Analytic Approximation Solution for Asian Options with Stochastic Volatility

Analytic Approximation Solution for Asian Options with Stochastic Volatility

Chung-Gee Lin

Associate Professor of Finance, Department of Business Mathematics, Soochow University, Taiwan

Address: 56, Kuei-Yang Street, Section 1, Taipei 100, Taiwan

Tel: +886-2-2311-1531 ext. 3629

Fax: +886-2-2381-2510

E-mail: [email protected]

Analytic Approximation Solution for Asian Options with Stochastic Volatility

Abstract

The valuation of Asian options is complicated because the arithmetic average of lognormal random variables is no longer lognormal. Furthermore, the stochastic volatility inherent to financial asset prices easy to observe. However, few academic works consider the pricing and hedging of Asian options with stochastic volatility, despite the popularity of such options in practical applications. Therefore, this article extends work by Hull and White (1987) and integrates the Taylor series expansion technique to derive an analytic solution for Asian options with stochastic volatility.

Numerical experiments show that the proposed analytic solution performs well and is computationally efficient compared with large sample simulations. The analytic solution provides a practical tool for pricing and hedging Asian options with stochastic volatility that is both easy to implement and desirable in terms of computing speed.

Keywords: Asian option, analytic solution, lognormal, stochastic volatility, Taylor series expansion.

JEL Classification: G13

1. Introduction

Asian options (average rate options) are path-dependent options whose payoff depends on the average value of the underlying assets during a specific set of dates across the life of the option. Because the payoff of Asian options depends on the average value of the underlying asset, volatility in the average value tends to be smoother and lower than that of the plain vanilla options. ¹ Therefore, Asian options, which tend to be less expensive than comparable plain vanilla puts or calls, appear very attractive. The two basic forms of averages in Asian options—arithmetic and geometric—both can be structured as puts or calls. A geometric average Asian option is easy to price and hedge because a closed- form solution is available, but arithmetic averages are the most commonly used, though an exact analytic solution for arithmetic average rate Asian options does not exist. This missing solution is primarily because the arithmetic average of a set of lognormal random variables is not lognormally distributed. ²

Developing a practical pricing formula for Asian options therefore is complicated for at least three reasons. First, these options are path-dependent, such that the value of an Asian option depends on not only the value of the underlying asset at that time but also the history of the underlying asset. Second, the arithmetic average is not lognormally distributed when the underlying asset follows a standard lognormal process, which makes it difficult to derive the probability distribution analytically. Third and most important,

1

Asian options are also popular in OTC markets, especially thinly traded markets such as crude oil (see Kemna and Vorst 1990; Turnbull and Wakeman 1991), in which the underlying asset price often is manipulated. It is more difficult to manipulate the average value of an underlying asset price over an extended period of time than at the expiration of an option.

2

The geometric average of lognormal random variables remains lognormal, so a closed-form

solution for a geometric Asian option can be derived straightforwardly by extending the Black-

Scholes (1973) formula. See Kemna and Vorst (1990) for related literature.

stochastic volatility, inherent to financial asset prices, gets extensively observed, ³ so incorporating a dynamic stochastic volatility process into the valuation of Asian options is critical for their practical use. The pricing of Asian options with constant volatility may cause some model risks, but few academic works attend to the pricing of Asian options with stochastic volatility, despite the popularity of these Asian options. Instead, numerical techniques often serve to value these options, even though pricing Asian options with numerical methods is very time consuming and tricky for practitioners.

Analytic approximations for Asian options have been proposed by Turnbull and Wakeman (1991), Levy (1992), Curran (1992), and Ju (2002), to name a few. Because the underlying asset is assumed to be lognormally distributed, its geometric average is also lognormal. Kemna and Vorst (1990) apply this principle to derive an analytic solution for a geometric average rate option, using that solution as the control variate for a Monte Carlo solution of the price of an arithmetic average rate option. Similarly, Hull and White (1993) discuss the use of binomial methods in pricing Asian options. Finally, Tsao et al. (2003) have developed an analytic approximation formula for pricing forward- starting Asian options.

