bounded frombelow).

I We can also deal with local volatility and non-zero correlationusing a gauge transformation (for the latter, we have to makerestrictions on µ).


Tail asymptotics: the SABR model with ρ 6= 0, andperturbations of stoc vol models (NEW)

I Consider the correlated SABR model St with β = 1:{dSt = StYtdW

1t ,

dYt = σYtdW2t

with dW 1t dW

2 = ρdt, ρ ≤ 0.I Using saddlepoint methods, when y0 = σ = 1, we have the following

right tail behaviour for the transition density of St :

pt(x) =1

2√πe− ρ

ρ2− t8−

log2 22t

x−1− 1

ρ2

C (x)32

[|ρ|D(x)

ρ2

] 12−

log 2t[

log

(|ρ|D(x)

ρ2

)]− 12

× exp

(− 1

2t

[log

(|ρ|D(x)

ρ2

)]2)(1 + O([log(|ρ|C (x)/ρ2)]−

12 ))

where u0(x) is the unique positive solution to ρρ2 D(x) = log ux

uxand

D(x) = C (x) + log x + ρ > 0, C (x) =√

(log x + ρ)2 + ρ2 > 0. Wecan easily adapt this for y0, σ 6= 0, which extends the result in[GS10] for the uncorrelated case (see Forde&Zhang[FZ14]).


I Using Girsanov’s theorem, we can extend the previous result tocharacterize the right-tail behaviour of transition density for aperturbed SABR model where dYt = g(Yt)dt + σYtdW

2t , and also

for the Heston and Sten-Stein models.


Large deviations and transaction costs

I Consider a market with one safe asset S0t = 1, and a risky asset

dSt = St(µdt + σdWt with ask (buying) price St .

I We assume that the bid (selling) price is (1− ε)St , where ε ∈ (0, 1)is the relative bid-ask spread.

I (ϕ0t , ϕt) is called an admissible self-financing trading strategy if

both processes are right cts and of a.s. finite variation (or else weincur infinite costs in finite time) which satisfy the self-financingcondition:

dϕ0t = −Stdϕ↑t + (1− ε)Stdϕ

↓t .

I The values of the safe position Xt and of the risky position Yt thenevolve as:

dXt = −Stdϕ↑t + (1− ε)Stdϕ↓t ,

dYt = µYtdt + σYtdWt + Stdϕ↑t − Stdϕ

↓t .

where ϕt = ϕ↑t − ϕ↓t is the difference of two increasing processes.


I Let Υt denote arithmetic Brownian motion with drift µ− 12σ

2,volatility σ and reflecting barriers at 0 and b:

dΥt = (µ− 1

2σ2)dt + σdWt + dLt − dUt

with Υ0 = x , and b > 0, where Lt and Ut are the associated localtime processes with {dLt > 0} ⊆ {Υt = 0} and{dUt > 0} ⊆ {Υt = b}.

I The liquidation value of the wealth associated to an admissiblestrategy is given by Ξϕt = ϕ0

t + ϕ+t (1− ε)St − ϕ−t St .

I It can be shown that under exponential utility U(x) = −e−θx forθ > 0, the ϕt process which maximizes

lim infT→∞

− 1

θTlogE(e−θΞϕT )

satisfies

dϕt/ϕt = d logϕt = dLt − dUt .

for some b > 0 (see [GM13]). Thus we buy when Υt = 0 and sellwhen Υt = b and do nothing in between, so the interval (0, b) iscalled the no-trade region (can show that width of the region is

O(ε13 ) for ε� 1).Martin Forde King’s College London Feb 2014 Large deviations and asymptotic methods in finance

I The relative share turnover is then defined as∫ t

0

d‖ϕs‖ϕs

= Lt + Ut

which is a measure of trading volume.

I Using martingale arguments, we compute Ex(e−α(Lτ+Uτ )) where τ isan independent Exp(λ) random variable.

