Option Pricing under Hybrid Stochastic and Local Volatility

Sun-Yong Choi1, Jean-Pierre Fouque,2 Jeong-Hoon Kim3

Abstract This paper proposes a new option pricing model which can be thought of as a hybridstochastic volatility and local volatility model. This model is built on the local volatility of theconstant elasticity of variance (CEV) model multiplied by stochastic volatility which is driven by afast mean-reverting Ornstein-Uhlenbeck process. The formal asymptotic approximations for optionprices are studied to extend the existing CEV pricing formula. From the results, we show that ourmodel improves the downside of the CEV model in terms of dynamics of implied volatilities withrespect to underlying asset price by resolving possible hedging instability problem. The impliedvolatility structure and calibration of our option pricing model are also discussed.

Keywords: stochastic volatility, constant elasticity of variance, asymptotic analysis, option pricing,implied volatility

1. Introduction

The geometric Brownian motion assumption for underlying asset price in the standard Black-Scholes model (1973) is well-known not to capture the accumulated empirical evidence in financialindustry. A main draw back in the assumption of the Black-Scholes model is in flat impliedvolatilities, which is contradictory to empirical results showing that implied volatilities of the equityoptions exhibit the smile or skew curve. For example, Rubinstein (1985) before the 1987 crash andby Jackwerth and Rubinstein (1996) after the crash belong to those representative results. Beforethe 1987 crash, the geometry of implied volatilities against the strike price was often observed tobe U-shaped with minimum at or near the the money but, after the crash, the shape of smile curvebecomes more typical.

Among those several ways of overcoming the draw back and extending the geometric Brownianmotion to incorporate the smile effect, there is a way that makes volatility depend on underlyingasset price as well as time. One renowned model in this category is the constant elasticity ofvariance diffusion model. This model was initially studied by Cox (1975), and Cox and Ross (1976)and was designed and developed to incorporate the negative correlation between the underlyingasset price change and volatility change. The CEV diffusion has been applied to exotic optionsas well as standard options by many authors. For example, see Boyle and Tian (1999), Boyle etal. (1999), Davydov and Linetsky (2001, 2003), and Linetsky (2004) for studies of barrier and/orlookback options. See Beckers (1980), Emanuel and MacBeth (1982), Schroder (1989), Delbaenand Shirakawa (2002), Heath and Platen (2002), and Carr and Linetsky (2006) for general optionpricing studies. The book by Jeanblanc, Yor, and Chesney (2006) may serve as a general reference.

One disadvantage in local volatility models is, however, that volatility and underlying risky assetprice changes are perfectly correlated either positively or negatively depending on the chosen elas-ticity parameter, whereas empirical studies show that prices may decrease when volatility increasesor vice versa but it seems that there is no definite correlation all the time. For example, Harvey

1Department of Mathematics, Yonsei University, Seoul 120-749, Korea2Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93016, USA3Department of Mathematics, Yonsei University, Seoul 120-749, Korea

1

(2001) and Ghysels et al. (1996) show that it is time varying. Based on this observation togetherwith the renowned contribution of stochastic volatility formulation to option pricing, we introducean external process, apart from the risky asset process itself, driving volatility. Then new volatilityis given by the multiplication of a function of the new process and the local CEV volatility so thatthe new option pricing formulation can be thought of as a hybrid stochastic volatility and localvolatility model.

Many financial models seem to like to have a feature that is given by a mean-reverting process.The process is characterized by its typical time to get back to the mean level of its long-rundistribution (the invariant distribution of the process). We take a fast mean-reverting stochasticprocess as the external process mentioned above so that if mean reversion is very fast, then theunderlying asset price process should be close to the CEV diffusion. This means that if meanreversion is extremely fast, then the CEV model is a good approximation. However, for fast butfinite mean reversion, the CEV model needs to be corrected to account for stochastic volatility.This is the main concern of this paper.