This paper extends work by Hull and White (1987) and integrates the Taylor series expansion technique to derive analytic solutions of Asian options with stochastic volatility. Numerical experiments show that our analytic solution performs very well and is computationally efficient compared with a benchmark of large sample simulation trials.

We examine stochastic volatility Asian call options with average strike price, whose counterpart put option value can be obtained straightforwardly without significant effort

3

Blattberg and Gonedes (1974), Castanias (1979), and Christie (1982) show that variances

change over time, and Johnson and Shanno (1987), Scott (1987), Wiggins (1987), and Stein and

Stein (1991), among others, have developed stochastic volatility option pricing schemes to

from the put-call parity principle.

The contributions of this research to both existing literature and practical trading of Asian options are twofold. First, we provide an analytic solution for Asian options that considers the stochastic volatility inherent in the underlying assets. To the best of our knowledge, no previous literature addresses this issue; instead, ours is the first study to derive a compact analytic solution for pricing and hedging Asian options with stochastic volatility. Second, the proposed analytic solution is far more efficient and accurate than alternative large sample simulation methods. Therefore, the analytic solution offers a practical tool for pricing and hedging stochastic volatility Asian options and is more desirable in terms of computing speed, which represents an important consideration.

The remainder of this article is organized as follows: Section 2 provides the analytic solution for pricing and hedging Asian options with stochastic volatility; we offer the proof of this analytic formula in the Appendix. Section 3 presents comparisons between the values of Asian options with and without stochastic volatility, as well as numerical analyses for the pricing and hedging of stochastic volatility Asian options in various scenarios. We finally draw some conclusions and discuss implications of our findings in Section 4.

2. The model

In the study, we first demonstrate the analytic solution of a European-style stochastic volatility Asian call option with an average strike price. Subsequently, we address the hedge ratio (delta) of this Asian option. The detailed derivation of the analytic solution appears in the Appendix.

2.1 Analytic solution for pricing Asian options with stochastic volatility

We assume a European-style Asian option, which is written on an asset with a maturity date T, and apply a Taylor series expansion up to the second order for the exponential function to derive analytic approximation formulae for the stochastic volatility Asian options. We assume Black and Scholes (1973) economics throughout this work.

The price of the underlying asset is assumed to obey a geometric Brownian motion.

Moreover, following Hull and White (1987), assuming volatility is tradable or uncorrelated with aggregate consumption, we can apply the Cox and Ross (1976) risk- neutral valuation for pricing options. The dynamic processes for stock at time t, ^S _t , and

its instantaneous variance, V =σ _{t t} ² , in a risk-neutral world are as follows:

t t t t

dS S = (r-d)dt+σdZ , and (1)

2 2

t t t

dσσ=μdt+ξdW , (2)

where ^S _t is the underlying asset price at time t, r is the annualized risk-free interest rate, d is the continuous dividend yield, σ t denotes the instantaneous volatility of the underlying asset at time t, ^μ is the drift term, and ^ξ represents the volatility term for the variance of the underlying asset. The latter terms are both constant over time. In addition, dZ t and ^dW _t are independent Wiener processes. We assume the option maintains an arithmetic average strike price K over the period [0, T]. It then follows that the strike price is

T 0 t

K= 1 S dt

T  . (3)

Let _V be the mean variance over the time interval [0, T]:

T 2 0 u

V = σdu 1

T  . (4)

The first three moments of _V at ^μ=0 can be derived as ⁴

E(V)=V

0

, (5)

ξT 2 2

2 2

4 2 0

2(e -ξT-1)

E(V )= V

ξT , and (6)

2 2

3ξTξT 2

3 3

6 3 0

e -9e +6ξT+8

E(V )= V

3ξT . (7)

It thus is straightforward to derive the variance and skewness of _V ,

4 k

0 4 2 0

2σ(e -k-1)

Var(V) = -σ

k , and (8)

3k k 2 3

6

0 3

e -(9+18k)e +(8+24k+18k +6k )

Skew(V)=σ[ ]

3k , (9)

where k=ξT

²

.

