ssrn-id1542077

Electronic copy available at: http://ssrn.com/abstract=1542077

Asymptotics of implied volatility in local volatilitymodels

Jim GatheralDepartment of Mathematics, Baruch College, CUNYOne Bernard Baruch Way, New York, NY 10010, USA

e-mail: [email protected]

Elton P. HsuDepartment of Mathematics, Northwestern University

Evanston, IL 60208, USAemail: [email protected]

Peter Laurence1

Dipartimento di Matematica, Universita di Roma 1Piazzale Aldo Moro, 2, I-00185 Roma, Italia

andCourant Institute of Mathematical Sciences, New York University

251 Mercer Street, New York, NY 10012, USAe-mail: [email protected]

Cheng OuyangDepartment of Mathematics, Purdue University

West Lafayette, IN 47907, USAemail: [email protected]

Tai-Ho WangDepartment of Mathematics, Baruch College, CUNYOne Bernard Baruch Way, New York, NY 10010, USA

e-mail: [email protected]

Keywords and phrases: Implied volatility, local volatility, asymptoticexpansion, heat kernels.

We thank the anonymous referees for their helpful comments andsuggestions to improve the paper and the interesting points raised.All errors are our responsibility. THW is partially supported by thegrant PSC-CUNY40-60125-39 of The City University of New York.

1Corresponding author

Electronic copy available at: http://ssrn.com/abstract=1542077

ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCALVOLATILITY MODELS

JIM GATHERAL, ELTON P. HSU, PETER LAURENCE, CHENG OUYANG,AND TAI-HO WANG

ABSTRACT. Using an expansion of the transition density functionof a 1-dimensional time inhomogeneous diffusion, we obtain thefirst and second order terms in the short time asymptotics of Euro-pean call option prices. The method described can be generalizedto any order. We then use these option prices approximations tocalculate the first order and second order deviation of the impliedvolatility from its leading value and obtain approximations whichwe numerically demonstrate to be highly accurate.

1. INTRODUCTION

Local volatility models are powerful tools for modeling the priceevolution of a financial asset consistent with known prices of Euro-pean options on that asset. In the driftless case applicable to the evo-lution of the asset price under the forward measure, local volatilitymodels take the form

(1.1) d ft = a( ft, t)dWt,

where Wt is a Brownian motion. In general, known European optionprices can be matched only by allowing an explicit dependence ofthe volatility of ft on t. We thus allow ft to evolve according to a timeinhomogeneous diffusion.

From a historical perspective, one of the earliest examples of a lo-cal volatility model is the CEV (constant elasticity of variance) modelintroduced by Cox (1975), in which a( f ) = f β, β < 1. Rubinsteinshowed that the CEV model could be used to match single-maturity

Key words and phrases. Implied volatility, local volatility, asymptotic expansion,heat kernels.

We thank the anonymous referees for their helpful comments and suggestionsto improve the paper and the interesting points raised. All errors are our respon-sibility. THW is partially supported by the grant PSC-CUNY40-60125-39 of TheCity University of New York.

Address correspondence to Peter Laurence, Dipartimento di Matematica, Uni-versita di Roma 1 “La Sapienza”, Rome, Italy; email: [email protected].

1

2 GATHERAL, HSU, LAURENCE, OUYANG, AND WANG

smiles in index option markets with fitted values of β typically lessthan zero. The CEV model and its close relative the CIR model (Cox,Ingersoll and Ross), as well as their multidimensional generaliza-tions, continue to be very popular to this day. Another special butquite flexible local volatility model that has gained increasing popu-larity recently is the quadratic model explored by Zuhlsdorff (2001),Lipton (2002), Andersen (2008) and others.

In pioneering contributions to mathematical finance, Dupire (1994)and Derman and Kani (1994) showed that, given a set of Europeanoptions with a continuum of strikes and expirations, it is possibleto back out a local volatility function such that an asset evolving ac-cording to (1.1) will generate call prices that exactly match the pricesof these options. Equivalently, given an implied volatility surface(a function returning implied volatilities for all strikes and expira-tions), the local volatility function may be computed using a simpleformula, see for example (1.10) in Gatheral (2006). Dupire and Der-man and Kani further showed that local variance (the square of localvolatility) may be represented as a conditional expectation of instan-taneous variance in a stochastic volatility model.

Shortly thereafter Dupire (2004) pointed out that in the context ofEuropean option pricing, local volatility models also arise naturally,as a mean to replace a multi-factor stochastic volatility model forthe asset ft and a possibly stochastic volatility and interest rate, bya less complex one-factor local volatility model, generating identi-cal European option prices. It turned out that this idea had, outsideof mathematical finance, been independently introduced in an evenmore general form by Gyongy (1986). Piterbarg (2007) took this onestep further by offering a systematic set of tools (dubbed Markovianprojection) to generate an approximate local volatility model givenany specific stochastic volatility model. Piterbarg then showed howMarkovian projection combined with his parameter averaging tech-nique can permit the efficient calibration of complex models of assetdynamics to European option prices. In the same paper, Piterbargsubsequently applied these techniques to the efficient approximationof the prices of spread and basket options. Using an alternative andtypically more accurate approach, Avellaneda, Boyer-Olson, Busca,and Friz (2002) showed how to generate a one factor local volatilityfunction that is effective in reproducing the option prices of a multi-dimensional index governed by a multi-dimensional local volatilitymodel. In summary, even though local volatility models as such arenot considered reasonable models for asset dynamics, they do arisenaturally in the context of fast approximations to option prices in

ASYMPTOTICS OF IMPLIED VOLATILITY 3

more complex (and presumably more reasonable) models of assetdynamics. A central question is then how to efficiently approximateimplied volatilities given a local volatility function.

In this paper, we show how to apply the heat kernel expansion tothe approximation of implied volatilities in a local volatility modelwhen time to expiration is small. This geometric technique, basedon a natural Riemannian metric, was introduced into mathematicalfinance in a Courant Institute lecture by Andre Lesniewski (2002).Thereafter it has been considerably extended and developed by Henry-Labordere (2005). The contribution of our paper is two-fold. First weprovide a rigorous derivation of this expansion up to first order inthe time to expiration. Second, we take the analysis one step furtherand determine, in addition to the first order correction, the secondorder correction in the case of interest rate r = 0, and moreover al-lowing time-dependent coefficients. The second order correction inthe case of interest rate r 6= 0 can then readily be obtained by a pro-cedure we will briefly describe.

In the case when the volatility does not depend explicitly on time(time-homogeneous models), our lowest order approximation to im-plied volatility agrees with the well-known formula of Berestycki,Busca, and Florent (2002). When the local volatility does dependexplicitly on time (time-inhomogeneous models), we find that theformula for the zeroth order term requires a small but key correctionrelative to the prior formulae of Hagan, Lesniewski, and Woodward(2004) and Henry-Labordere (2005).

Even in the time-homogeneous case, our first order correction termis different from and more accurate than the ones in Hagan andWoodward (1999), Hagan, Kumar, Lesniewski, and Woodward (2003),Hagan, Lesniewski, and Woodward (2004), and Henry-Labordere(2005). In all of these earlier contributions the first order correctioninvolves the first and second order derivatives of the local volatilityfunction. Our first order correction on the other hand does not in-volve any derivatives of the local volatility function. In addition weshow rigorously that our first order correction actually correspondsto the first order derivative of the implied volatility with respect tothe time to expiration, and similarly for the second order correction.This characterization implies that for very small times to maturitythe formula is optimal, being the unique limit of the ratio of differ-ence quotients, i.e., the ratio of change in implied volatility over asmall time interval to the length of the time interval. Our numeri-cal experiments show that there is a clear gain in accuracy even fortimes that are not small.


After this work was completed, it was brought to our attentionthat our first order correction, specialized to the time-homogeneouscase and to the case of interest rate r = 0, had already been presentedby Henry-Labordere (2008). There it is obtained by a heuristic pro-cedure, in which the nonlinear equation of Berestycki, Busca, andFlorent (2002) satisfied by the implied volatility is expanded in pow-ers of the time to maturity. Our approach allows us to rigorouslyjustify this formula. More recently still, we learned from Leif Ander-sen that the same formula, restricted to the time-homogeneous caseappears in a joint paper with Brotherton-Ratcliffe (see Proposition 1in Andersen and Brotherton-Ratcliffe 2001).