The methodology to be used here is asymptotic analysis based upon a mean-reverting Ornstein-Uhlenbeck (OU) diffusion which decorrelates rapidly while fluctuating on a fast time-scale. Thechoice of fast mean-reverting OU process provides us analytic advantage. This is basically relatedto averaging principle and ergodic theorem or, more directly, asymptotic diffusion limit theory ofstochastic differential equations with a small parameter which has been initiated by Khasminskii(1966) and developed by Papanicolaou (1978), Asch et al. (1991), Kim (2004) and Cerrai (2009).This type of theory has been effectively applied to some ‘pure’ stochastic volatility models infinancial mathematics. Fouque et al. (2000) considered a class of underlying asset price modelsin which volatility is a function of fast mean-reverting OU diffusion and give a certain modelindependent correction extending the standard Black-Scholes formula. This correction is universalin this class of models and it is given by two group parameters which can be easily calibrated fromthe observed implied volatility surface. The analysis has been effectively applied to many topics infinance including option pricing, interest rate derivatives and credit derivatives.

This paper is organized as follows. In Section 2, we review the CEV option price formula andformulate an option pricing problem as a stochastic volatility extension of the CEV model in termsof a singularly perturbed PDE. In Section 3, we use singular perturbation technique on the optionpricing PDE to derive the corrected price of the CEV option price and also give an error estimateto verify mathematical rigor for our approximation. In Section 4, we study the implied volatilitystructure and the calibration of our option pricing model. Concluding remarks are given in Section5.

2. The CEV Formula - Review

In this section we review briefly the well-known CEV option price formula.The dynamics of underlying asset price of the renowned CEV model is given by the SDE

dXt = µXtdt + σXθ2t dWt,(1)

where µ, σ and θ are constants. In this model, volatility is a function of underlying asset price

which is given by σXθ2−1

t . Depending upon the elasticity parameter θ2 , this model can reduce to the

well-known popular option pricing models. For example, If θ = 2, it reduces to the Black-Scholes2

model with constant volatility σ (1973). If θ = 1, it becomes the square root model or Cox-Rossmodel (1976). If θ = 0, then it is the absolute model. Note that if θ < 2, the volatility and thestock price move inversely. If θ > 2, the volatility and the stock price move in the same direction.

There is an analytic form of option price formula for the CEV diffusion solving (1). It was firstgiven by Cox (1975) using the transition probability density function for the CEV diffusion. Hedid not show the derivation of the transition probability density function but it is not difficult toobtain it. Basically, one transforms Xt into Yt via Yt = X2−θ

t , θ 6= 2 and then apply the Ito lemmaand the Feynman-Kac formula. Then using the well-known result on the parabolic PDE by Feller(1951) one can obtain the transition probability density function as follows:

f(x, T ; x, t) = (2− θ)k1

2−θ (uv1−θ)1

2(2−θ) e−u−vI 12−θ

(2(uv)12 ), θ 6= 2(2)

where

k =2r

σ2(2− θ)(er(2−θ)(T−t) − 1),

u = kx2−θer(2−θ)(T−t),

v = kx2−θ

and Iq(x) is the modified Bessel function of the first kind of order q defined by

Iq(x) =∞∑

n=0

(x2 )2n+q

r! Γ(n + 1 + q)

Once the transition probability density function is obtained, the call option pricing formula forthe CEV model will follow as sum of two integrals:

CCEV (t, x) =∫ ∞

−∞e−r(T−t)(x−K)+f(x, T ; x, t) dx

= e−r(T−t)

∫ ∞

Kxf(x, T ;x, t) dx− e−r(T−t)K

∫ ∞

Kf(x, T ; x, t) dx.

The computation of these integrals was done by Schroder(1989) and the result is given by thefollowing theorem. Of course, the put price can be given by the put-call parity.

Theorem 2.1: The call option price CCEV for Xt = x is given by

CCEV (t, x) = x∞∑

n=0

g(n + 1, u) G(n + 1 +1

2− θ, kK2−θ)

−Ke−r(T−t)∞∑

n=0

g(n + 1 +1

2− θ, u) G(n + 1, kK2−θ),(3)

where G(m, v) and g(m,u) are the complementary Gamma distribution and its density given by

G(m, v) =∫ ∞

vg(m,u) du, g(m,u) =

e−uum−1

Γ(m),

respectively.