Proposition. The analytic solution for pricing Asian options with stochastic volatility,

2

0 0

f(S ,σ) , can be approximated by the following formula:

4

As Hull and White (1987) note, for any nonzero μ , options of different maturities exhibit

markedly different implied volatilities. Because this difference has not been observed empirically,

Hull and White (1987) conclude that μ is at least close to 0. Inspired by their work, and to

simplify the derivation of the analytic solution for options, we only discuss the case in which

μ=0 . However, the case in which μ0  and the moments of V with nonzero μ _{can be}

derived without much effort; see Hull and White (1987).

2 - - 2 2

0 0 0

2 2

3

2 2 2 2

0 2 1 1

2 2

1 1 4

2 2 2 2 0

1 2

( , ) [ ( ) 2 ]

1 1 1

{ [ ( )

2 2 32

2 ( - -1)

1 1

]} (

8 8

 

 

    



 

     

   

  

 

  



 

 

rT m v

m m

v v

rT

m m

k

v v

f S S e m N m e

S e M N m e M A e v V

v

e v V B e v V e k

k

   

 



 

 

4 2 0

2 2

2 2

0 3 1 2

2

3 2 3

2 6

3 4 0 3

- )

1 1 1

{ [ ( )

6 2 32

1 - (9 18 ) (8 24 18 6 )

] }( [ ])

8 3

 

 

 



 



   



    

 

m m

v v

rT

m k k

v

S e M N m e e

v

e k e k k k

e k



 

 

  



(10)

where

N(.) = cumulative density function of standard normal distribution,

2 2 2 2 2

1 1 1 1

m T V - (r-d)T V+ (r-d) T + (r-d)T

12 3 2 2

  ,

3 3 3 2 2 3 2 2

7 7 1 7 3 1

= T V -[ T (r-d)+ T ]V +[ T (r-d) + T (r-d)+ T]V

60 15 8 15 4 3

  ,

1 3

2 2

1 1

m 1

A M v m v V

V v 2

 

      

         ,

2

1 2 2

1 1

m 1

B m v M m v V

V 2v 2

 



 

    

           ,

 

2 2

1

m 1 1

M T V r d T

V 6 3

 

   

 ,

2

2

2 2

m 1

M T

V 6

 

 

 ,

3

3 3

M m 0

V

 

 

 ,

    ²  

2 3 2 3 3 2

1

v 1 3 7 1 14 7

V [ T T r d T r d ] [ T T r d ]V T V

V 3 4 15 4 15 20

 

          

 ,

 

2

2 3 3

2 2

v 1 14 7

V [ T T r d ] T V

V 4 15 10

 

     

 ,

3

3

3 3

v 7

V T

V 10

 

 

 ,

1 3 3 5

2 2 2 2 2

1 2 1 1 2 1 1 2 1

1 3

α2M A M AB M v M v M V m v V m v V

2 4

   

 

     

       

  ,

5 3 3

3 2

2 2 2

2 1 1 1 2

αv V 3 v V B 2v V V

2

  

  

    ,

3 1 1

2 2

2 2 2

3 1 1 2

1

1 2 2 1 2 3 2 2

2 1 1 1 1 2 1 2

αv V B v V B 1 v V B

2

v V v M 2m v M V m v M m v V 1 m v V

2

  

     

  

   

 

         

         

, and

3 1 1

2 2 2

4 1 2 2 3

αv V V 1 v V B v V

2

  

  

    .

Proof. See the Appendix.