I Using this and a Tauberian result, in [FKZ13] we show that1t

∫ t

0d‖ϕs‖ϕs

ds satisfies the LDP on [0,∞) as t →∞ with a convexrate function given by the Fenchel-Legendre transform

Λ∗(x) = supα∈R

[αx − Λ(α)]

for x ≥ 0, where Λ(α) = (α∗)−1(α) and

α∗(λ) = γ coth(γb)−√γ2/ sinh2(γb) + δ2

and δ = µσ2 − 1

2 , γ =√δ2 + 2λ/σ2.


Asymptotics for the Brownian random bridge

I Let ξtT = tT X + βtT where βtT is a standard 1-d Brownian bridge

with β0T = βTT = 0 and X ∼ ν is independent of β.I Now let Ft = σ({ξsT}0≤s≤t) be the filtration generated by ξ.

Then X is FT -measurable and can be shown that ξtT is a Markovprocess wrt Ft but X is not Ft-measurable for t < T .

I X = ξTT ∼ ν at time 0, and X becomes “known” at T , but att < T we only know the value of ξtT but not X or βtT , so βtT is thenoise. Model asset price process as XtT = E(X | Ft) = E(X | ξtT ),

then dXtT = Var(X | ξtT )T−t dWt , for some Ft-Brownian motion.

I From Macrina et al.[HHM11], assuming ν has a smooth density:

P(X ∈ dy | ξtT = x) =e−

12 [ (y−x)2

ε − y2

T ]ν(y)dy∫e−

12 [ (z−x)2

ε − z2

T ]ν(z)dz∼ C (x)e

− (y−y∗(x))2

2 Ttε ν(y)dy (t → T )

Var(X | ξtT = x)

T − t= 1 + [

1

T+

v(y∗)v ′′(y∗)− v ′(y∗)2

v(y∗)2]ε + O(ε2) (3)

as t → T , where ε = T − t and y∗(x) = Tt x ∼ XtT is the

saddlepoint. Thus at leading order, XtT behaves like Brownianmotion for T − t � 1 *.


SPDEs

I Consider the stochastic Burgers equation with small-noise:

∂tuε(t, x) = ν∂2

xxuε(t, x) +

√εW (t, x) + uε(t, x)∂xu

ε(t, x) (4)

on [0,T ]× [0, 1], with Dirichlet boundary conditionuε(t, 0) = uε(t, 1) = 0 and uε(0, x) = u0(x) for u0 ∈ L2[0, 1]. W isspace-time white noise, i.e. a Gaussian random set function withWA ∼ N(0,Leb(A)) for A ∈ B([0,T ]× [0, 1]) andE(WAWB) = Leb(A ∩ B). ν is the viscosity and we set ν = 1.

I (4) arises in the study of turbulent fluid motion.

I W (t, x) := W[0,t]×[0,x] is the Brownian sheet.

I We can give a rigorous meaning to (4) by writing the solution as

uε(t, x) =

∫ 1

0

Gt(x , y)u0(y)dy +√ε

∫ t

0

∫ 1

0

Gt−s(x , y)W (ds, dy)

−√ε

∫ t

0

∫ 1

0

∂yGt−s(x , y)g(uε(t, y))dyds (5)

where g(r) = 12 r

2 and the stochastic integral is defined in similarway to the classical Ito integral, and Gt(x , y) is the Green kernel for∂t − 1

2∂2xx with the same Dirichlet boundary conditions (test

functions).Martin Forde King’s College London Feb 2014 Large deviations and asymptotic methods in finance

I The solution has a modification which lies in C ([0,T ]; L2[0, 1]), sothe modification is a continuous L2[0, 1]-valued stochastic process.

I Now let H = {h : h = ∂2h∂t∂x ∈ L2([0,T ]× [0, 1])} with norm

‖h‖ = (∫ t

0

∫ x

0∂2h∂t∂x (u, z)dudz)

12 and let Z h denote the map which

takes h to the solution of the associated (non-stochastic) PDE withε = 1:

∂tu(t, x) = ∂2xxu(t, x) + h(t, x) + u(t, x)∂xu(t, x) .