Proof: Refer to Schroder (1989) or Chen and Lee (1993).

3

Alternatively, the call option price can be obtained by solving the pricing PDE

∂tC +12σ2xθ∂2

xxC + r(x∂xC − C) = 0,(4)

C(T, x) = (x−K)+.(5)

This PDE can be solved, for example, by the Green’s function method after being reduced to asimple form. See Lipton (2001) for details.

Sometimes it is more convenient to rewrite the above option formula in the form which directlygeneralizes the standard Black-Scholes formula (see, for example, Lipton (2001)):

CCEV (t, x) = e−r(T−t)x

∫ ∞

K

(x

y

) 12(2−θ)

e−(x+y)I 12−θ

(2√

xy) dy

+ e−r(T−t)K

∫ ∞

K

(y

x

) 12(2−θ)

e−(x+y)I 12−θ

(2√

xy) dy,(6)

where

x =2xer(2−θ)(T−t)

(2− θ)2χ,

χ =σ2

(2− θ)r(er(2−θ)T − er(2−θ)t),

K =2K2−θ

(2− θ)2χ.

Note that by using the asymptotic formula

I 12β

(x

β2

)∼ βex/β2−1/8x

√2πx

, β → 0

for the Bessel function, one can show that the CEV call price (CCEV ) goes to the usual Black-Scholesprice (CBS) as θ goes to 2.

3. Problem Formulation

In this section, we establish a new option pricing model that extends the local volatility of theCEV model to incorporate stochastic volatility.

First, the local volatility term Xθ2t of the CEV model is changed into such a form that it is a

product of a function of external process Yt and Xθ2t so that the volatility of the asset price depends

on both internal and external noise sources. In this form, it would be natural to choose Yt as amean-reverting diffusion process since the newly formed volatility gets back to the mean level thatcorresponds to the CEV case. The mean-reverting Ornstein-Uhlenbeck process is an example ofsuch process.

As an underlying asset price model in this paper, therefore, we take the SDEs

dXt = µXtdt + f(Yt)Xθ2t dWt(7)

dYt = α(m− Yt)dt + βdZt,(8)4

where f(y) is a sufficiently smooth function and the Brownian motion Zt is correlated with Wt

such that d〈W, Z〉t = ρdt. In terms of the instantaneous correlation coefficient ρ, we write Zt =ρWt +

√1− ρ2Zt for convenience. Here, Zt is a standard Brownian motion independent of Wt.

It is well-known from the Ito formula that the solution of (8) is a Gaussian process given by

Yt = m + (Y0 −m)e−αt + β

∫ t

0e−α(t−s) dZs

and thus Yt ∼ N(m + (Y0 − m)e−αt, β2

2α(1 − e−2αt)) leading to invariant distribution given byN(m, β2

2α). Later, we will use notation 〈·〉 for the average with respect to the invariant distribution,i.e.,

〈g(Yt)〉 =1√

2πν2

∫ +∞

−∞g(y)e−

(y−m)2

2ν2 dy, ν2 ≡ β2

2α

for arbitrary function g. Notice that ν2 denotes the variance of the invariant distribution of Yt.Once an equivalent martingale measure Q, under which the discounted asset price e−rtXt is a

martingale, is provided, the option price is given by the formula

P (t, x, y) = EQ[e−r(T−t)h(XT )|Xt = x, Yt = y](9)

under Q. The existence of this martingale measure is guaranteed by the well-known Girsanovtheorem, which further provides that processes defined by

W ∗t = Wt +

∫ t

0

µ− r

f(Ys)Xθ−22

s

ds,

Z∗t = Zt +∫ t

0γs ds

are independent standard Brownian motion under Q, where γt is arbitrary adapted process to bedetermined. We will use notation Qγ in stead of Q to emphasize the dependence on γ. It may raisea question whether Novikov’s condition is satisfied on W ∗. This issue could be resolved by givingdependence of µ on the process (Xt, Yt). We, however, omit the details in this paper.