The form of the first order correction we give in the case r 6= 0appears to be new, as does the formula for the first order term σ1in the case of time-inhomogeneous diffusions. In the simplest case,when the interest rate r = 0 and time-homogeneous diffusions, thefirst order correction is given by

σ1 =σ3

0(ln K− ln s)2 ln

√σ(s)σ(K)σ0(T, s)

,

where

σ0 =

[1

ln K− ln s

∫ K

s

duuσ(u)

]−1

is the leading term of the implied volatility.Besides the results we present here, the only other rigorous results

leading to a justification of the zeroth order approximation in localand stochastic volatility models we are aware of were provided byBerestycki, Busca, and Florent (2002, 2004). Medvedev and Scaillet(2007) and Henry-Labordere (2008) have shown how to obtain sim-ilar results by matching the coefficients of the powers of the timeto expiration in the nonlinear partial differential equation satisfiedby the implied volatility. Also the work of Kunimoto and Taka-hashi (2001), beginning with a series of papers, was based on a rigor-ous perturbation theory of Malliavin-Watanabe calculus. Takahashi,Takehara, and Toda (2009) have recently applied this approach to theλ−Sabr model.

We derive our results up to first order in the time-homogeneouscase by two different methods, both ultimately resting on the sameheat kernel expansion. Indeed the heat kernel expansion has sinceits inception admitted both a probabilistic approach, pioneered byMolchanov (1975), and an analytic one going back to the work of


Hadamard (1952) and Minakshisundaram and Pleijel (1949). The lat-ter approach was refined in a form more suitable to our purposes afew years later by Yoshida (1953). The probabilistic method is de-scribed in SECTION 2. The second analytic approach is describedmore briefly in SECTION 3, can also be made rigorous in the case ofnon-degenerate diffusions, by coupling the “geometric expansion”with the Levi parametrix method to control the tail of the series. Asmentioned above, the PDE approach in this form was first discov-ered by Yoshida (1953).

In SECTION 2 we only carry out the expansion to order one fortime-homogeneous diffusions. However the second PDE approachof SECTION 3 is computationally more straightforward, so we use itto compute the first and second order corrections for the call pricesand for the implied volatilities in the more general setting of time-inhomogeneous diffusions. An appendix is devoted to the extensionof the results in the body of the paper to the case, which is almostuniversal in mathematical finance, where the volatility is degenerateon the boundary. The probabilistic approach is readily extended tothis case using the so-called principle of not feeling the boundary,first formulated by Kac (1951). Such a relatively straightforward ex-tension to the degenerate case is a strong point of the probabilisticapproach. Although Yoshida’s analytic approach described in SEC-TION 3 can easily be made rigorous in the case of a non-degeneratediffusion, it appears quite difficult to extend those proofs to the de-generate setting. However, in the end both approaches must lead tothe same formulae, resting as they do on the heat kernel expansion.And in addition to being computationally more convenient, we feelstrongly that Yoshida’s approach deserves to be more widely knownin the mathematical finance community. A numerical SECTION 4 ex-plores the accuracy of our results, comparing our approximationswith those appearing in the existent literature.

2. PROBABILISTIC APPROACH

2.1. Call price expansion. Suppose that the dynamics of the stockprice S is given by

dSt = St r dt + σ(St) dWt .

Then the stochastic differential equation for the logarithmic stockprice process X = ln S is

dXt = η(Xt)dWt +

[r− 1

2η(Xt)

2]

dt,


where η(x) = σ(ex). Denote by c(t, s) the price of the European calloption (with expiry T and strike price K understood) at time t andstock price s in the local volatility model. It satisfies the followingBlack-Scholes equation

ct +12

σ(s)2s2css + rscs − rc = 0, c(T, s) = (s− K)+.

It is more convenient to work with the function

v(τ, x) = c(T − τ, ex), (τ, x) ∈ [0, T]×R,

where τ = T − t is the time to expiration. The reason is that forthis function the Black-Scholes equation takes the simpler followingform

vτ =12

η(x)2vxx +

[r− 1

2η(x)2

]vx − rv, v(0, x) = (ex − K)+.

We now study the the asymptotic behavior of the modified call pricefunction v(τ, x) as τ ↓ 0. Our basic technical assumption is as fol-lows. There is a positive constant C such that for all x ∈ R,

(2.1) C−1 ≤ η(x), |η′(x)| ≤ C, |η′′(x)| ≤ C.

Since η(x) = σ(ex), the above assumption is equivalent to the as-sumption that there is a constant C such that for all s ≥ 0,

C−1 ≤ σ(s) ≤ C, |sσ′(s)| ≤ C, |s2σ′′(s)| ≤ C.

For many popular models (e.g., CEV model), these conditions maynot be satisfied in a neighborhood of the boundary points s = 0 ands = ∞. However, since we only consider the situation where thestock price and the strike have fixed values other than these bound-ary values, or more generally vary in a bounded closed subintervalof (0, ∞), the behavior of the coefficient functions in a neighborhoodof the boundary points will not affect the asymptotic expansions ofthe transition density and the call price. Therefore we are free tomodify the values of σ in a neighborhood of s = 0 and s = ∞ sothat the above conditions are satisfied. The principle of not feelingthe boundary (see APPENDIX A) shows that such a modification onlyproduces an exponentially negligible error which will not show upin the relevant asymptotic expansions.

Proposition 2.1. Let X = ln S be the logarithmic stock price. Denote thedensity function of Xt by pX(τ, x, y). Then as τ ↓ 0,

pX(τ, x, y) =u0(x, y)√

2πτe−

d2(x,y)2τ [1 + O(τ; x, y)] .


Hered(x, y) =

∫ y

x

duη(u)

and

(2.2) u0(x, y) = η(x)1/2η(y)−3/2 exp[−1

2(y− x) + r

∫ y

x

duη(u)2

].

Furthermore, the remainder satisfies the inequality |O(τ; x, y)| ≤ Cτ for aconstant C independent of x and y.

Proof. Recall that Xt satisfies the following stochastic differentialequation

dXt = η(Xt)dWt +

[12

η(Xt)− r]

dt,

where η(x) = σ(ex). Introduce the function

f (z) =∫ z

x

duη(u)

and let Yt = f (Xt). Then

dYt = dWt − h(Yt)dt.

Here

h(y) =η f−1(y) + η′ f−1(y)

2− r

η f−1(y).

It is enough to study the transition density function pY(τ, x, y) of theprocess Y.

Introduce the exponential martingale

Zτ = exp[∫ τ

0h(Ys) dWs −

12

∫ τ

0h(Ys)

2ds]

and a new probability measure P by dP = ZTdP. By Girsanov’stheorem, the process Y is a standard Brownian motion under P. Forany bounded positive measurable function ϕ we have∫

Ωϕ(Yτ)dP =

∫Ω

ϕ(Yτ)ZτdP.

Hence, denoting Ex,y · = Ex · |Yτ = y, we have∫ϕ(y)p(τ, x, y)dy =

∫ϕ(y)Ex,y(Zτ)pY(τ, x, y)dy,

where

p(τ, x, y) =1√2πτ

exp[− (y− x)2

2τ

]


is the transition density function of a standard one dimensional Brow-nian motion. It follows that

p(τ, x, y)pY(τ, x, y)

= Ex,y(Zτ).(2.3)

For the conditional expectation of Zτ, we have

Ex,yZτ =Ex,y exp[∫ τ

0h(Ys) dYs −

12

∫ τ

0

h′(Ys)− h(Ys)

2

ds]

= eH(y)−H(x)Ex,y exp[−1

2

∫ τ

0

h′(Ys)− h(Ys)

2

ds]

,

where H′(y) = h(y). The relation (2.3) now reads as

p(τ, x, y)eH(y)

pY(τ, x, y)eH(x)= Ex,y exp

[−1

2

∫ τ

0

h′(Ys)− h(Ys)

2

ds]

.(2.4)

From the assumption (2.1) it is easy to see that the conditional expec-tation above is of the form 1 + O(τ; x, y) and |O(τ; x, y)| ≤ Cτ witha constant C independent of x and y. From

h(y) =η f−1(y) + ηx f−1(y)

2− r

η f−1(y)

we have

H(y)− H(x) =f−1(y)− f−1(x)

2+

12

lnη( f−1(y))η( f−1(x))

−∫ f−1(y)

f−1(x)

rη2(v)

dv.

