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Pricing arithmetic Asian options under hybrid stochastic and local volatility.

1. Introduction

Since the well-known work of Black and Scholes [1] on the classical vanilla European option, there has been concern about the pricing of more complicated exotic options. An exotic option is a derivative which has a payoff structure more complex than commonly traded vanilla options. They are usually traded in over-the-counter market or embedded in structured products. Also, the pricing of them tends to require more complex methods than the classical Black Scholes approach.

This paper is concerned with one of the exotic options called an Asian option. This option is a path dependent option whose final payoff depends on the paths of its underlying asset. More precisely, the payoff is determined by the average value of underlying prices over some prescribed period of time. The name of "Asian" options is known to come from the fact that they were first priced in 1987 by David Spaughton and Mark Standish of Bankers Trust when they were working in Tokyo, Japan (cf. [2]). The main motivation of creating these options is that their averaging feature could reduce the risk of market manipulation of the underlying risky asset at maturity. Since Asian options reduce the volatility inherent in the option, the price of these options is usually lower than the price of classical European vanilla options. Note that there are two types of Asian options depending on the style of averaging: one is the arithmetic average Asian option and the other is the geometric average Asian option.

Since there is no general analytical formula for the price of Asian option, a variety of techniques have been developed to approximate the price of this option. Subsequently, there has been quite an amount of literature devoted to studying this option. For instance, Geman and Yor [3] computed the Laplace transform of the price of continuously sampled Asian options. However, there is a problem of slow convergence for low volatility or short time-to-maturity cases as indicated by Fu et al. [4]. Linetsky [5] derived a new integral formula for the price of Asian options but with the same convergence problem. Apart from this approximation technique, there are Monte Carlo simulation approach and partial differential equation (PDE) approach. Monte Carlo simulation methods typically require variance reduction techniques and also have to take into account the discretization error caused by discrete sampling. Refer to Kemna and Vorst [6] for a discussion of the pricing of Asian options with Monte Carlo methods. On the other hand, the PDE methods must deal with an extra state variable representing the running sum of the underlying process, which leads to an issue for reducing the dimension of the PDE. Refer to Ingersoll [7] and Rogers and Shi [8] and Vecer [9] for the PDE methods.

It is well known that the constant volatility assumption, on which a review of literature quoted above is based, for the underlying asset price is severely in contrast with many empirical studies which demonstrate the skew or smile effect of implied volatility, fat-tailed and asymmetric returns distributions, and the mean-reversion of volatility. Thus a number of alternative underlying models have been proposed. The constant elasticity of variance model by Cox [10], a stochastic volatility model by Heston [11] or Fouque et al. [12], and a Levy model by Carr et al. [13] are among those representative ones that can reproduce an empirically reasonable outcome.

So, it is desirable to study Asian options based on these alternative models. In fact, there are a number of recent studies along the lines of this type of extension. For instance, B. Peng and F. Peng [14] for the CEV model, Fouque and Han [15] for a stochastic volatility model, and Lemmens et al. [16] for a Levy type model are among those works extending the price for the Black-Scholes framework with constant volatility. As long as we understand, up to now, however, there has been no work for Asian options based on a hybrid stochastic and local volatility model. Recently, hybrid stochastic and local volatility models have become an industry standard for the pricing of derivatives and several financial institutions have incorporated those models into their systems [17]. Therefore, it is worth studying Asian options on the hybrid models.

This paper studies the pricing of an arithmetic Asian option under a hybrid stochastic and local volatility model which was introduced by Choi et al. [18], where the volatility is given by the product of a multiscale stochastic process and a power (the elasticity of variance) of the underlying's price. The hybrid nature of this volatility enables us to capture the leverage effect produced by the constant elasticity of variance (CEV) model as well as the smile effect of implied volatility, fat-tailed and asymmetric returns distributions, the tendency of the volatility process to revert towards a long-term mean at a certain rate, and a degree of correlation between the randomness of volatility and the randomness of underlying's price produced by a "pure" stochastic volatility (SV) model. This hybrid model is called the SVCEV model. So, this paper will generalize both [14, 15] into an approximation problem for the price of arithmetic average Asian options based on the SVCEV model.