We first change the subjective SDEs (7)-(8) into the risk-neutral version as follows. Under anequivalent martingale measure Qγ , we have

dXt = rXt dt + f(Yt)Xθ2t dW ∗

t ,(10)

dYt = (α(m− Yt)− βΛ(t, y)) dt + β dZ∗t ,(11)

where Z∗t = ρW ∗t +

√1− ρ2Z∗t and Λ (the market price of volatility risk) is assumed to be inde-

pendent of x.Then the Feynman-Kac formula leads to the following pricing PDE for the option price P .

Pt +12f2(y)xθPxx + ρβf(y)x

θ2 Pxy +

12β2Pyy

+rxPx + (α(m− y)− βΛ(t, y))Py − rP = 0.(12)5

4. Asymptotic Option Pricing

Now, to develop an asymptotic pricing theory on fast mean-reversion, we introduce a smallparameter ε to denote the inverse of the rate of mean reversion α:

ε =1α

(13)

which is also the typical correlation time of the OU process Yt. We assume that ν (the standarddeviation of the invariant distribution of Yt), which was introduced in Section 3 as ν = β√

2α, remains

fixed in scale as ε becomes zero. Thus we have α ∼ O(ε−1), β ∼ O(ε−1/2), and ν ∼ O(1).After rewritten in terms of ε, the PDE (12) becomes

(1εL0 +

1√εL1 + L2

)P ε = 0(14)

where

L0 = ν2∂2yy + (m− y)∂y,(15)

L1 =√

2ρνf(y)xθ2 ∂2

xy −√

2νΛ(t, y)∂y,(16)

L2 = ∂t +12f(y)2xθ∂2

xx + r(x∂x − ·).(17)

Here, the operator αL0 is the same as the infinitesimal generator of the OU process Yt acts only onthe y variable. The operator L1 is the sum of two terms; the first one is the mixed partial differentialoperator due to the correlation of the two Brownian motions W ∗ and Z∗ and the second one is thefirst-order differential operator with respect to y due to the market price of elasticity risk. Finally,the third operator L2 is a generalized version of the classical Black-Scholes operator at the volatilitylevel σ(x, y) = f(y)x

θ2−1 in stead of constant volatility.

Before we solve the problem (14)-(17), we write a useful lemma about the solvability (centering)condition on the Poisson equation related to L0 as follows.

Lemma 4.1: If solution to the Poisson equation

L0χ(y) + ψ(y) = 0(18)

exists, then the condition 〈ψ〉 = 0 must be satisfied, where 〈·〉 is the expectation with respect tothe invariant distribution of Yt. If then, solutions of (18) are given by the form

χ(y) =∫ t

0Ey[ψ(Yt)] dt + constant.(19)

Proof: Refer to Fouque et al. (2000). ¤

Although the problem (14)-(17) is a singular perturbation problem, we are able to obtain a limitof P ε as ε goes to zero and also characterize the first correction for small but nonzero ε. In orderto do it, we first expand P ε in powers of

√ε:

P ε = P0 +√

εP1 + εP2 + · · ·(20)

Here, the choice of the power unit√

ε in the power series expansion was determined by the methodof matching coefficient.

6

Substituting (20) into (14) yields

1εL0P0 +

1√ε

(L0P1 + L1P0)

+ (L0P2 + L1P1 + L2P0) +√

ε (L0P3 + L1P2 + L2P1) + · · · = 0,(21)

which holds for arbitrary ε > 0.Under some reasonable growth condition, one can show that the first corrected price P0 +

√εP1

is independent of the unobserved variable y as follows.

Theorem 4.1 Assume that P0 and P1 do not grow so much as

∂Pi

∂y∼ e

y2

2 , y →∞, i = 0, 1.

Then P0 and P1 do not depend on the variable y.

