small-time asymptotics for fast mean-reverting stochastic volatility models

7/27/2019 Small-time Asymptotics for Fast Mean-reverting Stochastic Volatility Models

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SMALL-TIME ASYMPTOTICS FOR FAST

MEAN-REVERTING STOCHASTIC VOLATILITYMODELS

By Jin Feng, Jean-Pierre Fouque and Rohini Kumar

University of Kansas and University of California Santa Barbara

In this paper, we study stochastic volatility models in regimeswhere the maturity is small but large compared to the mean-reversiontime of the stochastic volatility factor. The problem falls in the classof averaging/homogenization problems for nonlinear HJB type equa-tions where the fast variable lives in a non-compact space. We de-velop a general argument based on viscosity solutions which we applyto the two regimes studied in the paper. We derive a large deviation

principle and we deduce asymptotic prices for Out-of-The-Money calland put options, and their corresponding implied volatilities. The re-sults of this paper generalize the ones obtained in [13] (J. Feng, M.Forde and J.-P. Fouque, Short maturity asymptotic for a fast meanreverting Heston stochastic volatility model, SIAM Journal on Finan-cial Mathematics, Vol. 1, 2010) by a moment generating functioncomputation in the particular case of the Heston model.

1. Introduction. On one hand, the theory of large deviations has beenrecently applied to local and stochastic volatility models [14, 21], and hasgiven very interesting results on the behavior of implied volatilities nearmaturity (an implied volatility is the volatility parameter needed in theBlack-Scholes formula in order to match a call option price; it is commonpractice to quote prices in volatility through this transformation). In thecontext of stochastic volatility models, the rate function involved in thelarge deviation estimates is given in terms of a distance function, which ingeneral cannot be calculated in closed-form. For particular models, such asthe SABR model [20, 22], approximations obtained by expansion techniqueshave been proposed (see also [19, 23, 30]). Asymptotics for tail probabilitiesare studied in [14] and semi closed form expressions for short time impliedvolatilities have been obtained in [15].

Work supported in part by National Science Foundation grants DMS-0806434 andDMS-0806461. In addition, Jin Feng is supported by Funding from the State of Kansas -

Kan0064212.AMS 2010 subject classifications: Primary 60F10, 91B70, 49L25Keywords and phrases: Stochastic volatility, multi-scale asymptotic, large deviation

principle, implied volatility smile/skew.

1


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2 J. FENG, J.-P. FOUQUE, R. KUMAR

On the other hand, multi-factor stochastic volatility models have beenstudied during the last ten years by many authors (see for instance [8, 16,

19, 29, 31]). They are quite efficient in capturing the main features of impliedvolatilities known as smiles and skews, but they are usually not simple tocalibrate. In the presence of separated time scales, an asymptotic theory hasbeen proposed in [16, 17]. It has the advantage of capturing the main effectsof stochastic volatility through a small number of group parameters arisingin the asymptotic. The fast time scale expansion is related to the ergodicproperty of the corresponding fast mean reverting stochastic volatility factor.

It is natural to try to combine these two modeling aspects and limitingresults, by considering short maturity options computed with fast mean-reverting stochastic volatility models, in such a way that maturity is oforder 1 and the mean-reversion time, , of volatility is even smaller oforder = 2 (fast mean-reversion) or = 4 (ultra-fast mean-reversion).

In [13], the authors studied the particular case of the Heston model in theregime = 2 by an explicit computation of the moment generating functionof the stock price and its asymptotic analysis.

In this paper, we establish a large deviation principle for general stochas-tic volatility models in the two regimes of fast and ulta-fast mean-reversion,and we derive asymptotic smiles/skews. For such general dynamics, a mo-ment generating function approach is no longer available. Our problem fallsin the class of homogenization/averaging problems for nonlinear HJB typeequations where the fast variable lives in a non-compact space. We de-velop a general argument based on viscosity solutions which we apply to thetwo regimes studied in the paper. Viscosity solution techniques have been

used in averaging of nonlinear HJB equations over non-compact space in [5].However, the techniques in [5] were proved for a certain class of nonlinearHJB equations which does not include our case.

We start by considering the following stochastic differential equationsmodeling the evolution of the stock price (St) under a risk-neutral pricingprobability measure, and with a stochastic volatility determined by a process(Yt):

dSt = rStdt + (Yt)StdW(1)t ,(1.1a)

dYt =1

(m

Yt)dt +

Yt dW

(2)t ,(1.1b)

where m R, r, > 0, W(1) and W(2) are standard Brownian motions withW(1), W(2)t = t, with || < 1 constant. The process (Yt) is a fast mean-reverting process with rate of mean reversion 1/ ( > 0). The parameter


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ASYMPTOTICS FOR STOCHASTIC VOLATILITY MODELS 3

and (y) are chosen to satisfy the following.

Assumption 1.1.We assume that

1. {0} [ 12 , 1);2. in the case of = 1/2, we require m > 2/2 and Y0 > 0 a.s., in the

case of 1/2 < < 1, we require m > 0 and Y0 > 0 a.s.;3. (y) C(R;R+) satisfies

(y) C(1 + |y|),

for some constants C > 0 and with 0 < 1 .

These assumptions ensure existence and uniqueness of a strong solu-tion of (1.1). This can be seen as a combination of existence of martin-

gale problem solution (e.g. Theorem 5.3.10 in Ethier and Kurtz [9]) andthe Yamada-Watanabe theory for 1-D diffusions (e.g. Chapter 5, Karatzasand Shreve [24]). In particular, Assumption 1.1.2 ensures that, in the case [12 , 1), Yt > 0 a.s. for all t 0 (see Appendix A). In the case = 0, Y isan Ornstein-Uhlenbeck (OU) process with a natural state space (, ).In order to present both model cases using one simple set of notation, wedenote the state space for Y as E0 with E0 := R if = 0 and E0 := (0, )when [ 12 , 1).

Note that the Heston model, for which = 1/2 and (y) =

y, doesnot satisfy these assumptions, but it has been treated separately in [13] byexplicit computation of the moment generating function.

The infinitesimal generator of the Y process, when = 1, can be identifiedwith the following differential operator on the class of smooth test functionsvanishing off compact sets:

(1.2) B := (m y)y + 12

2|y|22yy .

Following the general theory of 1-D diffusion (e.g. page 221, Karlin andTaylor [26]), we introduce the so called scale and speed measure of the ( Yt)process:

s(y) := exp y

1

2(m z)2

|z|2

dz , m(y) :=1

2

|y|2s(y)

.

Denoting dS(y) := s(y)dy and dM(y) := m(y)dy, we then have

Bf(y) =1

2

d

dM

df(y)

dS

.(1.3)


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Under Assumption 1.1 there exists a unique probability measure

(1.4) (dy) := Z1

m(y)dy, Z :=E0 m(y)dy <

such that

Bfd = 0 for all f C2c (E0). See Appendix C.By a change of variable Xt = log St, we have

dXt =

r 1

22(Yt)

dt + (Yt)dW

(1)t .