This paper is structured as follows. In Section 2, we review the SVCEV model introduced by [18]. A multiscale partial differential equation for the price of an arithmetic average Asian option is obtained in Section 3. Section 4 is devoted to obtain an approximated option price under the condition of fast mean-reverting volatility. In Section 5, the effect of hybrid structure on the pricing of the Asian option is illustrated by a numerical study.

2. The SVCEV Model

In this section, we establish a partial differential equation for the price of Asian floating strike call option based on the SVCEV model.

As introduced by [18], the SVCEV model for the underlying's price is given by the stochastic differential equations (SDEs):

[dS.sub.t] = [mu]([S.sub.t])dt + ([[sigma].sub.t])[S.sup.1+[gamma].sub.t][dW.sub.t], [[sigma].sub.t] := f([Y.sub.t]), [dY.sub.t] = [alpha](m - [Y.sub.t])dt + [beta][dB.sub.t], (1)

where [mu] and m are some constants, [gamma] is greater than -1, [alpha] and [beta] are positive constants, and [W.sub.t] and [B.sub.t] are correlated Brownian motions such that d[<W, B>.sub.t] = [rho]dt for some [rho]. If [rho] > 0, then St may fail to be a true martingale since [Y.sub.t] may go infinite. Refer to [19]. Therefore, [rho] < 0 is assumed here. It is observed in most financial markets that there is a minus correlation or leverage effect between stock price and volatility impact. However, some commodity markets show the opposite effect (the inverse leverage effect). So, the model under the negative correlation condition can apply to many financial markets but has limits for some commodity markets. Also, generally speaking, the correlation [rho] may rely on time but it is assumed to be a constant for simplicity. In fact, in most real situations, it is taken to be such. We do not specify the function f but it has to satisfy a growth condition to avoid the nonexistence of moments of [S.sub.t]. It is assumed in this paper that 0 < [c.sub.1] [less than or equal to] f [less than or equal to] [c.sub.2] < [infinity] for some constants [c.sub.1] and [c.sub.2].

From the I to formula, the solution Y of the second equation in (1) is an ergodic process given by the Ornstein-Uhlenbeck process:

[Y.sub.t] = m +([Y.sub.0] - m) ([e.sup.-[alpha]t]) + [beta] ([[integral].sup.t.sub.0])([e.sup.-[alpha](t-s)])[dB.sub.s] (2)

and so [Y.sub.t] ~ N(m+([Y.sub.0]-m)[e.sup.-[alpha]t], ([[beta].sup.2]/2[alpha])([1-e.sup.-2[alpha]t])), which has an invariant distribution given by N(m, [[beta].sup.2]/2[alpha]). Later, we will use notation <*> for the average with respect to this invariant distribution; that is,


for arbitrary function g.

Now, we take the process [Y.sub.t] to be a fast mean-reverting process as in Fouque et al. [20]. This means that we take into account a fast time scale volatility factor as a major driving force for the volatility movement of the underlying asset. This assumption can be justified by an empirical analysis of high-frequency S&P 500 index data. The analysis confirms that volatility is fast mean-reverting when looked at over the time scale of a derivative contract although it reverts slowly to its mean in comparison to the tick-by-tick fluctuations of the index value. See [20] for details. Also, this assumption will provide us with analytic tractability for an approximation for the price of the Asian option as also shown in [15]. So, in terms of a parameter, say [epsilon], which is taken as positive and small, we assume that

[alpha] = [1/[epsilon]], [beta] = [[v[square root of 2]]/[square root of [epsilon]]], (4)

where v ~ O(1) which implies that the long run magnitude of volatility fluctuations remains fixed as a constant. Then, under a risk-neutral probability measure [P.sup.*], we have


where [W.sup.*.sub.t] and [B.sup.*.sub.t] are two Brownian motions under the measure [P.sup.*] whose correlation is given by d<[W.sup.*.sub.t], [B.sup.*.sub.t]> = [rho]dt and [LAMBDA](y) denotes the combined market price of risk given by

[LAMBDA](y) = [rho]([[[mu] - r]/f(y)]) + [lambda](y)[square root of 1 - [[rho].sup.2]]. (6)

Here, [lambda] is the market price of volatility risk, which is assumed to be a bounded function depending on y.