Proof: From the asymptotic expansion (21), we first have

L0P0 = 0.(22)

Solving this equation yields

P0(t, x, y) = c1(t, x)∫ y

0e

(θ−z)2

2ν2 dz + c2(t, x)

for some functions c1 and c2 independent of y. From the imposed growth condition on P0, c1 = 0can be chosen. This leads that P0(t, x, y) becomes a function of only t and x. We denote it by

P0 = P0(t, x).(23)

Since each term of the operator L1 contains y-derivative, the y-independence of P0 yields L1P0 =0. On the other hand, from the expansion (21), L0P1+L1P0 = 0 holds. These facts are put togetherand yield L0P1 = 0 so that P1 also is a function of only t and x;

P1 = P1(t, x).(24)

¤We next derive a PDE that the leading order price P0 satisfies based upon the observation that

P0 and P1 do not depend on the current level y of the process Yt.

Theorem 4.2: Under the growth condition on Pi (i = 0, 1) expressed in Theorem 4.1, the leadingterm P0(t, x) is given by the solution of the PDE

∂tP0 +12σ2xθ∂2

xxP0 + r(x∂xP0 − P0) = 0(25)

with the terminal condition P0(T, x) = h(x), where σ =√〈f2〉. So, P0 (Call) is the same as the

price in the CEV formula in Theorem 2.1 while only σ is replaced with the effective coefficient σ.

Proof: From the expansion (21), the PDE

L0P2 + L1P1 + L2P0 = 0(26)7

holds. Since each term of the operator L1 contains y-derivative, L1P1 = 0 holds so that (26) leadsto

L0P2 + L2P0 = 0(27)

which is a Poisson equation. Then the centering condition must be satisfied. From Lemma 3.1 withψ = L2P0, the leading order P0(t, x) has to satisfy the PDE

〈L2〉P0 = 0(28)

with the terminal condition P0(T, x) = h(x), where

〈L2〉 = ∂t +12〈f2〉xθ∂2

xx + r(x∂x − ·).

Thus P0 solves the PDE (25). ¤

Next, we derive the first correction P1(t, x) and estimate the error of the approximation P0+√

εP1.For convenience, we need notation

V3 =ρν√

2〈fψ

′〉,(29)

V2 = − ν√2〈Λψ

′〉,(30)

where ψ(y) is solution of the Poisson equation

L0ψ = ν2ψ′′

+ (m− y)ψ′= f2 − 〈f2〉.(31)

Note that V3 and V2 are constants.

Theorem 4.3: The first correction P1(t, x) satisfies the PDE

∂tP1 +12σ2xθ∂2

xxP1 + r(x∂xP1 − P1) = V3xθ2

∂

∂x(xθ ∂2P0

∂x2) + V2x

θ ∂2P0

∂x2(32)

with the final condition P1(T, x) = 0, where V3 and V2 are given by (29) and (30), respectively.Further, the solution P1(t, x) (θ 6= 2) of (32) is given by

P1(t, x) = −∫ τ

0

∫ +∞

−∞

1√2πσ2(τ − z)

exp{− (s− y)2

2σ2(τ − z)}

·exp{{−12(

a

y2− a

s2) +

12σ2

(a

y+ by)2 +

12σ2

(a

s+ bs)2}z +

1σ2

(a lny

s+

b

2(y2 − s2))}(33)

·h(y1

− 12 θ+1 , T − z) dy dz,

where τ = T − t, s = x−12θ+1 and

σ = σ(−12θ + 1), a =

12σ2(−1

2θ + 1)(−1

2θ), b = r(−1

2θ + 1),

h(x, t) = V3xθ2

∂

∂x(xθ ∂2P0

∂x2) + V2x

θ ∂2P0

∂x2.

8

Proof: From (17) and (28)

L2P0 = L2P0 − 〈L2P0〉=

12xθ

(f2 − 〈f2〉) ∂2

xxP0

holds. Then (27) and (31) lead to

P2 = −L−10 (L2P0)

= −12xθL−1

0

(f2 − 〈f2〉) ∂2

xxP0(34)

= −12

(ψ(y) + c(t, x))xθ∂2xxP0

for arbitrary function c(t, x) independent of y.On the other hand, from the expansion (21)

L0P3 + L1P2 + L2P1 = 0

holds. This is a Poisson PDE for P3. Thus the centering condition 〈L1P2 + L2P1〉 = 0 has to besatisfied. Then using the result (34) we obtain

〈L2〉P1 = 〈L2P1〉= −〈L1P2〉=

12〈L1ψ〉xθ∂2

xxP0.