In order to study small time behavior of the system, we rescale time t tfor 0 < 1; denoting the rescaled processes by X,,t and Y,,t, we havein distribution:

dX,,t =

r 1

2 2

(Y,,t)

dt + (Y,,t)dW(1)

t ,(1.5a)

dY,,t =

(m Y,,t)dt +

Y,,tdW

(2)t .(1.5b)

We are interested in understanding the two scale , 0 limit behaviorof option prices and its implication to implied volatility. In this paper, werestrict our attention to the following two regimes :

= 4 and = 2.

In view of [13], to obtain a large deviation estimate of option prices, it issufficient to obtain a large deviation principle (LDP) for

{X,,t : > 0

}. By

Brycs inverse Varadhan lemma [7][Theorem 4.4.2], we know that the keystep is proving convergence of the following functionals:

(1.6) u,(t,x,y) := log E[e1h(X,,t)|X,,0 = x, Y,,0 = y], h Cb(R),

to some quantity independent of y. The rate function in the LDP is thengiven in terms of a variational formula involving the limit of the functionalsu,.

For each h Cb(R), the function u, satisfies a nonlinear partial differ-ential equation given in (3.4). In section 3.2, we use heuristic arguments toobtain PDEs that characterize the limit of these u,. Proving this conver-gence rigorously however is non-trivial. Intuitively we know that, as Y hasa mean reversion rate 1/ and , the effect of the Y process shouldget averaged out. To be exact, the form of nonlinear operator (3.5) indicatesthat convergence ofu, is an averaging problem (over the fast y variable) forHamilton-Jacobi equations. Such problems, in the context of compact state


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space for the averaging variable, can be handled by extending standard lin-ear equation techniques using viscosity solution language. The Y process in

this article lies in E0, which isR

in the case of = 0 and (0, ) in othercases. E0 is a non-compact space, and therein lies an additional difficulty.We adapt methods developed in Feng and Kurtz [12]. Indeed, an abstract

method for large deviation for sequence of Markov processes, based on con-vergence of HJB equation, is developed fully in [12]. The two schemes treatedin this article are of the nature of Example 1.8 and Example 1.9 introducedin Chapter 1 and proved in details in Chapter 11 of [12]. In this article, wenot only present a direct proof, but also introduce some argument to furthersimplify [12] in the setting of multi-scale. This is possible in a large part dueto the locally compact state space and mean-reverting nature of the processY.

In particular, modulo technical subtleties in verification of conditions,the setup of Section 11.6 in [12] corresponds to the large deviation resultin our case of = 2. Since E0 is locally compact and we only deal withPDEs instead of abstract operator equations, great simplification of [12]can be achieved through the use of a special class of test functions. SeeConditions 4.1 and 4.2. The techniques we introduce (Lemmas 4.1 and 4.2)are not limited to averaging problems, but are also applicable to problemsof homogenization, which we will not delve into in this article. The rigorousjustification of convergence of u, is shown in section 5.

The main results of the paper are stated in Section 2. Theorem 2.1 is arare event large deviation type estimate corresponding to short time, out-of-the-money option pricing. Corollary 2.1 and Theorem 2.2 give asymptotics

of option price and implied volatility, respectively, for such situation. Theproofs are given in the sections that follow, starting with heuristic proofsin section 3.2 and finishing with rigorous justifications in sections 4 and 5.Technical results Lemmas 4.1 and 4.2 may be of independent interest.

2. Main results. Observe that in the SDE (1.5), while the scaled logstock price process runs on a time scale of order , the scaled Y processruns on a time scale of order /. This is due to the extremely short mean-reversion time, = r (r = 2, 4), of the Y,, process. Thus, as approacheszero, long time behavior of the unscaled Y process comes into play. Thislong time behavior of the Y process manifests itself in the large deviation

principle (LDP) of the scaled log stock price via the quantities 2

and

H0defined below. Define

(2.1) 2 :=

2(y)(dy);


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the average of the volatility function 2() with respect to the invariantdistribution ofY. Recall B, the generator of the Y process, defined in (1.2).

Define the perturbed generatorBpg(y) = Bg(y) + ypyg(y), g C2c (E0).(2.2)

Let Yp be the process corresponding to generator Bp and define

H0(p) : = limsupT+

supyE0

T1 log E[e12|p|2

T02(Yps )ds|Yp0 = y].(2.3)

Yp has strong enough ergodic properties that the limit above does not de-pend upon y even if we omitted the supyE0; and, in fact, the limsupTcan be replaced with limT in the above definition. We will justify thisfact in the rigorous derivations. By Girsanov transformation

(2.4) H0(p) = lim supT+

T1 log E[eT0 p(Ys)dW

(2)

(s)+(12)

2 |p|2 T

0 2

(Ys)ds]

where Y is the process with generator B. From this expression, we see thatH0 is convex and superlinear in p. H0(p) is the scaled limit of the log momentgenerating function of a function of occupation measures of the process Yp.As such, it has an equivalent representation in terms of the rate functionfor the LDP of occupation measures of Yp. This equivalent representationof H0 is given in (5.13) in section 5.2.

Having defined these crucial terms, we proceed to the statement of ourresults.

Theorem2.1 (Large Deviation)

.Assume X,

r

,0 = x0 and Y,r

,0 = y0where r = 2, 4 and suppose that Assumption 1.1 holds. For x R, let

I4(x; x0, t) :=|x0 x|2

22t, where is defined in (2.1) and(2.5)

I2(x; x0, t) := tL0

x0 x

t

,(2.6)

where L0 is the Legendre transform of H0 defined in (2.3).Then, for each regime r {2, 4}, for every fixed t > 0 and x0 R, y0

E0, a large deviation principle (LDP) holds for {X,r,t : > 0} with speed1/ and good rate function Ir(x; x0, t). In particular,

(2.7) lim0 log P(X,r

,t > x) = I(x; x0, t) when x > x0.Similarly, when x < x0, we have

(2.8) lim0

log P(X,r,t < x) = I(x; x0, t).


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Remark 2.1. The rate functionsIr(x; x0, t), in both regimes, are convex,continuous functions of x and Ir(x0; x0, t) = 0.

Remark 2.2. In the case = 4, observe that the rate function I4,in (2.5), is the same as the rate function for the Black-Scholes model withconstant volatility . In other words, in the ultra fast regime, to the leadingorder, it is the same as averaging first and then take the short maturity limit.

Remark 2.3. In the case = 2, no explicit formula for the rate functionis obtained. However, an explicit formula of the rate function is obtainedfor the Heston model in [13] which corroborates the formula in (2.6). TheHeston model per se does not fall in the category of stochastic volatilitymodels covered in this paper, but direct computation of H0, given by (2.3),and L0, its Legendre transform, is possible for this model.

Let S0 > 0 be the initial value of stock price and let X,r,0 = x0 = log S0.The asymptotic behavior of the price of out-of-the-money European calloption with strike price K and short maturity time T = t is given in thefollowing corollary. We only consider out-of-the-money call options by taking

(2.9) S0 < K or x0 < log K.

The other case, S0 > K, is easily deduced by considering out-of-the-moneyEuropean put options and using put-call parity.