This together with (2.4) gives us the asymptotics pY(τ, x, y) of Yt.Once pY(τ, x, y) is found, it is easy to convert it into the density ofXt = f−1(Yt) using the relation

pX(τ, x, y) = pY(τ, f (x), f (y)) f ′(y).

Now we compute the leading term of the call price v(τ, x) = c(T−τ, ex) as τ ↓ 0. For the heat kernel pX(τ, x, y) itself, the leading termis

u0(x, y)√2πτ

exp[−d(x, y)2

2τ

].

The in-the-money case s > K yielding nothing extra (see SECTION2.3), we only consider the out-of-the-money case s < K, or equiva-lently, x < ln K. We express the call price function v(τ, x) in terms of


the density function pX(τ, x, y) of Xt. We have

v(τ, x) =c(T − τ, ex)

= e−rτE[(ST − K)+|ST−τ = ex]

= e−rτE[(Sτ − K)+|S0 = ex]

= e−rτEx

[(eXτ − K)+

].

In the third step we used the Markov property of S. Therefore

v(τ, x) =1√2πτ

∫ ∞

ln K(ey − K)e−rτ pX(τ, x, y) dy(2.5)

=1√2πτ

∫ ∞

ln K(ey − K)e−rτu0(x, y)·

exp[−d(x, y)2

2τ

](1 + O(τ; x, y)) dy.

From the inequality |O(τ; x, y)| ≤ Cτ for some constant C, it is clearfrom (2.5) that remainder term will not contribute to the leading termof v(τ, x). For this reason we will ignore it completely in the subse-quent calculations. Similarly, because |e−rτ− 1| ≤ rτ, we can replacee−rτ in the integrand by 1. A quick inspection of (2.5) reveals that theleading term is determined by the values of the integrand near thepoint y = ln K. Introducing the new variable z = y− ln K, we con-clude that v(x, τ) have the same leading term as the function

v](τ, x) =K√2πτ

∫ ∞

0(ez − 1)u0(x, z + ln K)(2.6)

exp[−d(x, z + ln K)2

2τ

]dz.

The key calculation is contained in the proof of the following result.

Lemma 2.2. We have as τ ↓ 0∫ ∞

0zk exp

[−d(x, z + ln K)2

2τ

]dz

∼ k![

σ(K)τd(x, ln K)

]k+1

exp[−d(x, ln K)2

2τ

].

Proof. We follow the method in De Bruin (1999). Recall that

d(x, y) =∫ y

x

duη(u)

.


Let

f (z) = d(x, z + ln K)2 − d(x, ln K)2.

The essential part of the exponential factor is e− f (z)/2τ. For any ε > 0,there is λ > 0 such that f (z) ≥ λ for all z ≥ ε, hence

f (z)τ≥(

1τ− 1)

λ + f (z).

From our basic assumptions (2.1) we see that there is a positive con-stant C such that f (z) ≥ Cz2 for sufficiently large z. Since the integral∫ ∞

0zke−Cz2

dz

is finite, the part of the original integral in the range [ε, ∞) does notcontribute to the leading term of the integral. On the other hand,near z = 0 we have f (z) ∼ f ′(0)z with f ′(0) = 2d(x, ln K)/σ(K). Itfollows that the integral has the same leading term as

exp[−d(x, ln K)2

2τ

] ∫ ∞

0zk exp

[−d(x, ln K)

σ(K)τz]

dz.

The last integral can be computed easily and we obtain the desiredresult.

The main result of this section is the following.

Theorem 2.3. Suppose that the volatility function σ satisfies the basic as-sumption (2.1). If x < ln K we have as τ ↓ 0,

v(τ, x) ∼ Ku0(x, ln K)√2π

[σ(K)

d(x, ln K)

]2

τ3/2 e−d(x,ln K)2/2τ.

Proof. We have shown that v(x, τ) and v](x, τ) defined in (2.6) havethe same leading term. We need to replace the function before theexponential factor by the first order approximation of its value nearthe boundary point z = 0. First of all, we have

|ez − 1− z| ≤ z2ez

From the explicit expression (2.2) and the basic assumption (2.1) it iseasy to verify that

|u0(x, z + ln K)− u0(x, ln K)| ≤ zeCz


for some positive constant C. By the estimate in LEMMA 2.2 we ob-tain

v(τ, x) ∼ v](τ, x)

∼ Ku0(x, ln K)√2πτ

∫ ∞

0z exp

[−d(x, z + ln K)2

2τ

]dz

∼ Ku0(x, ln K)√2πτ

[σ(K)τ

d(x, ln K)

]2

e−d(x,ln K)2/2τ.

2.2. Implied volatility expansion. Using the leading term of the callprice function calculated in the previous section, we are now in aposition to prove the main theorem on the asymptotic behavior ofthe implied volatility σ(t, s) near expiry T. We will obtain this bycomparing the leading terms of the relation

c(t, s) = C(t, s; σ(t, s), r).

Here C(t, s; σ, r) is the classical Black-Scholes pricing function. Forthis purpose, we need to calculate the leading term of the classicalBlack-Scholes call price function. Our main result is the following.

Theorem 2.4. Let σ(t, s) be the implied volatility when the stock price is sat time t. Then as τ ↓ 0,

σ(t, s) = σ0 + σ1τ + O(τ2),

where

σ0 =

[1

ln K− ln s

∫ K

s

duuσ(u)

]−1

and

σ1 =σ3

0(ln K− ln s)2 ·(2.7) [

ln

√σ(s)σ(K)σ(T, s)

+ r∫ K

s

(1

σ2(u)− 1

σ2(T, s)

)duu

].

As we have mentioned above, the leading term σ(T, s) was ob-tained in Berestycki, Busca, and Florent (2002). They first deriveda quasi-linear partial differential equation for the implied volatilityand used a comparison argument. When the interest rate r = 0, thefirst order approximation of the implied volatility is given by

σ1(T, s) =σ3

0(ln K− ln s)2 ln

√σ(s)σ(K)σ(T, s)

.


This case has already appeared in Henry-Labordere (2008).For the proof of THEOREM 2.4 we first establish the following as-

ymptotic result for the classical Black-Scholes call price function.

Lemma 2.5. Let V(τ, x; σ, r) = C(T − τ, ex; σ, r) be the classical Black-Scholes call price function. Then as τ ↓ 0,

V(τ, x; σ, r) ∼ 1√2π

Kσ3τ3/2

(ln K− x)2 exp[− ln K− x

2+

r(ln K− x)σ2

]exp

[− (ln K− x)2

2τσ2

]+ R(τ, x; σ, r).

The remainder satisfies

|R(τ, x; σ, r)| ≤ C τ5/2 exp[− (ln K− x)2

2τσ2

],

where C = C(x, σ, r, K) is uniformly bounded if all the indicated parame-ters vary in a bounded region.

Proof. The result can be proved by the classical Black-Scholes for-mula for V(τ, x; σ, r). We omit the detail. Note that the leading termcan also be obtained directly from THEOREM 2.3 by assuming thatσ(x) is independent of x.

We are now in a position to complete the proof of the main THEO-REM 2.4. Set σ(τ, x) = σ(T − τ, s). From the relation

(2.8) v(τ, x) = V(τ, x; σ(τ, x), r)

and their expansions we see that the limit

σ0 = limτ↓0

σ(τ, x)

exists and is given by the given expression in the statement of thetheorem. Indeed, by comparing only the exponential factors on thetwo sides we obtain

d(x, K) =ln K− x

σ0,

from which the desired expression for σ0 follows immediately.Next, let

σ1(τ, x) =σ(τ, x)− σ(0, x)

τ.

We have obviously

(2.9) σ(τ, x) = σ(0, x) + σ1(τ, x)τ.


From this a simple computation shows that(2.10)

exp[− (ln K− x)2

2τσ(τ, x)2

]= exp

[− d2

2τ+

ρ2

σ30· σ1(τ, x)

][1 + O(τ)] ,

where for simplicity we have set

ρ = ln K− x, d = d(x, K), σ0 = σ(0, x)

on the right hand side. From (2.8) and (2.9) we have

v(τ, x) = V (τ, x; σ(0, x) + σ1(τ, x)τ, r) .