Applying (16), (17), (29) and (30) to this result, we obtain the PDE (32).Now, we solve the PDE (32) by following step. First, we change (32) with the terminal condition

P1(T, x) = 0 into the initial value problem

− ∂P1

∂τ+

12σ2 ∂2P1

∂s2+ (

a

s+ bs)

∂P1

∂s− rP1 = h(s, τ),

P1(s, 0) = 0.(35)

where h(s, τ) = h(s1

− 12 θ+1 , τ). Then by the change of dependent variables

P1(τ, s) = v(s)g(τ)u(τ, s),

where

v(s) = exp{− 1

σ2(a ln s +

b

2s2)

},

g(τ) = exp[{

12(

a

s2− b)− 1

2σ2(a

s+ bs)2 − r

}τ

],

the PDE (35) becomes

∂u

∂τ=

12σ2 ∂u2

∂s2+ F (s, τ),

u(s, 0) = 0,(36)

where

F (s, τ) = −exp{{−12(

a

s2− b) +

12σ2

(a

s+ bs)2 + r}τ +

1σ2

(a ln s +b

2s2)}h(s, τ).

9

Equation (36) is a well-known form of diffusion equation whose solution is given by

u(τ, s) =∫ τ

0

∫ +∞

−∞

1√2πσ2(τ − z)

exp{− (s− y)2

2σ2(τ − z)}F (y, z)dydz.

Going back to the original variables, therefore, the solution of the PDE (32) is given by (33). ¤

5. Accuracy - Error Estimate

Combining the leading order P0 and the first correction term P1 =√

εP1, we have the approx-imation P0 + P1 to the original price P . Naturally, the next concern is about accuracy of theapproximation P0 + P1 to the price P .

First, we write a well-known property of solutions to the Poisson equation (18) about theirboundedness or growth.

Lemma 5.1 If there are constant C and integer n 6= 0 such that

|ψ(y)| ≤ C(1 + |y|n),(37)

then solutions of the Poisson equation (18), i.e. L0χ(y) + ψ(y) = 0, satisfy

|χ(y)| ≤ C1(1 + |y|n)(38)

for some constant C1. If n = 0, then |χ(y)| ≤ C1(1 + log(1 + |y|)).

Proof: Refer to Fouque et al. (2000). ¤

Now, we are ready to estimate the error of the approximation P0 + P1 to the price P .

Theorem 5.1 If each of f and Λ is bounded by a constant independent of ε, then we have

P ε − (P0 + P1) = O(ε).(39)

Proof: We first define Zε(t, x, y) by

Zε = εP2 + ε√

εP3 − [P ε − (P0 + P1)].

Once we show that Zε = O(ε), the estimate (39) will follow immediately.In general, h is not smooth as we assume in this proof. However, the generalization of the proof

to the case of European vanilla call or put option is possible as shown in Fouque et al. (2003). Thedetailed proof is omitted here.

Using the operators in (15)-(17), we define

Lε =1εL0 +

1√εL1 + L2

10

and apply the operator Lε to Zε to obtain

LεZε = Lε(P0 +√

εP1 + εP2 + ε√

εP3 − P ε)

= Lε(P0 +√

εP1 + εP2 + ε√

εP3)

=1εL0P0 +

1√ε(L0P1 + L1P0) + (L0P2 + L1P1 + L2P0)

+√

ε(L0P3 + L1P2 + L2P1) + ε(L1P3 + L2P2) + ε√

εL2P3

= ε(L1P3 + L2P2) + ε√

εL2P3.(40)

Here, we have used LεP ε = 0 and the fact that the expansion (21) holds for arbitrary ε. The result(40) yields LεZε = O(ε).