COROLLARY 2.1 (Option Price). For fixed t > 0 ,

lim0+

log E

ert (S,r,t K)+

= Ir(log K; x0, t),

for r = 2, 4.

Denote the Black-Scholes implied volatility for out-of the-money Euro-pean call option, with strike price K, by r,(t, log K, x0), where r = 2, 4correspond to the two regimes. By the same argument used in [13], we getan asymptotic formula for implied volatility:

Theorem 2.2 (Implied Volatilities).

lim0+

2r,(t, log K, x0) =(log K x0)2

2Ir(log K; x0, t)t.


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Remark 2.4. In the case = 4, the implied volatility is , which isobtained by averaging the volatility term 2(y) with respect to the equilib-

rium measure for Y. It is likely that more features of the Y process, beyondits equilibrium, will be manifested in higher order terms of implied volatil-ity. Studying the next order term of implied volatility is a topic for futureresearch.

Remark 2.5. The limit of at-the-money implied volatility i.e. lim0 2r,(t, x0, x0)

is obtained as in [13][Lemma 2.6]. However, the continuity of the limitingimplied volatility at log K = x0 is not obvious in the r = 2 case. We discussthis at the end of section 6.3.

3. Preliminaries. The process (X, , Y,) is Markovian, and can beidentified through a martingale problem given by generator

A,f(x, y) =

(r 12

2(y))xf(x, y) +1

22(y)2xxf(x, y)

(3.1)

+

Bf(x, y) +

(y)y2xyf(x, y),

where f C2c (R E0). Recall that B is given by (1.2). Let g Cb(R) anddefine

(3.2) v,(t,x,y) := E[g(X,,t)|X,,0 = x, Y,,0 = y] .

In general, v, Cb([0, T] R E0). If moreover v, C1,2([0, T] RR),then it solves the following Cauchy problem in classical sense:

tv = A,v, in (0, T] R E0;(3.3a)v(0, x , y) = g(x), (x, y) R E0.(3.3b)

3.1. Logarithmic transformation method. Recall the definition of u, in

(1.6). That is, u, := log v, when g(x) = e1h(x), h Cb(R) in (3.2). By

(3.3) and some calculus, at least informally, (3.4) below is satisfied. Thisis the logarithmic transform method by Fleming and Sheu. See ChaptersVI and VII in [18]. In general, in the absence of knowledge on smoothnessof v,, we can only conclude that u, solves the Cauchy problem (3.4) inthe sense of viscosity solution (Definition 4.1). In addition to Fleming andSoner [18], such arguments can also be found in Section 5 of Feng [11].

Lemma 3.1. Forh Cb(R), u, defined as in (1.6), is a bounded contin-uous function satisfying the following nonlinear Cauchy problem in viscosity


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solution sense:

tu = H

,u, in (0, T]

R

E0

;(3.4a)

u(0, x , y) = h(x), (x, y) R E0.(3.4b)

In the above,

H,u(t,x,y) = e1uA,e

1u(t,x,y)

=

(r 12

2(y))xu +1

22(y)2xxu

+

1

2|(y)xu|2

+2

e

1uBe1u + (y)y(

2xyu +1

xuyu),

(3.5)

where

2

e

1uBe1u =

Bu + 1

1

2|yyu|2.

Note that H, only operates on the spatial variables x and y.

3.2. Heuristic expansion. By Brycs inverse Varadhan lemma (e.g. The-orem 4.4.2 of [7]), we know that convergence of u, is a necessary conditionto obtain the LDP for {X,,t : > 0}. In this section, we describe heuris-tically PDEs characterizing u, in the limit and the nature of convergenceitself.

Henceforth, for notational simplicity, we will drop the subscript andwrite u and H for u, and H, respectively. We begin by the followingheuristic expansion of u in integer powers of

(3.6) u = u0 + u1 + 2u2 +

3u3 + 4u4 + . . .

in both regimes. The ui, i = 0, 1, . . . are functions of t,x,y. In this heuristicsection, we make reasonable choices of ui which a posteriori, following arigorous proof of the convergence of u in section 5, are shown to be theright choice.

3.2.1. The case of = 4. Computation ofHu (see (3.5)) reveals that,

in this scale, the fast process Y oscillates so fast that averaging occurs up toterms of order 2. Viz., u0 = u0(t, x), u1 = u1(t, x) and u2 = u2(t, x) will notdepend on y. To see this, we equate coefficients of powers of in tu = Hu.


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Terms of O14

satisfy

0 = 12 2y2(yu0)2;

so we choose u0 independent of y. With this choice of u0 the equation for

the coefficients of the next order terms, which is of O12

, reduces to

0 = Bu1 +1

22y2(yu1)

2.

This equation is satisfied by choosing u1 independent ofy. With this choice

ofu1, the equation for coefficients of the next order terms, ofO1

, becomes

0 = Bu2.

By choosing u2 independent of y the last equation is satisfied.Thus, by these choices of u0, u1 and u2 independent of y, it follows that

Hu(x, y) =1

2|(y)xu0|2 + Bu3

+ (2(y)xu0xu1 +1

22(y)xxu0 + (r 1

22(y))xu0

+ (y)yxu0yu3 + Bu4)

+ o().

The 0 order terms then satisfy

tu0(t, x) =1

2|xu0(t, x)|22(y) + Bu3(t,x,y)

i.e.

Bu3(t,x,y) = tu0(t, x) 12|xu0(t, x)|22(y).

The above is a Poisson equation for u3 with respect to the operator B inthe y variable. We impose the condition that the right hand side is centeredwith respect to the invariant distribution (given in (1.4)). This ensures asolution to the Poisson equation, which is unique up to a constant in y. See

Appendix B for growth estimates of the solution. Therefore we get

tu0(t, x) =1

2|xu0(t, x)|2;


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where

2 = 2(y)(dy).

Thus the leading order term in the heuristic expansion satisfies

tu0 = H0u0(x), t > 0;(3.7a)

u0(0, x) = h(x),(3.7b)

where

H0u0(x) :=1

2|xu0(x)|2.

3.2.2. The case of = 2. When goes to zero at a slower rate 2, limitsbecome very different and more features in the Y process (rather than justits equilibrium) is retained. We observe that while u0 is independent ofy asin the faster scaling regime, u1 may now depend on y. Equating coefficientsof O

2

in tu = Hu we get

0 =1

22y2(yu0)

2,

and so we choose u0 = u0(t, x) independent of y. Then Hu reduces to

Hu(t,x,y) =1

2|(y)xu0|2 + (y)yxu0yu1 + eu1Beu1

+

2(y)xu0xu1 +1

22(y)xxu0 + (r 1

22(y))xu0

+Bu2 + y2yu1yu2 + (y)yxyu1 + (y)yxu1yu1

+(y)yxu0yu2

+o().

The leading order terms should satisfy

tu0(t, x) =12 |xu0(t, x)|22(y) + (y)yxu0(t, x)yu1(t,x,y)(3.8)

+eu1Beu1(t,x,y).