We now use the asymptotic expansions for v(τ, x) and V(τ, x; σ, r)given by THEOREM 2.3 and LEMMA 2.5, respectively, and then apply(2.10) to the second exponential factor in the equivalent expressionfor V(τ, x; σ, r). After some simplification we obtain

u0 σ(K)2(1 + O(τ)) = σ0 exp

[−ρ

2+

rρ

σ20

]exp

[− ρ2

σ30· σ1(τ, x)

],

where u0 = u0(x, ln K). Letting τ ↓ 0, we see that the limit

σ1 = limτ↓0

σ(τ, x)− σ(0, x)τ

exists and is given by

σ1 =rρ−

σ30

2ρ2 −σ3

0ρ2 ln

u0σ(K)2

σ0.

Using the expression of u0 = u0(x, ln K) in (2.2) we immediatelyobtain the formula for σ1.

2.3. In-the-money case. For the in-the-money case s > K, from

(s− K)+ = (s− K) + (s− K)−,

we have

u(τ, x)− ex

= Ex

[eXτ − K

]+e−rτ − ex

= Ex

[eXτ e−rτ − ex

]+ e−rτEx

[eXτ − K

]−− Ke−rτ.

The process eXτ e−rτ = Sτe−rτ is a martingale in τ starting from ex,hence the first term immediately after the second equal sign van-ishes. The calculation of the leading term of the second term is sim-ilar to that of the case when s < K. Therefore for s > K we have the


following asymptotic expansion for u(τ, x):

ex − Ke−rτ − Ku0(x, ln K)√2π

[σ(K)

d(x, ln K)

]2

τ3/2 exp[−d(x, ln K)2

2τ

].

It is easy to see that this case produces nothing new.

3. YOSHIDA’S APPROACH TO HEAT KERNEL EXPANSION

3.1. Time-inhomogeneous equations in one dimension. In this sec-tion we review an expression of the heat kernel for a general non-degenerate linear parabolic differential equation due to Yoshida (1953).We will only work out the one dimensional case but in a form thatis more general than we actually need in view of possible futureuse. In our opinion, this form of heat kernel expansion is more effi-cient for applications at hand than the covariant form pioneered byAvramidi and adapted by Henry-Labordere. The latter, being intrin-sic, is preferable for higher order corrections. However, the Yoshidaapproach is completely self-contained and, especially when the coef-ficients in the diffusions depend explicitly on time, introduces someclear simplifications of the necessary computations. Moreover, asmentioned in the introduction, Yoshida’s heat kernel expansion isfully rigorous and leads to a uniformly convergent series on anycompact subdomain of R. Thus, in the non-degenerate case, thismethod will yield an asymptotic expansion of the implied volatilityin the form

σBS(t, T) ∼n

∑i=0

σBS,iτi,(3.1)

as the time to expiration τ = T − t goes to zero. In this paper wewill limit ourselves to describing in detail only the zeroth, first, andsecond coefficient. It should be pointed out that Yoshida’s approachdoes not easily extend to the degenerate (on the boundary) case.

Consider the following one dimensional parabolic differential equa-tion

(3.2) ut +L u = ut +12

a(s, t)2uss + b(s, t)us + c(s, t)u = 0,

where subscripts refer to corresponding partial differentiations. Inour case, a(s, t) = sσ(s, t), where σ(s, t) is the local volatility func-tion. Note that a(s, t) vanishes at s = 0 so it is not non-degenerate atthis point. For the applicability of Yoshida’s method in this case seeREMARK 3.5. We seek an expansion for the kth order approximation


to the fundamental solution p(s, t, K, T) in the form

(3.3) p(s, t, K, T) ∼ e−d(K,s,t)2/2τ

√2πτa(K, T)

k

∑i=0

ui(s, K, t)τi,

where

d(K, s, t) =∫ s

K

dη

a(η, t)(3.4)

is the Riemannian distance between the points K and s with respectthe time dependent Riemannian metric ds2/a(s, t)2, and τ = T − t.Yoshida (1953) established that the coefficients ui have the followingform:

u0(s, K, t) =

√a(s, t)a(K, t)

exp[−∫ s

K

b(η, t)a(η, t)2 dη −

∫ s

K

dt(K, η, t)a(η, t)

dη

].

and

ui(s, K, t) =u0(s, K, t)d(K, s, t)i ·(3.5) ∫ s

K

d(K, η, t)i−1

u0(η, K, t)

[L ui−1 +

∂ui−1

∂t

]dη

a(η, t).

The function u0 is given explicitly and ui can be calculated recur-sively using mathematical software packages such as Mathematica orMaple.

The case b = c = 0 in the equation (3.2) is of special interest to us.There

u0(s, K, t) =

√a(s, t)a(K, t)

exp[−∫ s

K

dt(K, η, t)a(η, t)

dη

].

(3.6)

and

u1(s, K, t) =u0(s, K, t)d(K, s, t)

∫ s

K

1u0(η, K, t)

[a2

2∂2u0

∂s2 +∂u0

∂t

]dη

a(η, t).

(3.7)


In particular, when in addition a is independent of time, we have

u0(s, K) =

√a(s)a(K)

,

u1(s, K) =1

4d(K, s)

√a(s)a(K)

[a′(s)− a′(K)− 1

2

∫ s

K

a′(η)2

a(η)dη

].

Remark 3.1. In the Black-Scholes setting, a(s, t) = σBSs and uBS0 and

uBS1 are given explicitly as

uBS0 (s, K) =

√sK

, uBS1 (s, K) = −

σ2BS8

√sK

.

In fact, every uBSk can be calculated explicitly:

uBSk (s, K) =

(−1)k

k!

(σ2

BS8

)k√sK

,

which in turn yields the following heat kernel expansion for Black-Scholes’ transition probability density

(3.8) pBS(s, K, t) =1√

2πtσBSK

√sK

exp

[− (ln s− ln K)2

2σ2BSt

−σ2

BSt8

].

This formula can also be verified directly.

3.2. Option price calculations. We follow the approach adopted byHenry-Labordere (2005) based on the earlier work of Dupire (2004)and Derman and Kani (1994), who used the following method toobtain the call prices directly from the probability density functionwithout requiring a space integration. Unlike the method describedin SECTION 2, the result can be obtained without using Laplace’smethod. Thus an additional approximation is avoided at this stage.

Suppose that b = c = 0 in 3.2. The Carr-Jarrow formula in Carrand Jarrow (1990) for the call prices C(s, K, t, T) reads

C(s, K, t, T) = (s− K)+ +12

∫ T

ta(K, u)2p(s, t, K, u)du.

We use the Yoshida expansion (3.3) for the heat kernel p(s, t, K, u),which gives

C(s, K, t, T)− (s− K)+

∼ 12√

2π

k

∑i=0

[∫ T

ta(K, u)e−d(K,s,t)2/2(u−t)(u− t)i− 1

2 du]

ui(s, K, t).


When we calculate the coefficient σBS,2 in the expansion (3.1) for theimplied volatility σBS, we need an expansion for the call price func-tion in the following form:[

C(s, t, K, T)− (s− K)+]

ed(K,s,t)2/2τ(3.9)

= C(1)(s, K, t)τ3/2 + C(2)(s, K, t)τ5/2 + o(τ5/2).

The key step in making the approach rigorous is to show that theremainder in (3.9) truly is o(τ5/2). This in turn will follow from thefact that the first few terms in the “geometric series expansion” canbe complemented by the Levi parametrix method to ensure that wehave

p(s, t, K, T) =e−d(K,s,t)2/2τ

√2πa(K, T)

[2

∑i=0

ui(s, K, t)τi− 12 + o(τ

32 )

],

i.e., that the suitably modified preliminary approximation of the heatkernel can actually give a convergent series with a tail of order smallerthan the last term in the geometric series. Note that the theory re-quires us to proceed till order k = n− 1 (or k = n in the at-the-moneycase, see below) in the series in order to calculate σBS,n in the expan-sion for the implied volatility. If we wish to use the series to calculatea first order Greek, like the Delta, we would need to expand up toorder k = 2 before using the Levi parametrix. See SECTION 3.4 forsome additional details.

We now proceed with some additional approximations that willbe necessary to obtain the expansion for the implied volatility. Wewill set τ = T − t as before and use d = d(K, s, t) for simplicity. Wehave∫ T

ta(K, u)e−d2/2(u−t)(u− t)i− 1

2 du

∼∫ T

t[a(K, t) + at(K, t)(u− t) +

12

att(K, t)(u− t)2]e−d2/2(u−t)(u− t)i− 12 du

= a(K, t)∫ T

te−d2/2(u−t)(u− t)i− 1

2 du

+ at(K, t)∫ T

te−d2/2(u−t)(u− t)i+ 1

2 du

+12

att(K, t)∫ T

te−d2/2(u−t)(u− t)i+ 3

2 du

= a(K, t)Ui(d, τ) + at(K, t)Ui+1(d, τ) +12

att(K, t)Ui+2(d, τ),


where

(3.10) Ui(ω, τ) =∫ τ

0ui− 1

2 e−ω2/2udu.