3. A PDE for Option Price

In this section, we utilize the generalization of Vecer's dimension reduction technique given by Fouque and Han [15] to derive a two-space dimensional PDE representation for the option price.

In this paper, a payoff function for arithmetic average Asian options is given by

h([1/T]([[integral].sup.T.sub.0]) [S.sub.t]dt - ([K.sub.1])[S.sub.T] - [K.sub.2]), (7)

where h is a function satisfying

h([alpha]x) = [alpha]h(x), (8)

for any real number [alpha]. Note that, when [K.sub.1] = 0, it becomes a payoff for fixed strike Asian options, whereas, when [K.sub.2] = 0, it becomes a payoff for floating strike Asian options. So, the option price P(t, x, y; T, [K.sub.1], [K.sub.2]) is defined by


under a risk-neutral measure [P.sup.*].

First, we would like to replicate the averaged process (1/t) [[integral].sup.t.sub.0] [S.sub.u]du with a portfolio

[X.sub.t] = [[alpha].sub.t]([S.sub.t]) + [[beta].sub.t]([e.sup.rt]). (10)

Here, [[alpha].sub.t] is assumed to be a nonrandom function. By the self-financing strategy, we note that


The assumption of the nonrandomness of [[alpha].sub.t] yields

d([e.sup.r(T-t)]([[alpha].sub.t])[S.sub.t]) = [e.sup.rT-t]([[alpha].sub.t])([dS.sub.t] - [rS.sub.t]dt) + [e.sup.t(T-t)]([S.sub.t])[d[alpha].sub.t]. (12)

Thus, using (11) and (12), one can obtain


By integration of (13), we obtain

[X.sub.T] - [e.sup.rT]([X.sub.0]) = [[integral].sup.T.sub.0] d([[alpha].sub.t][e.sup.r(T-t)][S.sub.t]) - ([[integral].sup.T.sub.0])[e.sup.r(T-t)][S.sub.t][d[alpha].sub.t], (14)

which can be written as

[X.sub.T] = - ([[integral].sup.T.sub.0])[e.sup.r(T-t)]([S.sub.t])[d[alpha].sub.t] + ([e.sup.rT][X.sub.0]) + ([[alpha].sub.T][S.sub.T]) - [[alpha].sub.0][e.sup.rT][S.sub.0]. (15)

For technical reason, if we choose the trading strategy [[alpha].sub.t] as

[[alpha].sub.t] = [[1 - [e.sup.-r(T-t)]]/rT] (16)

and the initial portfolio price [X.sub.0] as

[X.sub.0] = x = [[alpha].sub.0]([S.sub.0]) - [e.sup.-rT]([K.sub.2]), (17)

then the final portfolio price [X.sub.T] becomes (1/T) [[integral].sup.T.sub.0][S.sub.t]dt - [K.sub.2] and so the general payoff function of arithmetic average Asian options given by (8) becomes h([X.sub.T] - [K.sub.1][S.sub.T]). Refer to [9, 21] or [15] for details. In terms of the portfolio process [X.sub.t], the price of an arithmetic average Asian option at time t = 0 is given by

P(0, s, y; T, [K.sub.1], [K.sub.2]) = [E.sup.*][[e.sup.-rT]h([X.sub.T] - ([K.sub.1])[S.sub.T]| [S.sub.0] = s, [Y.sub.0] = y]. (18)

Next, we change the probability measure [P.sup.*] into the probability measure [[??].sup.*] defined by


Then we are ready to obtain the option price as the solution of a partial differential equation.