We will rewrite the above equation as an eigenvalue problem. Recall B, thegenerator of the Y process, defined in (1.2) and the perturbed generator Bp

defined in (2.2). Then

(3.9) eu1Beu1 + (y)yxu0yu1 = eu1Bxu0(t,x)eu1 .


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Fix t and x, and rewrite (3.8) in terms of the perturbed generator (3.9):

eu1

Bxu0(t,x)

eu1

(t,x,y) +

1

2 |xu0(t, x)|2

2

(y) = tu0(t, x).

Multiplying the above equation by eu1, we get the eigenvalue problem:

(3.10)

Bxu0 + V

g(y) = g(y),

where V() = 12 |xu0(t, x)|22() is a multiplicative potential operator, g() =eu1(t,x,) and (t, x) = tu0(t, x). Choose u1 such that (, g) is the solution tothe principal (positive) eigenvalue problem (3.10). Note that the dependenceof the eigenvalue, , on t and x is only through xu0. If (3.10) can be solvedwith a nice g, then we have

(t, x) = H0(xu0),(3.11)

where H0 was defined in (2.3). The leading order terms then satisfy

tu0(t, x) = H0(xu0(t, x)).(3.12)

Constructing a classical solution for (3.10) is a considerably hard problemeven in the 1-D situation. If (3.10) can be solved with a nice g, then (2.3)always holds with the H0 given by (3.11). The converse is not always true.Especially, (2.3) says nothing about the eigenfunction g. However, we onlyneed the definition in (2.3) in rigorous treatment of the problem. We willshow (in section 5.2) that (3.12) is the limit equation where H0 is given

by (2.3) irrespective of whether a solution to the eigenvalue problem (3.10)exists or does not.

To summarize,

tu0(t, x) = H0(xu0(t, x)); t > 0,(3.13a)

u0(0, x) = h(x),(3.13b)

where H0 is given by (2.3) or (2.4).

4. Convergence of HJB equations. The results of this section canbe independently read from the rest of the article.

We reformulate and simplify some techniques, regarding multi-scale con-vergence of HJB equations, introduced in [12]. Compared with [12], the sim-plification makes ideas more transparent and readily applicable. These aremade possible because we are dealing with Euclidean state spaces which arelocally compact. All these results are generalizations of Barles-Perthames


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half-relaxed limit argument first introduced in single scale, compact statespace setting.

Let E Rm

, E0 Rn

and E

:= E E0 Rd

where d = m + n. Atypical element in E is denoted as x and a typical element in E is denotedas z = (x, y) with x E and y E0. We denote a class of compact sets inE

Q := {K K : compact K E, compact K E0}.We specify a family of differential operators next. Let be an index set and

Hi(x,p,P; ) : ERm Mmm R, i = 0, 1;H(z ,p ,P) : E Rd Mdd R

be continuous. For each f C2(Rd), let f(x) Rd and D2f(x) Mddrespectively denote gradient and Hessian matrix evaluated at x. We considersequence of differential operators

Hf(z) := H(z, f(z), D2f(z)),

for f belongs to the following two domains

D,+ := {f : f C2(E), f has compact finite level sets}D, := D,+ := {f : f C2(E), f has compact finite level sets}.

We will separately consider these two domains depending on the situation ofsub- or super-solution. We also define domains D+, D similarly replacing

E by E.We will give conditions where u(t, z) = u(t,x,y) solving

(4.1) tu(t, z) = H(z, u(t, z), D2u(t, z)),

converges to u(t, x) which is a sub-solution to

(4.2) tu(t, x) inf

H0(x, u(t, x), D2u(t, x); ),

and a super-solution to

(4.3) tu(t, x) sup

H1(x, u(t, x), D2u(t, x); ).

The meaning of sub- super-solutions is defined as follows (as for instance inFleming and Soner [18]).


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Definition 4.1 (viscosity sub- super- solutions). We call a boundedmeasurable function u a viscosity sub-solution to (4.2) (respectively super-

solution to (4.3) ), if u is upper semicontinuous (respectively lower semicon-tinuous), and for each

u0(t, x) = (t) + f0(x), C1(R+), f0 D+,and each x0 E satisfying u u0 has a local maximum (respectively each

u1(t, x) = (t) + f1(x), C1(R+), f1 D,and each x0 E satisfying u u1 has a local minimum) at x0, we have

tu0(t0, x0) inf

H0(x0, u0(t0, x0), D2u0(t0, x0); ) 0;

(respectively

tu1(t0, x0) sup

H1(x0, u1(t0, x0), D2u1(t0, x0); ) 0).

If a function is both a sub- as well as super- solution, then it is a solution.

We will assume the following two conditions.

Condition 4.1 (limsup convergence of operators). For each f0 D+and each , there exists f0, D,+ (may depend on ) such that

1. for each c > 0, there exists K K Q satisfying{(x, y) : Hf0,(x, y) c} {(x, y) : f0,(x, y) c} K K;

2. for each K K Q,(4.4) lim

0sup

(x,y)KK

|f0,(x, y) f0(x)| = 0;

3. whenever (x, y) K K Q satisfies x x,(4.5) lim sup

0Hf0,(x, y) H0(x, f0(x), D2f0(x); ).

Condition 4.2 (liminf convergence of operators). For each f1

D

and each , there exists f1, D, (may depend on ) such that1. for each c > 0, there exists K K Q satisfying

{(x, y) : Hf1,(x, y) c} {(x, y) : f1,(x, y) c} K K;


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2. for each K K Q,

lim0

sup(x,y)KK|

f1

(x)

f1,

(x, y)|

= 0;

3. whenever (x, y) K K Q, and x x,

liminf0

Hf1,(x, y) H1(x, f1(x), D2f1(x); ).

Let u be viscosity solutions to (4.1), we define

u3(t, x) := sup{lim sup0+

u(t, x, y) :(t, x, y) [0, T] K K,

(t, x) (t, x), K K Q},

u4(t, x) := inf{lim inf0+

u(t, x, y) :(t, x, y) [0, T] K K,(t, x) (t, x), K K Q},

and u = u3 the upper semicontinuous regularization of u3 and u = (u4) thelower semicontinuous regularization of u4.

Lemma 4.1. Suppose that sup>0 u < . Then,1. under Condition 4.1, u is a sub-solution to (4.2);2. under Condition 4.2, u is a supersolution to (4.3).

Proof. Let u0(t, x) = (t) + f0(x) for a fixed C1(R+) and f0 D+.Let (t0, x0) be a local maximum of u u0, t0 > 0. We can modify f0 and if necessary so that (t0, x0) is a strict global maximum. For instance, bytaking f0(x) = f0(x) + k|x x0|4 and (t) = (t) + k|t t0|2 for k > 0 largeenough. Note that such modification has the property that

lim0+

sup|xx0|


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and

(4.7) t(t

)

Hf0,

(z)

0.