Now, let F ε and Gε denote

F ε(t, x, y) = L1P3 + L2P2 +√

εL2P3,(41)

Gε(x, y) = P2(T, x, y) +√

εP3(T, x, y),(42)

respectively. Then from (40)-(42) we have a parabolic PDE for Zε of the form

LεZε − εF ε = 0

with the final condition Zε(T, x, y) = εGε(x, y). Applying the Feynman-Kac formula to this PDEyields

Zε = εE∗[e−r(T−t)Gε −

∫ T

te−r(s−t)F ε ds |Xε

t = x, Y εt = y

]

in a risk-neutral world. On the other hand, since P2 and P3 composing F ε and Gε are solutions ofthe Poisson equations L0P2 +L2P0 = 0 and L0P3 +(L1P2 +L2P1) = 0, respectively, F ε and Gε arebounded uniformly in x and at most linearly growing in |y| from Lemma 5.1. Consequently, thedesired error estimate Zε = O(ε) follows. ¤

6. Implied Volatilities and Calibration

To obtain the first correction term P1, whose PDE is given by Theorem 4.3, it is required tohave the parameters V3 and V2. Practically, the value of the parameter θ is close to 2. So, in thissection, we employ another perturbation for θ and fit the parameters to observed prices of Calloptions for different strikes and maturities.

For convenience, we use the implied volatility I defined by

CBS(t, x; K, T ; I) = P (Approximated Observed Price).

After we choose I0(K, T ) such that

CBS(t, x; K, T ; I0) = P0(t, x) = CCEV (σ),

if we use expansion I = I0+√

εI1+··· on the left-hand side and our approximation P = P0+P1+···on the right-hand side, then

CBS(t, x;K,T ; I0) +√

εI1∂CBS

∂σ(t, x; K, T ; I0) + · · · = P0(t, x) + P1(t, x) + · · ·,

11

Number Maturity(day) θ1 70 2.01382 130 2.00633 220 2.01114 490 2.00755 860 2.0121

Table 1. The Elasticity for SPX

which yields that the implied volatility is given by

I = I0 + P1(t, x)(

∂CBS

∂σ(t, x; K, T ; I0)

)−1

+ O(1/α).(43)

where ∂CBS∂σ (Vega) is well-known to be given by

∂CBS

∂σ(t, x; K, T ; I0) =

xe−d21/2√

T − t√2π

,

d1 =log(x/K) + (r + 1

2I20 )(T − t)

I0

√T − t

.

On the other hand, we note that when θ = 2, P0 is the Black-Scholes Call option price withvolatility σ, denoted by CBS(σ), and P1 is explicitly given by

P1 = −(T − t)H0,

where

H0 = A(V ∗3 , V ∗

2 )CBS(σ)(44)

with

V ∗3 =

ρν√2α〈fψ

′〉,

V ∗2 = − ν√

2α〈Λψ

′〉.

Here, we used a differential operator A(α, β) defined by

A(α, β) = αx∂

∂x

(x2 ∂2

∂x2

)+ βx2 ∂2

∂x2,

and ψ(y) as a solution of the Poisson equation (31). Note that to compute A(V ∗3 , V ∗

2 ) it is requiredto have Gamma and Speed of the Black-Scholes price. So, fitting (43) to the implied volatilitiesone can obtain V ∗

3 and V ∗2 .

Practically, the elasticity θ is close to 2. See Table 1. So, from now on we approximate P0 andP1 near θ = 2. We use notation

φ = 2− θ

here. We expand P0 as

P0 = CCEV (σ) = CBS(σ) + φP0,1 + φ2P0,2 + · · ·12

and H, defined by H(t, x) = A(V ∗3 , V ∗

2 )P0, and P1, respectively, as

H = H0 + φH1 + φ2H2 + · · ·P1 = −(T − t)H0 + φP1,1 + φ2P1,2 + · · ·.