Expanding a(K, u) to the second order is sufficient for computing theimplied volatility expansion up to the second order of τ. Insertingthis into (3.9) we get

C(s, K, t, T)− (s− K)+

(3.11)

∼ 12√

2π

k

∑i=0

[a(K, t)Ui(d, τ) + at(K, t)Ui+1(d, τ) +

12

att(K, t)Ui+2

]ui(s, K, t).

We need to take k = 1 for the case s 6= K and k = 2 for the case s = K(at the money).

Remark 3.2. In the Black-Scholes setting the above asymptotic rela-tion reads

CBS(s, K, t, T)− (s− K)+ ∼√

sK2√

2π

[σBSU0(dBS, τ)−

σ3BS8

U1(dBS, τ)

],

where

dBS = dBS(K, s) =1

σBSln

sK

is the distance between K and s in the Black-Scholes’ setting. In fact,the complete series can be obtained by using the general formula(3.8) in REMARK 3.1:

CBS(s, K, t, T)− (s− K)+ =

√sKσBS

2√

2π

∞

∑k=0

(−1)k

k!

(σ2

BS8

)k

Uk(dBS, τ).

The leading term of the call price away from the money is τ3/2e−d2/2τ.Besides the leading term, we also need the next term, which has theorder τ5/2e−d2/2τ. After canceling some common factors and drop-ping higher order terms we have arrived at the following balancerelation between the call prices from the local volatility model and


from the Black-Scholes setting:

√sK

[σBSU0(dBS, τ)−

σ3BS8

U1(dBS, τ)

](3.12)

∼[

a(K, t)U0(d, τ) + at(K, t)U1(d, τ) +12

att(K, t)U2(d, τ)

]u0(s, K, t)

+ [a(K, t)U1(d, τ) + at(K, t)U2(d, τ)] u1(s, K, t).

We now consider two regimes separately.REGIME 1: s 6= K fixed (away from the money). We shall use the

following asymptotic formulas for U0 and U1 that are easily obtainedfrom the well known asymptotic formula for the complementary er-ror function. As τ ↓ 0, we have

U0(ω, τ) = 2√

τe−ω2/2τ − 2ω∫ ∞

ω√τ

e−x2/2dx

= 2

[τ3/2

ω2 −3τ5/2

ω4 + O(

τ7/2)]

e−ω2/2τ,

U1(ω, τ) =2τ3/2

3e−ω2/2τ − ω2

3U0(ω, τ)

=

[2τ5/2

ω2 + O(

τ7/2)]

e−ω2/2τ

These expansions will be applied to ω = d(K, s, t) and ω = dBS(K, s),respectively. We now let

ξ = lnsK

.

The relation between dBS and σBS is

dBS =ξ

σBS.(3.13)

In the time-inhomogeneous case we seek an expansion for σBS in theform (3.1). Note the dependence of the coefficients in the expan-sion on the spot variable t. This dependence is absent in the time-homogeneous case, as will be clear from the result of the expansion.Its presence in the case of time-inhomogeneous diffusions is naturalsince already the transition probability density depends jointly ont and T and not only on their difference. We seek natural expres-sions for the coefficients which do not depend on the expiry explic-itly. Mathematically it is of course also possible to expand around


t = T, but in this case more terms are needed to recover the sameaccuracy.

We now use the above expansions for U0(ω, τ) and U1(ω, τ) and(3.13) in (3.12). After removing the factor τ3/2/2

√2π, we see that

the left hand side of (3.12) becomes

√sK

ξ2 exp

[− ξ2

2σ2BS,0τ

+ξ2σBS,1

σ3BS,0

](3.14)

σ3BS,0 +

[3σ2

BS,0σBS,1 −ξ2σBS,0

2

(3(

σBS,1

σBS,0

)2

− 2σBS,2

σBS,0

)

−σ5BS,0

(3ξ2 +

18

)]τ

.

Here we have used the following expansion for expanding the expo-nent in the exponential term:

1σ2

BS∼ 1

σ2BS,0

[1− 2σBS,1

σBS,0τ +

(3σ2

BS,1

σ2BS,0− 2σBS,2

σBS,0

)τ2

].

On the local volatility side, in the present regime (away from themoney) the terms involving att in (3.12) can be ignored because theirorders in τ is at least 7/2. After again removing the factor τ3/2/2

√2π,

we see that the terms of up to order τ are(3.15)

exp[− d2

2τ

] [a(K, t)u0

d2 +

(at(K, t)u0

d2 − 3a(K, t)u0

d4 +a(K, t)u1

d2

)τ

].

We are ready to identify the corresponding terms in (3.14) and (3.15).

• by matching the exponents, we obtain

σBS,0 =ξ

d(K, s, t)=

[1

ln s− ln K

∫ s

K

dη

a(η, t)

]−1

.(3.16)

• by matching the constant terms, we obtain(3.17)

σBS,1 =σ3

BS,0

ξ2 ln

[u0(s, K, t)a(K, t)ξ2√

sKd2σ3BS,0

]=

ξ

d(K, s, t)3 ln[

u0(s, K, t)a(K, t)d(K, s, t)ξ√

sK

],


• by matching the first order term, we obtain

σBS,2 = −3σBS,1σ2

BS,0

ξ2 +3σ2

BS,1

2σBS,0+

σ3BS,0

ξ2[3σ2

BS,0

ξ2 +σ2

BS,0

8+

at(K, t)a(K, t)

− 3d2(K, s, t)

+u1(s, K, t)u0(s, K, t)

]

= −3σBS,1σBS,0

ξ2 +3σ2

BS,1

2σBS,0+

σ5BS,0

8ξ2 +σ3

BS,0

ξ2

[at(K, t)a(K, t)

+u1(s, K, t)u0(s, K, t)

].

(3.18)

In the above formulas we need the expressions for u0 and u1 ob-tained earlier in (3.6) and (3.7).

REGIME 2: s = K > 0 (at the money). We use again the expansion(3.11) for the call price. After setting s = K there we find that up tothe order τ5/2 the call price C(K, K, t, T) is equivalent to(3.19)

12√

2π

2

∑i=0

[aUi(0, τ) + atUi+1(0, τ) +

12

attUi+2(0, τ)

]ui(K, K, t).

With ω = 0 in (3.10) we have trivially

U0(0, τ) = 2√

τ, U1(0, τ) =23

τ3/2, U2(0, τ) =25

τ5/2.

Therefore in the expansion (3.19) we only for the case i = 0 the terminvolving the second derivative att needs to be kept. On the Black-Scholes side we then have, after dropping the factor 1/2

√2π,

K[

2(

σBS,0 + τσBS,1 + τ2σBS,2

)√τ − 1

12

(σBS,0 + σBS,1τ + σBS,2τ2

)3τ3/2

+1

320

(σBS,0 + σBS,1τ + σBS,2τ2

)5τ

52 + O(τ7/2)

]Grouping the terms according to powers of τ, we obtain

2KσBS,0√

τ + K(

σBS,1 −1

12σ3

BS,0

)τ3/2

+

[2KσBS,2 −

K4

σ2BS,0σBS,1 + K

σ5BS,0

320

]τ5/2 + O(τ7/2).


On the local volatility side we have

2au0√

τ +23(atu0 + au1)τ

3/2 +25(attu0 + atu1 + au2)τ

5/2 + O(τ7/2),

where we have omitted the dependence on the independent vari-ables, in all of which we replace s by K. Matching the coefficients ofthe powers of τ and using u0(K, K, t) = 1, we obtain

KσBS,0 = a,

K(

2σBS,1 −1

12σ3

BS,0

)=

23(at + au1),

2KσBS,2 −K4

σ2BS,0σBS,1 +

K320

σ5BS,0 =

25(attu0 + atu1 + au2).