I Under suitable regularity on the initial datum and coefficients, uε

satisfies the LDP on C ([0,T ], L2[0, 1]) with rate function

I (f ) =

{inf{ 1

2

∫[0,T ]×[0,1]

( ∂2h

∂s∂t )2 dsdt : Z h = f } , f ∈ Im(Z h)

+∞ (otherwise)

( 12

∫[0,T ]×[0,1]

( ∂2h

∂s∂t )2 dsdt is the rate function for the small-noise

Brownian sheet).


I Can also derive large-time Donsker-Varadhan-type LDP for theoccupation measure of the stochastic Burger eq. The stationarymeasure and the realized occupation measure are now probabilitymeasures on H = L2[0, 1].


References

Armstrong, J., M.Forde and H.Zhang, “Small-time asymptotics for ageneral local-stochastic volatility model: curvature and the heatkernel expansion”, submitted.

Benaim, S., Lee, R. and P Friz, “On the Black-Scholes Im pliedVolatility at Extreme Strikes”, Frontiers in Quantitative Finance:Volatility and Credit Risk Modeling.

Cardon-Weber, C., “Large deviations for a Burgers’-type SPDE”,Stochastic Processes and their Applications 84 (1999) 5370

Dembo, A. and O.Zeitouni, “Large deviations techniques andapplications”, Jones and Bartlet publishers, Boston, (1998).

Deuschel, J.D., P.K.Friz, A.Jacquier, S.Violante, “Marginal densityexpansions for diffusions and stochastic volatility, Part II: Theoreticalfoundations ”, 2013, forthcoming in Comm. Pure Appl. Math.

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


Donsker, M.D., and S.R.S.Varadhan, “Asymptofic evaluation ofcertain Markov process expectations for large time-I”, Comm. PureAppl. Math., Vol. 27, 1975, pp. 1-47.

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).

Friz, P. S.Gerhold, A.Gulisashvili, S.Sturm, “On Refined VolatilitySmile Expansion in the Heston Model”, Quantitative Finance, 11(8),pp. 1151-1164, 2011.

Forde, M. and R.Kumar, “Large-time asymptotics for a generalstochastic volatility model with a stochastic interest rate, using theDonsker-Varadhan LDP”, submitted.

Ethier, S.N. and T.G.Kurtz, “Markov Processes: characterizationand convergence”, John Wiley and Sons, New York, 1986.


Forde, M., R.Kumar and H.Zhang, “Large deviations for boundarylocal time and trading volume under proportional transaction costs”,submitted.

Forde, M. and H.Zhang, “Sharp tail estimates for the correlatedSABR model”, working paper.

Hoyle, E., L.P. Hughston and A. Macrina (2011), “Levy RandomBridges and the Modelling of Financial Information”, StochasticProcesses and Their Applications, 121, 856-884.

Guasoni, P. and J.Muhle-Karbe, “Long Horizons, High Risk Aversion,and Endogeneous Spreads”, to appear in Mathematical Finance.

Gulisashvili, A. and E.M. Stein, “Asymptotic Behavior OfDistribution Densities In Models With Stochastic Volatility. I”,Mathematical Finance, Volume 20, Issue 3, pages 447477, July 2010.

Liu, R. and J.Muhle-Karbe, “Portfolio Choice with StochasticInvestment Opportunities: a User’s Guide”, 2013.

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

Varadhan, S.R.S., “Large Deviations and Applications”. SIAM,Philadelphia, 1984.Martin Forde King’s College London Feb 2014 Large deviations and asymptotic methods in finance

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Large deviations and asymptotic methods in finance · 2015. 12. 10. · Large-time asymptotics: the Donsker-Varadhan LDP for the occupation measure of the Ornstein-Uhlenbeck process