The above implies inf Hf0,(z) > . We verify next that f0,(z) 0

u + sup

>0

u0,(t, z) 0 f0,(z) < follows.Since K K is compact in E, there exists a subsequence of{(t, z)} (to

simplify, we still use the to index it) and a (t0, x0) [0, T] E such thatt t0 and x x0. Such (t0, x0) has to be the unique global maximizer(t0, x0) for u u0 that appeared earlier. This is because, by using x x0and z = (x, y), the definition of u and (4.4), from (4.6) we have

(u u0)(t0, x0) (u u0)(t, x), (t, x).(4.8)

Now, from (4.7) and (4.5), we also have

tu0(t0, x0) H0(x0, f0(x0), D2f0(x0); ).

Note that t0, x0 and u0 are all chosen prior to, and independent of, . Wecan take inf on both sides to get

tu0(t0, x0) inf

H0(x0, u0(t0, x0), D2u0(t0, x0); ) 0.

The proof that u is a super-solution of (4.3) under Condition 4.2 followssimilarly.

Lemma 4.2. Suppose that the conditions in Lemma 4.1 hold and that

there exists h Cb(E) such thatlim0

sup(x,y)KK

|h(x) u(0, x , y)| = 0, K K Q.


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Further suppose that for any sub-solution u0(t, x) of (4.2) with u0(0, x) =h(x) and super-solution u1 of (4.3) with u1(0, x) = h(x), we have

u0(t, x) u1(t, x), (t, x) [0, T] E.That is, a comparison principle holds for sub-solutions of (4.2) and super-solutions of (4.3) with initial data h.

Then u = u = u and

lim0

supt[0,T]

sup(x,y)KK

|u(t, x) u(t,x,y)| = 0, K K Q.

5. Rigorous justification of expansions. To rigorously prove theconvergence of operators H given by (3.5) to operators H0 obtained byheuristic arguments in section 3.2, we rely on, and extend, results developedin [12]. An exposition of the relevant results from [12] was laid out in section

4. In this section we verify conditions 4.1 and 4.2 and prove the comparisonprinciple in Lemma 4.2. We will adhere to notation used in section 4.

Conditions 4.1 and 4.2 require us to carefully choose a class of perturbedtest functions with an index set , and a family of operators{H0(; ), H1(; ); } to obtain viscosity sub- and super- solution esti-mates of u0, the limit of u. This technique was first introduced in [12] andillustrated through examples in Chapter 11 of that book. Our presentationsimplifies the technique in the context of application here. We will makethe sub-solution estimate given by H0(, ) tight, by inf-ing over . Henceintroducing yet another operator H0. Similarly, we sup over to tightenup the super-solution type estimate provided by H1(, ) which introducesoperator H1.Let

(5.1) (y) := |y m|,where > 0 is any number satisfying 2 < < 2(1 ) with and givenas in Assumption 1.1. Throughout the two regimes ( = 4, 2), we take theindex set

:= { = (, ) : C2c (E0), 0 < < 1};and define two domains

D+ := {f : f(x) = (x) + log(1 + |x|2); C2c (R), > 0},and

D := {f : f(x) = (x) log(1 + |x|2); C2c (R), > 0}.A collection of compact sets in R E0 is defined by

Q := {K K : compact K R, K E0}.


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5.1. Case = 4. For each f = f(x) D+, and each = (, ) , welet

g(y) := (y) + (y),and define perturbed test function

f(x, y) := f(x) + 3g(y) = f(x) + 3(y) + 3(y).

Note that xf + 2xxf < . Then

Hf(x, y) =

(r 1

22(y))xf +

1

22(y)2xxf

+

1

22(y)|xf|2

+B(y) + B(y) +1

222y2|y(y) + y(y)|2

+(y)yxf(y(y) + y(y)).

The choice of the number in definition of the function (y) in (5.1) guar-antees that B(y) C(y). Moreover, with earlier assumption that 0 < 1 , the growth of (y) as |y| dominates the growth in y of allother terms in Hf. Therefore, there exist constants c0, c1 > 0 with

Hf(x, y) 12|(y)xf(x)|2 + B(y) c0(y) + c1

In addition,f(x, y) = f(x) +

3g(y) f(x) 3.Furthermore, for each c > 0, we can find K

K

Q, such that

{(x, y) : Hf(x, y) c} {(x, y) : f(x, y) c} K K,(5.2)

verifying Condition 4.1.1. The rest of Condition 4.1 can be verified by taking

H0(x, p; , ) = supyE0

12|(y)p|2 + B(y) c0(y)

.

We define

H0f(x) : = inf

H0(x, xf(x); )

= inf 0


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Then Condition 4.2 holds for the choice of

H1(x, p; , ) = infyR1

2 |(y)p|2

+ B(y) + c0(y)

.

We define

H1f(x) : = sup

H1(x, xf(x); )

= sup0


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C(E0) be such that (y) = 1 when |y| 1 and 0 when |y| > 2. We take asequence of n(y) = (

yn)(y) which are truncated versions of . Then

Bn(y) =

yn

B(y) + (m y)(y)n1

yn

+1

22y2(y)n2

y

n

+ 2y2(y)n1

y

n

Suppose > 0. Noting that |(y)|, |(y)| and |(y)| are uniformly boundedand are 0 when |y| > 2, and using the growth estimates (5.4) for and ,we get,

|Bn(y)| cy2

1 +(m y)

n+

y

n

2n22 + y1

y

n

n1

1{ y

n2}

cy2

for all n.In the above, we used the fact that yn 2 and 1 < 0. Similarly, if(y) is bounded i.e. = 0, we get |Bn(y)| is uniformly bounded for all n.Therefore, for large y, (y) dominates Bn(y) uniformly in n in the followingsense: there exists a sub-linear function : R R+ such that

supn=1,2,...

|Bn(y)| ((y)).

With the above estimate, we have

H0f(x) lim supn

inf0


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Lemma 5.2.

lim0+

sup|t|+|x|+|y| 0

where u0 is the solution of (3.7) and is given by (5.6).

5.2. Case = 2. For each f = f(x) D+, and = (, ) , wechoose our perturbed test function as

f(x, y) := f(x) + g(y)

where g(y) = (1 )(y) + (y); (y) is defined as before in (5.1). Then,

Hf(x, y) = (r 1

22(y))xf +

1

22(y)2xxf +

1

22(y)|xf|2

+ eg(y)Bxf(x)eg(y)

(r 12

2(y))xf +1

22(y)2xxf

+

1

22(y)|xf|2

+ (1 )eBxfe(y) + eBxfe(y),

where Bxf(x) is the perturbed generator defined in (2.2). Recall that xf+2xxf < by the choice of domain D+. We can thus find a constantc0 > 0 such that

Hf(x, y) 12|(y)xf(x)|2 + (1 )eBxfe(y) + eBxfe(y) + c0.

Note that

eBxf(x)e(y) = B(y) + (y)yxf(x)y(y) +1

22y2|y(y)|2,

where

B(y) = |y m|+ 12

2y2( 1)|y m|2.(5.7)

The term (y) in B(y) dominates growth in y from all other terms in Hfas |y| . Since (y) as |y| , Hf(x, y) as |y| . Wealso have f(x, y) = f(x) + g(y)

f(x)

. Therefore, for each c > 0,

we can find K K Q, such that{(x, y) : Hf(x, y) c} {(x, y) : f(x, y) c} K K,(5.8)

verifying Condition 4.1.1.