They yield consecutively,

σBS,0 =a(K, t)

K,

σBS,1 =at + au1

3K+

a(K, t)3

24K3 ,

σBS,2 =attu0 + atu1 + au2

5K+

σ2BS,0σBS,1

8−

σ5BS,0

640.(3.20)

Remark 3.3. We have checked by explicit computation that in thetime-homogeneous case, these at-the-money expressions for the σBS,i,i = 0, 1, 2, coincide with limits as s → K of the out-of-the-money ex-pressions (3.16), (3.17) and (3.18).

Remark 3.4. One possible application of our approximate formula(3.20) for Black-Scholes implied volatility is to generate an impliedvolatility smile when good prices for out-of-the-money options arenot readily available, as is often the case with less liquid underly-ings. Supposing that the prices (and therefore implied volatilities)of at-the-money options are available, we may use (3.20) to build animplied volatility surface that matches the empirically observed im-plied volatility surface at the money (ATM).

Specifically, suppose we posit a functional form for the local volatil-ity function a(η, t), motivated for example either by a stochastic volatil-ity model or from a fit to another (better resolved) volatility surface.Denote observed ATM implied volatilities by σ

empBS (K, T) and as be-

fore, the approximate implied volatilities obtained from (3.20) byσBS(K, T). Then we may define a time-change T 7→ θ(T) throughthe relation

σBS(K, θ(T))2 θ(T) = σempBS (K, T)2 T


The generated volatility surface given by

σBS(x, T) = σBS(x, θ(T))

would then by construction match perfectly ATM with a smile con-sistent with the posited functional form for the local volatility func-tion a(η, t). Moreover, with the above specification of the time change,if σBS(·) is arbitrage free, σBS(·) will be as well.

3.3. Case of non-zero interest rate. It is straightforward to combinethe Yoshida approach with non-zero interest rates and dividends toaccount for the presence of the r dependent term in (2.7). Note thatif the stock satisfies the time-homogeneous equation

dSt = St r dt + σ(St) dWt ,

call it Problem (I), then the forward price ft = erτSt satisfies the drift-less but time-inhomogeneous equation

d ft = ftσ(e−rτ ft

)dWt = ftσ( ft, t)dWt,

call it Problem (II). The relationship between the implied volatilitiesfor these two problems is easily seen to be

σrBS(s, t, K, T) = σ

fBS

(s, t, Ke−r(τ), T

).

From this it follows that

σrBS,0(s, K) = σ

fBS,0(s, K) =

[1

ln s− ln K

∫ s

K

duuσ(u)

]−1

,

σrBS,1(s, K) = σ

fBS,1(s, K) +

∂

∂t

[σ

fBS,0

(s, Ker(τ)

)]∣∣∣∣t=T

= σfBS,1(s, K) + r

[∫ s

K

duuσ(u)

]−1

− r log(s/K)σ(K)

[∫ s

K

duuσ(u)

]−2

,

(3.21)

where in the first term above we need to determine σfBS,1(y, x) for

what is now a time-inhomogeneous problem (II). Since the volatilityσ now depends explicitly on time, in contrast (3.17), now there is anadditional term linear in r with the coefficient

− lnsK

[∫ s

K

duuσ(u)2

] [∫ s

K

duuσ(u)

]−3

+1

σ(K)ln

sK

[∫ s

K

duuσ(u)

]−2

.

The second term in the above expression cancels the third term in(3.21) to produce exactly the expression (2.7).


The procedure we have outlined above that allows us to pass fromthe zero to the nonzero interest case in the case of the first order coef-ficient can also be repeated to determine the second order correction.

3.4. Yoshida’s construction of the heat kernel. As mentioned ear-lier, if the diffusion coefficient is non-degenerate and the domain iscompact, Yoshida (1953) has shown how to use the so-called geo-metric series in (3.3) to obtain the exact fundamental solution to thebackward Kolmogorov equation. In this section we give an outlineof Yoshida’s method.

Consider the parabolic equation

ut +L u = 0,

where

L u =12

a2uyy + buy + cu.

Denote the geometric approximation of k + 1 terms in (3.3) by pk.Choose a function of one variable η(x) : [0, ∞) 7→ [0, ∞), whichequals 1 on [0, ε], and equals zero for x > 2ε. Define the truncatedgeometric approximation

pk(y, s, x, t) = η(d(x, y, s))pk(y, s, x, T),

where d(x, y, s) is the distance function (3.4). Define the space-timeconvolution operator

F⊗ G(y, s, x, t) =∫ τ

0dτ∫R

F(y, s, z, τ)G(z, τ, x, t)dz.

Starting from

K(k) = −∂ pk∂s−L pk,

we have a series of convolutions

K(k)n = K(k) ⊗ K(k)

n−1.

Let

Q(y, s, x, t) =∞

∑n=1

(−1)n+1K(k)n (y, s, x, t).

Then the fundamental solution is given by

P(y, s, x, t) = pk(y, s, x, t)− ( pk ⊗Q)(y, s, x, t).


Yoshida’s proof of convergence of the series representing P, assumesthe underlying domain is compact (actually a mild generalizationthereof). The key step is to show by induction that

|K(k)n (y, s, x, t)| ≤ DCn(t− s)k+n− 3

2 ,

where C and D are constants. The proof can easily be extended to un-bounded domains if proper conditions are imposed on the distancefunction d(x, y, s).

Remark 3.5. Yoshida’s approach requires the equation to be non-degenerate:

mint∈[0,T],S∈R

a(S, t) = c > 0.

Models like the CEV model, which will be considered in the numer-ical section, and even the Black-Scholes model itself, do not satisfythis condition since they are degenerate when S = 0. There mayalso be problems at S = ∞. We have encountered similar problemin the probabilistic approach in SECTION 2. However, as we haveexplained in SECTION 2, as long as we keep away from these twoboundary points, the behavior of the coefficient functions in a neigh-borhood of the boundary points are irrelevant as long as we alsoimpose some moderate conditions on the growth of a(S, t). In par-ticular the call price expansion will not be affected. This principle ofnot feeling the boundary is explained in detail in APPENDIX A.

4. NUMERICAL RESULTS

We have derived an expansion formula for implied volatility upto second order in time-to-expiration in the form

σBS(t, T) = σBS,0(t) + σBS,1(t)τ + σBS,2(t)τ2 + O(

τ3)

.

where the coefficients σBS,i(t) are given by (3.16), (3.17) and (3.18).To test this expansion formula numerically, we use well-known ex-

act formulas for option prices in two specific time-homogeneous lo-cal volatility models: the CEV model and the quadratic local volatil-ity model, as developed by Zuhlsdorff (2001), Lipton (2002), Ander-sen (2008), and others.

Time dependence is modeled as a simple time-change so that theseexact time-independent solutions may be re-used. Specifically, thetime change is

τ(T) =∫ T

0e−2 λ t dt =

12 λ

(1− e−2 λ T

)


Throughout, for simplicity, we assume zero interest rates and divi-dends so that b(s, t) = 0 in equation (3.2).

4.1. Henry-Labordere’s approximation. Henry-Labordere presentsa heat kernel expansion based approximation to implied volatility inequation (5.40) on page 140 of his book Henry-Labordere (2008):(4.1)

σBS(K, T) ≈ σ0(K, t)

1 +T3

[18

σ0(K, t)2 +Q( fav) +34G( fav)

]with

Q( f ) =C( f )2

4

[C′′( f )C( f )

− 12

(C′( f )C( f )

)2]

and

G( f ) = 2 ∂t log C( f ) = 2∂t a( f , t)

a( f , t),

where C( f ) = a( f , t) in our notation, fav = (s + K)/2 and the termσ0(K, t) is our lowest order coefficient (3.16) originally derived inBerestycki, Busca, and Florent (2002):

σ0(K, t) =[

1ln s− ln K

∫ s

K

dη

a(η, t)

]−1

On page 145 of his book, Henry-Labordere presents an alternativeapproximation to first order in τ, matching ours exactly in the time-homogeneous case and differing only slightly in the time-inhomogeneouscase. In SECTION 2 we have shown that our approximation is the op-timal one to first order τ.

4.2. Model definitions and parameters.

4.2.1. CEV model. The SDE is

d ft = e−λ t σ√

ft dWt

with σ = 0.2. In the time-independent version, λ = 0 and in thetime-dependent version, λ = 1. For the CEV model therefore,

Q( f ) = − 332

σ2

fand G( f ) = −2 λ

so the Henry-Labordere approximation (4.1) becomes

σBS(K, T) ≈ σ0(K, t)

1 +T3

[18

σ0(K, t)2 − 332

σ2

fav− 3

2λ

].