Theorem 1. Let [[psi].sub.t] := [X.sub.t]/[S.sub.t]. Then the Asian option price at time t = 0 can be expressed by

P(0, s, y; T, [K.sub.1], [K.sub.2]) = s[[??].sup.*][h([[psi].sub.T] - [K.sub.1]) | [[psi].sub.0] = [psi], [Y.sub.0] = y] (20)

under the measure [[??].sup.*], where

[psi] = [x/s] = [[1 - ([e.sup.-rT]]/rT]) - ([[[K.sub.2]]/s]) [e.sup.-rT]. (21)

If u(t, [psi], y; T, [K.sub.1], [K.sub.2]) := [[??].sup.*][h([[psi].sub.T] - [K.sub.1]) | [[psi].sub.t] = [psi], [Y.sub.t] = y], then u is given by the solution of the PDE:


with the terminal condition

u(T, [psi], y; T, [K.sub.1], [K.sub.2]) = h([psi] - [K.sub.1]). (23)

Proof. By the Ito formula, we obtain the following two results:


Then we have



[[??].sup.*.sub.t] := [W.sup.*.sub.t] - ([[integral].sup.t.sub.0])f([Y.sub.u])[S.sup.[gamma].sub.u]du, (26)

which is a Brownian motion under the measure [[??].sub.*] by the Girsanov theorem (cf. [22]).

From (8) and (18), we have



[psi] = [x/s] = ([[1 - [e.sup.-rT]]/rT]) - [[K.sub.2]/s][e.sup.-rT]. (28)

If u(t, [psi], y; T, [K.sub.1], [K.sub.2]) := [[??].sup.*][h([[psi].sub.T] - [K.sub.1]) | [[psi].sub.t] = [psi], [Y.sub.t] = y], the Asian option price at time t = 0 is given by

P{0, s, y; T, [K.sub.1], [K.sub.2]) = su(0, [psi], y; T, [K.sub.1], [K.sub.2]). (29)

Then, by the Feynman-Kac formula (cf. [22]) and (25), one can obtain (22).

Once the solution P(0, s, y; T, [K.sub.1], [K.sub.2]) is determined, from the result of Fouque and Han [15], the price P(t, s, y; T, [K.sub.1], [K.sub.2]) is given by

P(t, s, y; T, [K.sub.1], [K.sub.2]) = ([[T - t]/T]) P(0, s, y; T, [K.sub.1], [K.sub.2]) = ([[T - t]/T]) su (0, [psi], y; T, [K.sub.1], [K.sub.2]), (30)

where u is the solution of the PDE (22) at time t = 0.

4. Multiscale Analysis

In this section, we are interested in the solution of the multiscale PDE (22) in the asymptotic form u = ([u.sub.0]) + ([square root of [epsilon]][u.sub.1]) + ... +[[epsilon].sup.1/2]([u.sub.i]) + .... Then one can come up with single scale PDEs much easier to solve than the PDE (22) itself as follows.

Substituting the asymptotic form of u into the PDE (22) yields


with the terminal condition

[[infinity].summation over (i=0)]([[epsilon].sup.1/2])[u.sub.1] = h([psi] - [K.sub.1]), (32)



Note that [L.sub.0] is the infinitesimal generator of the Ornstein-Uhlenbeck process [Y.sub.t]. Then we have a hierarchy of PDEs as follows:


with the following terminal condition:

[u.sub.0](T, [psi], y; T, [K.sub.1], [K.sub.2]) = h([psi] - [K.sub.1]), [u.sub.i](T, [psi], y; T, [K.sub.1], [K.sub.2]) = 0, [for all]i [greater than or equal to] 1. (35)

In the following two theorems, we obtain PDEs for the leading order term [u.sub.0] as well as the first correction term [u.sub.1].

Theorem 2. Assume that the term [u.sub.i] does not grow as much as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with respect to the variable y for i = 0, 1. Then the leading order term [u.sub.0] is independent of y and satisfies the PDE problem:




Proof. From the O(1/[epsilon]) term in (34) and the fact that [L.sub.0] is the generator of [Y.sub.t], the growth condition of [u.sub.0] with respect to the variable y leads to the fact that [u.sub.0] does not depend on y. So, it is represented as

[u.sub.0] = [u.sub.0](t, [psi]; T, [K.sub.1], [K.sub.2]) (38)

without dependence on the variable y. From this fact, equation [L.sub.0][u.sub.1] + [L.sub.1][u.sub.0] = 0 becomes [L.sub.0][u.sub.1] = 0 and so, by the same reason as for [u.sub.0], [u.sub.1] also does not depend on the variable y:

[u.sub.1] = [u.sub.1] (t, [psi]; T, [K.sub.1], [K.sub.2]). (39)

Since [L.sub.1] is an operator in which every term contains the derivative with respect to the variable y, ([L.sub.0][u.sub.2]) + ([L.sub.1][u.sub.1]) + [L.sub.2][u.sub.0] = 0 becomes [L.sub.0][u.sub.2] + [L.sub.2][u.sub.0] = 0. By applying the Fredholm alternative (cf. Ramm [23]) to this Poisson equation with respect to the operator [L.sub.0], we have <[L.sub.2][u.sub.0]> = 0; that is,


By adding the boundary condition (35) for [u.sub.0] to this PDE, we obtain the PDE problem (36) as desired.

Next, we obtain a PDE for the first correction term [u.sub.1].

Theorem 3. Assume that the term [u.sub.i] does not grow as much as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with respect to the variable y for i = 0, 1. Then the first order correction term [u.sub.1] is independent of y and satisfies the PDE problem:


where [V.sub.2] and [V.sub.3] are constants, respectively, given by

[V.sub.2] = ([v/[square root of 2]])(<[LAMBDA][phi]'> - [rho]<f[phi]'>), [V.sub.3] = -([[[rho]v]/[square root of 2]])<f[phi]>. (42)

Here, [phi] is the solution of

[L.sub.0][phi](y) = (f[(y).sup.2]) - [[bar.[sigma]].sup.2]. (43)

Proof. Since <[L.sub.2][u.sub.0]> = 0 and [L.sub.0][u.sub.2] + [L.sub.2][u.sub.0] = 0, we obtain


Then, from the definition of [phi], we obtain

[u.sub.2](t, [psi], y; T, [K.sub.1], [K.sub.2]) = -[1/2][([psi] - [[alpha].sub.t]).sup.2]([phi](y) + c(t, [psi]))[s.sup.2[epsilon]][[[[partial derivative].sup.2] [u.sub.0]]/[partial derivative][[psi].sup.2]] (45)

for some function c(t, [psi]) independent of y.

By applying the Fredholm alternative to the Poisson equation [L.sub.0][u.sub.3] + [L.sub.1][u.sub.2] + [L.sub.2][u.sub.1] = 0, we have equation (L1 u2 + L2uf) = 0; that is,

[L.sub.2]([bar.[sigma]])[u.sub.1] = -<[L.sub.1](f(*))[u.sub.2]>. (46)

Combining (45) and (46), we obtain (41).

5. Approximate Option Price

From the results in Section 4, we formally obtain the first order approximation:

u(t, [psi], y; T, [K.sub.1], [K.sub.2]) [approximately equal to] [u.sub.0] (t, [psi]; T, [K.sub.1], [K.sub.2]) + [square root of [epsilon]][u.sub.1](t, [psi]; T, [K.sub.1], [K.sub.2]), (47)

where [u.sub.0] and [u.sub.1] are independent of the unobservable variable y and they are given by the PDEs in Theorems 2 and 3, respectively. Note that [u.sub.0] solves a homogeneous equation with the nonzero final condition while [u.sub.1] solves a nonhomogeneous equation with the zero final condition. In terms of accuracy of the approximation (47), if the payoff function h is smooth enough, it follows straightforwardly from [20] that the approximation is of order e in the pointwise convergent sense. Otherwise, it requires a regularization of the payoff as done for European vanilla options in [24]. Since the relevant argument (with an exact form of the density function for [[alpha].sub.T]) is omitted in this paper, we limit ourselves to the case of regularized payoffs here.

If we define [P.sub.0] and [P.sub.1] by

[P.sub.0](t, s; T, [K.sub.1], [K.sub.2]) := ([[T - t]/T])[su.sub.0](0, [psi]; T, [K.sub.1], [K.sub.2]), [P.sub.1](t, s; T, [K.sub.1], [K.sub.2]) := ([[T - t]/T])[su.sub.1](0, [psi]; T, [K.sub.1], [K.sub.2]), (48)

respectively, then, from (30), the option price P(t, s, y; T, [K.sub.1], [K.sub.2]) has an approximation given by

P(t, s, y; T, [K.sub.1], [K.sub.2]) [approximately equal to]([P.sub.0])(t, s; T, [K.sub.1], [K.sub.2]) + [square root of [epsilon]][P.sub.1](t, s; T, [K.sub.1], [K.sub.2]). (49)

In this section, we compute numerically the leading order price [P.sub.0] and the first correction term [[??].sub.1] := [square root of [epsilon]][P.sub.1] by using the finite difference method (the Crank-Nicolson method). The solution has the truncation error O([([DELTA]t).sup.2]) + O([([DELTA][psi]).sup.2]), where [DELTA]t = 0.005 and [DELTA][psi] = 0.0104.