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Super-solution case follows similarly, where we define the perturbed testfunction as f(x, y) = f(x) + (1 + )(y) (y), for each f D and(, ) .Take

H0(x, p; , ) := supyE0

12|(y)p|2 + (1 )eBpe(y) + eBpe(y)

,

H1(x, p; , ) := inf yE0

12|(y)p|2 + (1 + )eBpe(y) eBpe(y)

and

H0f(x) := inf0


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Denote p = xf(x). Then(5.9)

H0f(x) inf0


24/39


is viewed as +. Again through Theorem 7.44 of [32], we also get that H0,defined in (2.3), can be expressed as

H0(p) = supP(R+)

|p|22

R+

2d IB(;p)

= suphL2(p),hL2(p)=1

|p|22

R+

2(y)h2(y)p(dy)

2

2

0

y2|h(y)|2p(dy)

.

(5.13)

As in Lemma 11.35 of [12],

inf0


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mollification and truncation argument, we can find a sequence n(dy) =ehn(y)

ehndp dp(y) with hn + cn Cc (E0) for some constant cn, such that

limn IB(n;p) = IB(;p). Second, for every y E0 and every h Cc (E0), the following ergodic theorem holds

limt

1

tE[

t0

2(Yhs )ds|Yh0 = y] =

2(z)dh(z),(5.16)

where

dYhs =

(m Yhs ) + p(Yhs )(Yhs )+ 2(Yhs )2h(Yhs )

ds + (Yhs )dW2s ,

and where h is the unique stationary distribution of Yh. We will prove(5.16) in Lemma 5.5.

The process Yh is Yp under the Girsanov transformation of measures

dPh

dP|Ft = exp{h(Yt) h(Y0)

t0

ehBpeh(Ys)ds},

where P and Ph refer to the probability measures of the processes Yp andYh respectively. The invariant distribution of Yh is then

dh =e2hdpe2hdp

.

We can write

lim inft+

1

tlog EP exp

1

2|p|2

t

02(Yps )ds |Y

p0 = y

= limt

1

tlog EP

h

exp

12|p|2

t0

2(Yhs )ds

h(Yht ) h(Yh0 ) t0

ehBpeh(Yhs )ds

Yh0 = y

limt

1

tEP

h

1

2|p|2

t0

2(Yhs )ds

h(Yht ) h(Yh0 )

t0

ehBpeh(Yhs )dsYh0 = y

(by Jensens inequality)

=1

2|p|2

2(z)dh(z) +

ehBpeh(z)dh(z)


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(by ergodicity of Yh)

= 12|p|2

2(z)dh(z) Ep

dhdp

,

dhdp

=

1

2|p|2

2(z)dh(z) I(h;p).

By arbitrariness of h, (5.15) follows. To complete the proof, we finally checkthat

Lemma 5.5. (5.16) holds.

Proof. By Itos formula,

E[(Yht )] = E[(Yh0 )] + E[t0

Bh(Yhs )]ds,

where Bh(y) =

m y + p(y)y+ 2y2yh(y)

(y) +1

22y2(y). As

in (5.7), (y) is the dominating growth term in Bh(y). Therefore, defininga family of mean occupation measure

h(t,y,A) := E[t1t0

1{Yhs A}ds|Yh0 = y],

we have that

supt>0

z

(z)h(t,y,dz) = supt>0

t1E[t0

(Yhs )ds|Yh0 = y] C(y; h()) < .

Hence {h(t,y, ) : t > 0} is tight and along convergent subsequences andcorresponding limiting point h, we have

E[t1t0

(Yhs )ds|Yh0 = y] z

dh, Cb(E0).(5.17)

Such h is necessarily a stationary distribution satisfying

Bhdh = 0for all C2c (E0). Uniqueness of such probability measure can be provedby an argument similar to the one in Appendix C. We thus conclude that

there is only one such

h

and that convergence (5.17) occurs along the wholesequence, not just subsequences. Furthermore, the growth of2 is dominatedby , and so by uniform integrability argument, (5.16) holds.


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Now (5.10), (5.14) and (5.15) together give us the estimate for H1 inLemma 5.3.

From (2.4), we see that

H0(p) is convex in p R. Let us denote itsLegendre transform as L0, then we have the following.

Lemma 5.6. The unique viscosity solution to (3.13) is:

u0(t, x) := supxR

h(x) tL0

x xt

.(5.18)

Moreover, u converges uniformly over compact sets in [0, T]RE0 to u0.

Proof. We know that u0 defined by (5.18) solves (3.13) by the Lax for-mula. That u0 is the unique solution follows from standard viscosity com-

parison principle with convex Hamiltonians. The convergence result followsfrom multi-scale viscosity convergence results developed in section 4 Lem-mas 4.1 and 4.2.

6. Large deviation, asymptotic for option prices and implied

volatilities. We finish the proof of Theorem 2.1, Corollary 2.1 and Theo-rem 2.2.

6.1. A large deviation theorem.

Proof of Theorem 2.1. From the previous section we have u(t,x,y) u0(t, x) as

0 for each fixed (t,x,y)

[0, T]

R

E0. All we need is

exponential tightness of{X,,t} to apply Brycs lemma and to conclude ourproof. This is obtained as follows.

Let f(x) = log(1 + x2) and (y) be defined as in (5.1). Take

f(x, y) =

f(x) + 3(y) for the case = 4

f(x) + (y) for the case = 2.

Note that f(x) is an increasing function of |x| and () 0, therefore,for any c > 0 there exists a compact set Kc R such that f(x, y) > cwhen x / Kc. We next compute Hf(x, y) (see (3.5)). Observe that sincexf+2xxf < , by our choice of(), Hf(x, y) as |y| .Therefore supxR,yR Hf(x, y) = C < . For simplicity, we denote X,,tby X,t. The P and E below denote probability and expectation conditioned


28/39


on (X, Y) starting at (x, y).

P(X,t /

Kc)e(cf(x,y)tC)/

E

exp

f(X,t, Y,t)

f(x, y)

t0

ef(X,s,Y,s)

Aef(X,s,Y,s)

ds

1.

In the above inequalities, the term within expectation in the second line is anon-negative local martingale (and hence a supermartingale), see [9] [Lemma4.3.2]. We apply the optional sampling theorem to get the last inequalityabove. Therefore

log P(X,t / Kc) tC+ f(x, y) c const c,

giving us exponential tightness of X,t.Let uh,r0 denote the limit of u, when u,(0, x , y) = h(x) and =

r,r = 2, 4. Applying Brycs lemma we get, {X,r,t} for r = 2, 4 satisfies aLDP with speed 1/ and rate function

(6.1) Ir(x; x0, t) := suphCb(R)

{h(x) uh,r0 (t, x0)}.

In Appendix D we check that I2(x; x0, t) = tL0x0x

t

where L is the Leg-

endre transform of H0 defined in (2.3); and I4 =|x0x|2

22t .