The closed-form solution for the square-root CEV model is well-known and can be found, for example, in Shaw (1998).


4.2.2. Quadratic model. The SDE is

d ft = e−λ t σ

ψ ft + (1− ψ) +γ

2( ft − 1)2

dWt

with σ = 0.2, ψ = −0.5, and γ = 0.1. Again in the time-independentversion, λ = 0 and in the time-dependent version, λ = 1. Then forthe quadratic model,

Q( f ) =132

σ2( f − 1)3 (3 f + 1) γ2 + 24 (1− ψ) γ f

+ 12 ψ γ f 2 − 4[(4− 3 ψ) γ + ψ2

]and again

G( f ) = −2 λ.

The closed-form solution for the quadratic model with these param-eters1 is given in Andersen (2008).

4.3. Results. In TABLES 1 and 2, respectively, we present the errorsin the above approximations in the case of time-independent CEVand quadratic local volatility functions (with λ = 0 and T = 1).We note that our approximation does slightly better than Henry-Labordere’s, although the errors in both approximations are negli-gible. In Figures 1 and 2 respectively, these errors are plotted.

In Figures 3 and 4, we plot results for the time-dependent casesλ = 1 with T = 0.25 and T = 1.0 respectively, comparing our ap-proximation to implied volatility with the exact result. To first orderin τ = τ (with only σ1 and not σ2), we see that our approximation isreasonably good for short expirations (λ T 1) but far off for longerexpirations (λ T > 1). The approximation including σ2 up to order τ2

is almost exact for the shorter expiration T = 0.25 and much closer tothe true implied volatility for the longer expiration T = 1.0. On thisscale, Henry-Labordere’s first order approximation is indistinguish-able from our first order approximation (including σ1 but excludingσ2), consistent with the tiny errors shown in Figures 1 and 2.

APPENDIX A. PRINCIPLE OF NOT FEELING THE BOUNDARY

Consider a one-dimensional diffusion process

dXt = a(Xt) dWt + b(Xt) dt

1The solution is more complicated for certain other parameter choices


on [0, ∞), where the continuous function a(x) > 0 for x > 0. Weassume that b is continuous on R+. We do not make any assump-tion about the behavior of a(y) as y ↓ 0. Let d(a, b) be the distancebetween two points a, b ∈ R+ determined by 1/a. If say a < b, then

d(a, b) =∫ b

a

dxa(x)

.

Letτc = inf t ≥ 0 : Xt = c .

Lemma A.1. Suppose that x > 0 and c > 0. Then

limτ↓0

τ ln Px τc ≤ τ ≤ −d(x, c)2

2.

Proof. Let Yt = d(Xt, x). By Ito’s formula we have

dYt = dWt + θ (Yt) dt,

where

θ (y) =b(z)a(z)

− 12

a′(z), y = d(z, x).

Without loss of generality we assume that c > x. It is clear that

Px

τX

c ≤ τ= P0

τY

D ≤ τ

, D = d(x, c).

Let θ be the lower bound of the function θ(z) on the interval [0, D].Then Yt ≤ D for all 0 ≤ t ≤ τ implies that Wt ≤ D − θτ for all0 ≤ t ≤ τ. It follows that

P0

τY

D ≤ τ≤ P0

τW

D−θτ ≤ τ

.

The last probability is explicitly known:

P0

τW

λ ≤ τ=

λ√2π

∫ τ

0t−3/2e−λ2/2t dt.

Using this we have after some routine manipulations

limτ↓0

τ ln P0 τD−θτ ≤ τ ≤ −D2

2.

The desired result follows immediately.Note that we do not need to assume that X does not explode. By

convention τc = ∞ if X explodes before reaching c, thus making theinequality more likely to be true.


Let 0 < a < x < b < ∞. Let f be a nonnegative function on R+

and suppose that f is supported on x ≥ b, i.e., f (y) = 0 for y ≤b. This corresponds to the case of an out-of-the-money call option.Consider the call price

v(x, τ) = Ex f (Xτ).

We compare this with

v1(x, τ) = Ex f (Xτ); τ < τa .

Note that v1 only depends on the values of a on [a, ∞), thus he be-havior of a near y = 0 is excluded from consideration. We have

v(x, τ)− v1(x, τ) = Ex f (Xτ); τa ≤ τ def= v2(x, τ).

By the Markov property we have

v2(x, τ) = Ex Ea f (Xs)|s=τ−τa ; τa ≤ τ .

Now since f (y) = 0 for y ≤ b, we have

Ea f (Xs) = Ea f (Xs); τb ≤ s .

Using the Markov property again we have

Ea f (Xs) = Ea Eb f (Xt)|t=s−τb ; τb ≤ s .

We assume thatsup

0≤t≤1Eb f (Xt) ≤ C.

This assumption satisfied if we bound the growth rates of f , a and bat infinity appropriately. A typical case is when f grows exponen-tially (call option), and a and b grow at most linearly. These con-ditions are satisfied by all the popular models we deal with. It isclear that we have to make some assumption about the behavior ofthe data at infinity, otherwise the problem may not even make anysense. Under this hypothesis we have

Ea f (Xs) ≤ CPa τb ≤ s .

Now we havev2(x, τ) ≤ CPa τb ≤ τ .

It follows from LEMMA A.1 that

limτ↓0

τ ln v2(x, τ) ≤ −d(a, b)2

2.

Recall thatv(x, τ) = v1(x, τ) + v2(x, τ).


The function v1(x, τ) does not depend on the values of a near y = 0.We can alter the values of a near y = 0 and the resulting error isbounded asymptotically by exp

[−d(a, b)2/2τ

]. Now if the support

of f (as a closed set) contains y = b, then we can prove, assuming abehaves nicely near y = 0 if necessary, that

limτ↓0

τ ln Ex f (Xτ) = −d(x, b)2

2.

Since d(x, b) < d(a, b), we have proved the following principle ofnot feeling the boundary.

Theorem A.2. Let X1 and X2 be (a1, b1)- and (a2, b2)-diffusion processeson R+, respectively, f a nonnegative function on R+, and 0 < a < x < b.Suppose that ai, bi, f satisfy the conditions stated above. Suppose furtherthat a1(y) = a2(y) for y ≥ a. Then

lim supτ↓0

τ ln∣∣∣Ex f (X1

τ)−Ex f (X2τ)∣∣∣ ≤ −d(a, b)2

2

and

limτ↓0

τ ln Ex f (Xiτ) = −

d(x, b)2

2.

Corollary A.3. Under the same conditions, we have

limτ↓0

Ex f (X1τ)

Ex f (X2τ)

= 1.

See Hsu (1990) for a more general principle of not feeling the bound-ary for higher dimensional diffusions.

REFERENCES

[1] ANDERSEN, L., and R. BROTHERTON-RATCLIFFE (2001): Extended LIBORMarket Models with Stochastic Volatility, Working paper, Gen Re Securities.

[2] ANDERSEN, L.B.G. (2008): Option Pricing with Quadratic Volatility: A Re-visit, Discussion paper, Bank of America Securities.

[3] AVELLANEDA, M., D. BOYER-OLSON, J. BUSCA, and P. FRIZ (2002): Recon-structing Volatility, Risk Magazine 15, 87-91.

[4] BERESTYCKI, H., J. BUSCA, and I. FLORENT (2002): Asymptotics and Calibra-tion of Local Volatility Models, Quantitative Finance 2, 61-69.

[5] BERESTYCKI, H., J. BUSCA, and I. FLORENT, I (2004): Computing the ImpliedVolatility in Stochastic Volatility Models, Communications on Pure and AppliedMathematics 57(10), 1352-1373.

[6] DE BRUIN, N. G. (1999): Asymptotic Methods in Analysis, Dover Publications.[7] CARR, P., and R. JARROW (1990): The Stop-Loss Start-Gain Paradox and Op-

tion Valuation: A New Decomposition into Intrinsic and Time Value, Reviewof Financial Studies 3(3), 469-492.


[8] COX, J. (1975): Notes on Option Pricing I: Constant Elasticity of Diffusions,unpublished draft. Palo Alto, CA: Stanford University, September 1975.