Then from Theorem 4.2 P0,1 satisfies the Black-Scholes equation with a non-zero source but witha zero terminal condition which is given by

LBS(σ)P0,1 =12σ2x2 lnx

∂2CBS

∂x2,

P0,1(T, x) = 0,(45)

and LBS(σ)H0 = 0 as it should be, and from Theorem 4.3 P1,1 is given by

LBS(σ)P1,1 = H1 − 12σ2(T − t)x2 ln x

∂2H0

∂x2,

P1,1(T, x) = 0,

(46)

where H0 is (44) and H1 is given by

H1 = A(V ∗3 , V ∗

2 )P0,1.(47)

Up to order φ2, therefore, P0 is given by

P0 = CBS(σ) + φP0,1,

where P0,1 is the solution of (45), and the correction P1 is given by

P1 = −(T − t)A(V ∗3 , V ∗

2 )CBS(σ) + φP1,1,

where P1,1 is the solution of (46), respectively.Synthesizing the discussion above, pricing methodology can be implemented as follows in the

case near θ = 2. After estimating σ (the effective historical volatility) from the risky asset pricereturns and confirming from the historical asset price returns that volatility is fast mean-reverting,fit (43) to the implied volatility surface I to obtain V ∗

i , i = 2, 3. Then the Call price of a Europeanoption corrected for stochastic volatility is approximated by

P0 + P1 = CBS(σ)− (T − t)A(V ∗3 , V ∗

2 )CBS(σ) + (2− θ)(P0,1 + P1,1),(48)

where P0,1 and P1,1 are given (in integral form) by

P0,1 =12σ2E∗[−

∫ T

tX2

s lnXs∂2CBS(σ)

∂x2ds],

P1,1 = E∗[− 1√α

∫ T

tA(V ∗

3 , V ∗2 )P0,1ds

+12σ2

∫ T

t(T − s)X2

s ln Xs∂2

∂x2

(A(V ∗

3 , V ∗2 )CBS(σ)

)ds],

respectively.Our corrected price P0 + P1 has been constructed based upon the new volatility given by the

multiplication of a function of the OU process and the local CEV volatility. So, let us call ourmodel as SVCEV model. Figure 1 and 2 show the proof of our new option pricing formulation interms of implied volatilities. Here, the red line is market data. The blue line is SVCEV fit. Also,Figure 3 demonstrates that the implied volatilities move in the desirable direction with respect to

13

−0.1 −0.05 0 0.05 0.1 0.150.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

log(K/x)

Impl

ied

Vol

atili

ty

Days to Maturity =70

Market DataSVCEV fit

Figure 1. SPX Implied Volatility from August 11,2010

−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.30.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

log(K/x)

Impl

ied

Vol

atili

ty

Days to Maturity = 220

Market DataSVCEV fit

Figure 2. SPX Implied Volatility from August 11,2010

the underlying asset price, which would be contrary to the CEV model. In figure 3, we plot theimplied volatility curve using calibration results σ = 0.11, V2 = −0.0611, V3 = −0.1472, θ = 2.011for SPX (Days to Maturity 220).

7. Conclusion

Our new option pricing model based upon a hybrid structure of stochastic volatility and constantelasticity of variance has demonstrated the improvement of the downside of the CEV model in thatthe implied volatilities move in accordance with the real market phenomena with respect to theunderlying asset price. This is an important result in the sense that otherwise it may create apossible hedging instability problem. Also, the geometric structure of the implied volatilities showsa smile fitting the market data well. Consequently, the new underlying asset price model may serve

14

1000 1020 1040 1060 1080 1100 1120 1140 1160 1180 12000.126

0.128

0.13

0.132

0.134

0.136

0.138

0.14

0.142

0.144

Strike K

Impl

ied

Vola

tility

Line1Line2Line3Line4Line5

Figure 3. Line 1 : x = 1008, Line 2 : x = 1064, Line 3 : x = 1121, Line 4 :x = 1177 , Line 5 : x = 1233

as a sound alternative one replacing the CEV model for other financial problems such as credit riskand portfolio optimization apart from option pricing discussed in this paper.

AcknowledgmentsThe work of the first and third author was supported by Basic Science Research Program throughthe National Research Foundation of Korea funded by the Ministry of Education, Science and Tech-nology (NRF-2010-0008717) and also in part by the Ministry of Knowledge Economy and KoreaInstitute for Advancement in Technology through the Workforce Development Program in StrategicTechnology.

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