6.2. Option prices.

Proof of Corollary 2.1. We follow the proof of Corollary 1.3 in [13]

and show that lim0+

log E

(S,t K)+

is bounded above and below by

Ir(log K; x0, t).Recall that we are considering out-of-the-money call options and hence

x0 < log K (see (2.9)). Since our rate functions Ir(x; x0, t), for both r = 2, 4,are non-negative, convex functions with Ir(x0; x0, t) = 0, they are conse-quently monotonically increasing functions of x when x x0. Using thisfact and the continuity of the rate functions, the proof of the lower boundfollows verbatim from the proof in [13]. We refer the reader to [13] for details.

The upper bound follows from [13] once we justify the following limit: forany p > 1,

(6.2) lim0+

log E[Sp,,t] = 0 for both = 4 and = 2.


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Recall the operator A, defined at the beginning of section 3. By a slightabuse of notation, we can use A, to denote the operator acting on the

unbounded function epx

given below:

A,epx =

(r 1

22(y))pepx +

1

22(y)p2epx

.

Let

Mt := exp

pX,,t pX,,0

t0

epX,,sA,epX,,sds

.

Then Mt is a non-negative local martingale (supermartingale), this followsfrom the proof of [9] [Lemma 4.3.2]. By the optional sampling theorem

EMt

1.

Recall that X,,t = log S,,t, then

E[Sp2,,t] = E[e

p2X,,t]

(EMt)1/2

E

exp

+pX,,0 +

t0

epX,,sA,epX,,sds

1/2

(by Holders inequality)

1 e1

2px

0

E

expt

0 epX

,,sA,epX

,,sds1/2

.

(6.3)

We simplify and bound the right hand side of the above inequality:

E

exp

t0

epX,,sA,epX,,sds

= E

exp

t0

(r 12

2(Y,,s))p +1

22(Y,,s)p

2

ds

= erptE

exp{(p2 p)

t/0

2(Y,,u)du}

(by change of variable u = s; recall that = 2 or 4)

= erptE

exp{(p2 p)

t/0

2(Yu)du}


30/39


where Yu is the process with generator B given in (1.2). By convexity ofexponential functions we get

E

exp

t

0epX,,sA,e

pX,,sds

erptE

t

t/0

exp{t(p2 p)2(Yu)}du

.

(6.4)

Since = 2 or 4, / as 0. Therefore, by the ergodicity of Yand exp{t(p2 p)2(y)} L1(d) (this follows from an argument similar toproof of Lemma 5.5; note that < 1 by Assumption 1.1.3), the righthand side of the above inequality (6.4) is uniformly bounded for all > 0.Putting this together with (6.3) we get (6.2).

6.3. Implied volatilities.

Proof of Theorem 2.2. Recall that X,t = log S,t and x0 = log S0.Note that we have dropped the subscript in the notation and the depen-dence on = 4 or 2 should be understood by context. Our first step is toshow that

(6.5) lim0+

(t, log K, x0)

t = 0.

Once we have shown this, the rest of the proof is identical to that of Corollary1.4 in [13].

By definition of implied volatility

E[(S,t K)+] =ertS0

x0 log K+ rt + 122 t

t

K

x0 log K+ rt 122 t

t

(6.6)

where is the Gaussian cumulative distribution function. Let l 0 be thelimit of

t along a converging subsequence. If lim0+ of the left-hand-

side of (6.6) is 0, then l satisfies

S0

x0 log K

l+

l

2

K

x0 log K

l l

2

= 0.

The only solution of the above equation is l = 0 and thus we get (6.5).We therefore need to prove

(6.7) lim0+

E

(S,t K)+

= 0.


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By (1.5a) we have

S,t K = S0 K+ t

0 rS,tdt + t0 S,t(Y,t)dW

(1)t .

It can be verified that E[(S,t K) (S0 K)]2 0, as 0, for bothcases = 4 and = 2. Therefore

lim0+

E[(S,t K)+] = E[(S0 K)+] = 0

as S0 < K (this is an out-of-the-money call option).The same formula is obtained when S0 > K by considering out-of-the-

money put options. We finally turn our attention to at-the-money impliedvolatility. The asymptotic limit of At-The-Money(ATM) volatility can be

shown to be 2

i.e.

lim0

2r,(t, log K, x0) = 2 when x0 = log K; r = 2, 4,

by a similar argument as in[13] [Lemma 2.6]. The continuity, at-the-money,of the limiting implied volatility i.e.

lim| logKx0|0

(log K x0)22Ir(log K, x0, t)t

= 2,

is obvious in the r = 4 regime, but is more involved in the r = 2 regime. Weconjecture that it is true, i.e.

(6.8) limz0

z2

2t2L0zt

= 2,and we briefly indicate an outline of the proof. Let

T(p) := T1 log E

eT0p(Ys)dW

(2)s +

(12)2

|p|2T02(Ys)ds

,

so that H0(p) = limT (p). The result (6.8) follows if H0(p) is twice

differentiable in a neighborhood of p = 0 and H0 (0) =2

2 . It can easily be

checked that limT T(0) =

2

2 . The main difficulty is to get a uniform

bound on T(p) for all T and in a neighborhood of p = 0. Obtaining such

a uniform bound on

T(p) involves tedious calculations but should followfrom the multiplicative ergodic properties of the Y process (see [28]).


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Acknowledgment: The authors are greatly indebted to two anonymousreferees, whose suggestions greatly improved both the content and readabil-

ity of this paper.

In the following appendix, we collect some material regarding 1-D diffu-sions Y and technical but elementary estimates.

APPENDIX A: POSITIVITY OF THE Y PROCESS

In this section we prove positivity of the Y process when 12 < < 1 in(1.1b). Assume m > 0 and Y0 > 0. Recall the scale function s(y) defined inthe introduction and let S(y) =

y1 s(y)dy. By Lemma 6.1(ii) in Karlin and

Taylor [26], to prove that Yt remains positive a.s. for all t 0, it is sufficientto show that

lim0+ S() = .For 0 < 1,

S() =1

s(y)dy =

1

exp

y1

2(m z)2|z|2 dz

dy

= C

1

exp

2m

2(2 1)y21 +y22

2(1 )

dy

(where C is a positive constant and 2 1, 1 > 0)

=12 (positive integrand)dy + C

2 exp

2m2(2 1)y21 +

y22

2(1 ) dy C exp

2m

2(2 1)(2)21

+

as 0+, provided m > 0. Therefore lim0+ S() = .

APPENDIX B: GROWTH ESTIMATES FOR SOLUTIONS TOPOISSON EQUATIONS

Assume satisfies the Poisson equation

B(y) =1

2 |p

|2(2

2(y)),

where 2, defined in (2.1), is the average of2(y) with respect to the invari-ant distribution (dy), given in (1.4), of the Y process. In this section wefind growth estimates for .


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The right hand side of the above Poisson equation is centered with respectto the invariant distribution (dy) = m(y)Z dy (given in (1.4)) and so

(B.1)

0m(z)(2 2(z))dz = 0;

where

m(y) =1

2y2exp

y1

2(m z)2z2

dz

.