Figure 1 shows [P.sub.0] and [[??].sub.1] at time t = 0 for three different values of [gamma] (the elasticity parameter). The parameter values used in this figure are r = 0.06, [bar.[sigma]] = 0.5, T = 1, [K.sub.1] = 0, [K.sub.2] = 2, [[??].sub.2] := [square root of [epsilon]][V.sub.2] = -0.01, and [[??].sub.3] := [square root of [epsilon]][V.sub.3] = 0.004. Note that the parameter [epsilon] is absorbed by [[??].sub.2] and [[??].sub.3]. We call the price [P.sub.0] + [[??].sub.1] the (approximate) SVCEV price for the arithmetic average Asian option. It can reduce to the price for the well-known models. In particular, the price [P.sub.0] with [gamma] = 0 is the Black-Scholes price. The price [P.sub.0] + [[??].sub.1] with [gamma] = 0 corresponds to the stochastic volatility model studied by Fouque and Han [15]. Let us call it the SV price. When the leverage effect ([gamma] < 0) takes place, the SVCEV price becomes lower than the SV price. When the inverse leverage effect ([gamma] > 0) takes place, the SVCEV price becomes higher than the SV price. Regardless of the value of [gamma], the first order stochastic volatility correction term [[??].sub.1] is positive and has the maximum value near s = [K.sub.2].

Figure 2 shows the correction term [[??].sub.1] with respect to the group parameters [[??].sub.2] and [[??].sub.3]. This figure shows that the correction term [[??].sub.1] decreases as the parameter [[??].sub.2] increases whereas it increases as the [[??].sub.3] increases. Figure 2 also shows that as the parameter [gamma] becomes larger, the correction term [[??].sub.1] becomes larger.

6. Conclusion

A frequent criticism of the stochastic volatility or local volatility models for path dependent options is that they do not produce deltas precise enough for hedging purposes. So, relevant industry experts recommend using a hybrid stochastic local volatility model of their own development for best pricing option products. See, for instance, [25]. In this paper, by transforming the path dependent problem for Asian options in the hybrid SVCEV model into a European vanilla style problem, we approximate the volatility dependent price by the nonvolatility dependent price. The approximate price provides not only a correction to the price of [15] for different values of the elasticity of variance parameter but also a correction to the CEV price based on the assumption of fast mean-reverting stochastic volatility. The elasticity of variance plays an important role in characterizing volatile markets as well as differentiating commodity markets from financial markets. Refer to [26]. The numerically solved Asian option prices show that both the leading order option price (the CEV price) and the corrected price (the approximate SVCEV price) increase as the elasticity of variance goes up.

This result can provide more efficient risk hedging by choosing an appropriate elasticity parameter based on the observed market volatility or the chosen commodity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. The research was supported by the National Research Foundation of Korea NRF 2013R1A1A2A10006693.


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Min-Ku Lee, (1) Jeong-Hoon Kim, (2) and Kyu-Hwan Jang (2)

(1) Department of Mathematics, Sungkyunkwan University, Suwon, Gyeonggi-do 440-746, Republic of Korea

(2) Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea

Correspondence should be addressed to Jeong-Hoon Kim;

Received 31 July 2013; Revised 24 November 2013; Accepted 26 November 2013; Published 8 January 2014

Academic Editor: K. S. Govinder
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Title Annotation:Research Article
Author:Lee, Min-Ku; Kim, Jeong-Hoon; Jang, Kyu-Hwan
Publication:Journal of Applied Mathematics
Article Type:Report
Date:Jan 1, 2014
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