[9] DERMAN, E., and I. KANI (1994): Riding on a Smile, Risk Magazine 7, 32-39.[10] DUPIRE, B. (1994): Pricing with a Smile, Risk Magazine 7, 18-20.[11] DUPIRE, B. (2004): A Unified Theory of Volatility, Discussion paper Paribas

Capital Management. Reprinted in ”Derivative Pricing: the classic collec-tion”, edited by Peter Carr, Risk Books.

[12] GATHERAL, J. (2006): The Volatility Surface: A Practitioner’s Guide, WileyFinance Series.

[13] GYONGI, I. (1986): Mimicking the One Dimensional Marginal Distributionsof Processes having an Ito Differential, Probability Theory and Related Fields 71,501-516.

[14] HADAMARD, J. (1952): Lectures on Cauchy’s problem in linear partial differ-ential equations, Dover Publications.

[15] HAGAN, P., and D. WOODWARD (1999): Equivalent Black Volatilities, AppliedMathematical Finance 6, 147-157.

[16] HAGAN, P., A. LESNIEWSKI, and D. WOODWARD (2004): Probability Distri-bution in the SABR Model of Stochastic Volatility, Working Paper.

[17] HAGAN, P., D. KUMAR, A. LESNIEWSKI, and D. WOODWARD (2003): Man-aging Smile Risk, Wilmott Magazine, 84-108.

[18] HENRY-LABORDERE, P. (2005): A General Asymptotic Implied Volatility forStochastic Volatility Models, Preprint.

[19] HENRY-LABORDERE, P. (2008): Analysis, Geometry, and Modeling in Fi-nance, Chapman& Hall/CRC, Financial Mathematics Series.

[20] HSU, E. P. (1990): Principle of Not Feeling the Boundary, Indiana UniversityMathematics Journal 39(2), 431-442.

[21] KAC, M. (1951): On Some Connections between Probability Theory and Dif-ferential and Integral Equations, Proc. Second Berkeley Symposium on Math.Stat. and Prob., University of California Press, 189-215.

[22] KUNIMOTO, N., and A. TAKAHASHI (2001): The Asymptotic Expansion Ap-proach to the Valuation of Interest Rate Contingent Claims, Mathematical Fi-nance 11(1), 117-151.

[23] LESNIEWSKI, A. (2002): WKB Method for Swaption Smile, Courant InstituteLecture.

[24] LIPTON, A. (2002): The Volatility Smile Problem, Risk Magazine, February.[25] MEDVEDEV, A., and O. SCAILLET (2007): Approximation and Calibration

of Short-Term Implied Volatilities under Jump-Diffusion Stochastic Volatility,The Review of Financial Studies 20, 427-459.

[26] MINAKSHISUNDARAM, S, and A. PLEIJEL (1949): Some Properties of Eigen-functions of the Laplace Operatoron Riemannnian Manifolds, Canadian Jour-nal of Mathematics 1, 242-256.

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[32] ZUHLSDORFF, C. (2001): The Pricing of Derivatives on Assets with QuadraticVolatility, Applied Mathematical Finance 8, 235-262.


TABLE 1. CEV model implied volatility errors for vari-ous strike prices in the Henry-Labordere (HL) approxi-mation and our first and second order approximationsrespectively. The exact volatility in the last column isobtained by inverting the closed-form expression forthe option price in the CEV model.

Strikes ∆σHL ∆σ1 ∆σ2 σexact0.50 2.12e-05 1.31e-06 1.98e-08 0.23680.75 3.46e-06 7.98e-07 9.87e-09 0.21481.00 5.68e-07 5.68e-07 6.03e-09 0.20011.25 1.52e-06 4.21e-07 4.08e-09 0.18911.50 3.45e-06 3.33e-07 2.96e-09 0.18051.75 5.45e-06 2.73e-07 2.18e-09 0.17342.00 7.27e-06 2.29e-07 1.70e-09 0.1674

TABLE 2. Quadratic model implied volatility errors forvarious strike prices in the Henry-Labordere (HL) ap-proximation and our first and second order approxi-mations respectively. The exact volatility in the last col-umn is obtained by inverting the closed-form expres-sion for the option price in the quadratic model.

Strikes ∆σHL ∆σ1 ∆σ2 σexact0.50 -8.83e-05 -1.04e-05 -1.08e-07 0.31290.75 -3.42e-05 -3.05e-06 -1.94e-08 0.24511.00 -2.14e-06 -1.09e-06 -4.58e-09 0.20031.25 1.99e-05 -4.31e-07 -1.30e-09 0.16751.50 3.32e-05 -1.80e-07 -3.92e-10 0.14181.75 4.13e-05 -7.59e-08 -5.28e-11 0.12092.00 4.56e-05 -3.16e-08 9.57e-12 0.1032


JIM GATHERAL, DEPARTMENT OF MATHEMATICS, BARUCH COLLEGE, CUNY,ONE BERNARD BARUCH WAY, NEW YORK, NY 10010, USA

E-mail address: [email protected]

ELTON P. HSU, DEPARTMENT OF MATHEMATICS, NORTHWESTERN UNIVER-SITY, EVANSTON, IL 60208


PETER LAURENCE, DIPARTIMENTO DI MATEMATICA, UNIVERSITA DI ROMA 1,PIAZZALE ALDO MORO, 2, I-00185 ROMA, ITALIA, AND, COURANT INSTITUTEOF MATHEMATICAL SCIENCES, NEW YORK UNIVERSITY, 251 MERCER STREET,NEW YORK, NY10012, USA


CHENG OUYANG, DEPARTMENT OF MATHEMATICS, PURDUE UNIVERSITY, WESTLAFAYETTE, IN 47907


TAI-HO WANG, DEPARTMENT OF MATHEMATICS, BARUCH COLLEGE, CUNY,ONE BERNARD BARUCH WAY, NEW YORK, NY10010, USA



FIGURE 1. Approximation errors in implied volatil-ity terms as a function of strike price for the square-root CEV model with the time-homogeneous (λ =0) version of the local volatility function of Section4.2.1. The solid line corresponds to the error in Henry-Labordere’s approximation (5.40), and the dashed anddotted lines to our first and second order approxima-tions respectively. Note that the error in our secondorder approximation is zero on this scale.

0.5 1.0 1.5 2.00.0e+00

5.0e-06

1.0e-05

1.5e-05

2.0e-05

Strike

App

roxi

mat

ion

erro

r

H-Lσ1σ2


FIGURE 2. Approximation errors in implied volatilityterms as a function of strike price for the quadraticmodel with the time-homogeneous (λ = 0) version ofthe local volatility function of Section 4.2.2. The solidline corresponds to the error in Henry-Labordere’s ap-proximation (5.40), and the dashed and dotted linesto our first and second order approximations respec-tively. Note that the error in our second order approx-imation is zero on this scale.

0.5 1.0 1.5 2.0

-8e-05

-4e-05

0e+00

4e-05

Strike

App

roxi

mat

ion

erro

r

H-Lσ1σ2


FIGURE 3. Implied volatility approximations in theCEV model with the strongly time-inhomogeneous(λ = 1) version of the local volatility function of Sec-tion 4.2.1 for two expirations: τ = 0.25 on the left andτ = 1.0 on the right. The solid line is exact impliedvolatility, the dashed line is our approximation to firstorder in τ = τ (with only σ1 and not σ2), the dottedline is our approximation to second order in τ (includ-ing σ2 and almost invisible in the left panel). On thisscale, Henry-Labordere’s first order approximation isindistinguishable from our first order approximation(the dashed line).

0.7 0.8 0.9 1.0 1.1 1.2 1.3

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Strike

Impl

ied

vola

tility

0.5 1.0 1.5 2.0

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Strike

Impl

ied

vola

tility


FIGURE 4. Implied volatility approximationsin the quadratic model with the strongly time-inhomogeneous (λ = 1) version of the local volatilityfunction of Section 4.2.2 for two expirations: τ = 0.25on the left and τ = 1.0 on the right. The solid lineis exact implied volatility, the dashed line is ourapproximation to first order in τ = τ (with only σ1and not σ2), the dotted line is our approximation tosecond order in τ (including σ2 and almost invisible inthe left panel). On this scale, Henry-Labordere’s firstorder approximation is indistinguishable from ourfirst order approximation (the dashed line).

0.7 0.8 0.9 1.0 1.1 1.2 1.3

0.05

0.10

0.15

0.20

Strike

Impl

ied

vola

tility

0.5 1.0 1.5 2.0

0.05

0.10

0.15

0.20

Strike

Impl

ied

vola

tility

ssrn-id1542077

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