By (1.3),

(y) :=

dS(y)

y0

|p|2(2 2(z))dM(z)

=

1y2m(y)

y |p|2m(z)(2 2(z))2

dz

dy

is a solution up to a constant, and so

(y) =|p|2

2y2m(y)

y0

m(z)(2 2(z))dz

= |p|2

2y2m(y)

y

m(z)(2 2(z))dz

.

The last equality is by the centering condition (B.1). Given the bounds on(y) in Assumption 1.1.3, we can compute the following bounds where the

constants, denoted by c, are positive and vary from line to line.

|(y)| c|p|2

2y2m(y)

y

z2m(z)dz

=c|p|2ey12

2e y

22

2(1)

y

z22ez12

e z

22

2(1) dz

where = 2m2(21) > 0. Bounding ez12 above by 1 we get

|(y)

|

c|p|2ey12

2ey22

2(1)

y

z22e z

22

2(1) dz

=c|p|2ey12

2e y

22

2(1)

y22

u2122 exp

u

2(1 )

du


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(by change of variable u = z22)

c|p|2

ey12

2e y

22

2(1)

y21 exp

y222(1 )

.

In the last inequality we useda

u2122 e

u2(1)

du 2(1 )a2122 e

a

2(1) (since 2122 < 0). There-

fore

|(y)| c|p|2ey

12

2y21 c|p|2y21 as y ,

since ey12 O(1) as y .

APPENDIX C: YP PROCESS

Fix p R. Denote p(y) := (my)+p(y)y and let Yp be the processwith generator

Bpg = p(y)yg +1

22y22yyg, g C2c (E0).

In this section we calculate the unique stationary distribution and Dirichletform of the process Yp and we show that it is a reversible process. To thisend, we first compute the scale function and speed measure.

The scale function and speed measure for the Yp process are given by:

sp(y) = exp

y

1

2p(z)

2z2

and mp(y) =2

2y2sp(y) .

Evaluating the integral in sp(y) we get (the C below denotes a positiveconstant that varies from line to line):

sp(y) =

Cexp2m log y2 + y

22

2(1) 2p J

if = 12

Cexp

2m2(21)y21

+ y22

2(1) 2p J

if 0 (12 , 1).

where

J(y) =

y (z)z

dz.

Due to bounds on given in Assumption 1.1.3, there exist C1, C2 > 0 suchthat

C1y1 J(y) C2y1+,


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where

0 < 1 1 + 1 if 12 < 11 = 1 1 + < 2 if = 0 . Therefore

(C.1)

1sp(y)

0 when y 0 or y if 12 < 11

sp(y) 0 when |y| if = 0.

Define for y E0,

Sp(y) : =

y1

sp(z)dz.

Observe that Sp(y) as y approaches the left end point of E0 andSp(y) + as y .

C.1. Stationary distribution. Let p

be an invariant distribution ofthe process Yp. Suppose it has density function (y), i.e. dp(y) = (y)dy,then is uniquely determined as the solution of

1

2

2

y2

2y2(y)

y(p(y)(y)) = 0,

satisfying (y) 0 for all y and E0 (y)dy = 1. Solving the above dif-ferential equation, we get (y) = mp(y) [C1Sp(y) + C2]. Since is non-negative and Sp(y) as y approaches the left boundary of E0, wetake C1 = 0. The other constant C2 is uniquely determined by the condition

E0 (y)dy = 1. Therefore p is the unique invariant distribution of Yp and

is given by

(C.2) dp(y) = (y)dy =mp(y)

Z1dy =

2

Z12y2sp(y)dy for y E0,

where Z1 =E0

mp(y)dy.

C.2. Reversibility. Let , C2c (E0), thenE0

Bpdp =1

Z1

E0

1

22y2 + p(y)

2

2y2sp(y)dy

=1

Z1E0

ey 2p(y)

2y2 +2p

2y2ey 2p(y)

2y2

dy

=1

Z1

E0

d

dy

sp(y)

dy.


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Integrating by parts twice and using the boundary conditions (C.1), we get

E0

Bpdp =1

Z1E0

d

dy

sp(y)

dy =E0

Bpdp.

C.3. Dirichlet form. By similar calculations as before, when provingreversibility, we get: for f, g L2(p),

Ep(f, g) := E0

f Bpgdp

= 1Z1

E0

f(y)d

dy

g(y)

sp(y)

dy

=1

Z1 E0 f(y)g(y)

1

sp(y)dy

=2

2

E0

y2f(y)g(y)dp(y),

where we integrated by parts once and used (C.1) in the second last line.

APPENDIX D: RATE FUNCTION FORMULAS

Recall the following characterization of the rate functions given in (6.1):

Ir(x; x0, t) = suphCb(R)

{h(x) uh,r0 (t, x0)},

where r = 2, 4 correspond to the two regimes = 2

and = 4

respectively.The uh,r0 are given in (5.18) and (5.6) respectively as

uh,20 (t, x0) = supxR

h(x) tL

x0 x

t

,

uh,40 (t, x0) = supxR

h(x)

|x0 x|2

22t

.

For notational convenience, we will drop the subscript r in Ir, and, in

the case r = 4 we will denote the term|x0x|2

22t

by tL

x0x

t

. The rate

functions can then be rewritten as

I(x; x0, t) = suphCb(R)

infxR

h(x) h(x) + tL

x0 x

t

for both regimes r = 2 and r = 4.


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Lemma D.1.

I(x; x0, t) = tLx0 x

t .Proof. Note that for both cases r = 2, 4, L0 is convex, L0(0) = 0 and

L0 is a non-negative function. This is obvious for the case r = 4. We candeduce this in the r = 2 case since H0(p) (defined in (2.3)) is convex andH0(0) = 0.

Re-write

I(x; x0, t) = tL0

x0 x

t

+ suphCb(R)

infxR

h(x) h(x)

+ tL0

x0 x

t

tL0

x0 x

t

= tL0

x0 xt

+ J

where J = suphCb(R)

Jh and

Jh = infxR

h(x) h(x) + tL0

x0 x

t

tL0

x0 x

t

. Taking x = x

in the inf we get Jh 0 and therefore

(D.1) J 0.

Note that x0 and x are fixed. Define a function h Cb(R) as follows:

h(x) = tL0

x0 xt

tL0

x0 x

t

.

ThenJh = 0,

and consequently

(D.2) J 0.

By (D.1) and (D.2), J = 0 and we get

I(x; x0, t) = tL0

x0 x

t

.


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Jin Feng,

Department of Mathematics,

University of Kansas, Lawrence, KS 66045

E-mail: [email protected]: http://www.math.ku.edu/jfeng/

Jean-Pierre Fouque,

Department of Statistics and Applied Probability,

University of California, Santa Barbara, CA 93106

E-mail: [email protected]: http://www.pstat.ucsb.edu/faculty/fouque/

Rohini Kumar,

Department of Statistics and Applied Probability,

University of California - Santa Barbara, CA 93106

E-mail: [email protected]: http://www.pstat.ucsb.edu/faculty/kumar/

small-time asymptotics for fast mean-reverting stochastic volatility models

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