2.2 Analytic solution for hedging Asian options with stochastic volatility

Because the analytical solution for pricing Asian option is derived, the hedge ratio (

2

0 0

0

f(S ,σ)

Δ S

 

 ) can be obtained directly by differentiating the option price f(S ,σ) ₀ ² ₀

with respect to the asset price S ₀ at the present time:

2

0 0

0

-rT -m 2 2v

2 2

m m 3

2v 2v

rT 2 2

2 1 1

2 2

m 1 m 1 4 k

2v 2 2v 2 0 4

1 2 2 0

f (S ,σ)

Δ

S

e [m N(mν) eν2π]

m

1 1 1

{e [M N( ) e M A e v V

2 v 2π32π

2σ(e - k -1)

1 1

e v V B e v V ]} ( -σ)

8π8π k 1 {

6

 

 

    



 

     

 



   

  

 

  



 

 



 

2 2

m m

2v 2v

rT

3 1 2

m 2 3k k 2 3

2v 6

3 4 0 3

m 1 1

e [M N( ) eαeα

v 2π32π

1 e - (9 18k)e (8 24k 18k 6k )

eαα] }(σ[ ])

8π 3k

 

 

 



 



  



    

 

(11)

According to equation (11), the delta of the Asian option with stochastic volatility is invariant to the underlying asset price, which is constant over time. Thus, we can derive the second derivative of the stochastic volatility Asian option price with respect to the

asset price (

2 2

0 0

2 0

f(S ,σ)

Γ 0

S

  

 ), which is negligible over time.

The constant delta and zero gamma for an Asian option with stochastic volatility is good for the option issuer, who therefore does not need to rebalance the underlying asset position continuously to maintain a risk-free portfolio. In addition, the issuer of an Asian option with stochastic volatility does not suffer the risks induced from the first and second movements of the underlying asset price to the option value.

3. Numerical analyses

In this section, we highlight the need for pricing Asian options with stochastic volatility by comparing the values of Asian options with and without stochastic volatility effects.

We also demonstrate the efficiency and accuracy of our analytic solution for stochastic

volatility Asian options with a benchmark against a large sample Monte Carlo simulation.

Finally, we compare the hedge ratios of Asian options with stochastic versus constant volatilities.

3.1 Stochastic volatility effects on Asian options

Because the stochastic volatility features of financial assets are witnessed extensively, the pricing of Asian options with constant volatility is apt to cause model risks. To highlight the differences between the values of Asian options with and without stochastic volatility, we use a large sample Monte Carlo simulation for pricing Asian options in different scenarios. ⁵

As we show in Table 1, ⁶ columns 1 and 2, with no exceptions, the option values become more expensive as the drift of the asset variance ^μ increases. Both Asian option values (i.e., with stochastic or constant volatility) increase as the maturity lengthens. According to column 3, the differences between the Asian option values with and without stochastic volatility are highly significant; furthermore, as ^μ and T increase, the differences in the options magnify even more.

We also note that as the volatility of asset variance ^ξ increases, the values of Asian options with stochastic volatility are not necessary greater than those of options with constant volatility. This finding is consistent with Hull and White’s (1987) results. In short, from this table, we conclude that ξ leads to significant impacts on the value of Asian options. Thus, the results displayed in this table verify the need to integrate a stochastic volatility dynamic process into Asian options pricing.

5

To be consistent with reality, we assume 252 trading days in a year. We divide the simulations for each pricing path according to this rule; for example, we acknowledge 63 trading days (stages) in 0.25 years.

6

We use the C/C++ computer language for coding. The computations for the pricing and hedging

options were implemented on a desktop computer with 3.2 GHz CPU.

【Insert Table 1 about here】

3.2 Performance of the analytic solution

By benchmarking against Asian option values derived from a Monte Carlo simulation with 100,000 paths, repeated 50 times in various scenarios, we investigate the performance of the proposed analytic solution for stochastic volatility Asian options, as summarized in equation (10).

【Insert Table 2 about here】

In Table 2, we reveal that the values of the analytic solution are very close to the benchmarked large sample simulation prices for different initial stock prices and maturities. According to the results in column 3, the relative errors are all less than a penny, which demonstrates that our analytic solution is very accurate and trustworthy.

From the figures in column 6, we note that the proposed analytic solution is 300,000 times faster, in terms of the CPU time, than the use of simulations to price Asian options in different scenarios. These results confirm the efficiency of our analytic solution compared with numerical methods.

Because we divide the simulation path by the number of trading days, simulations for pricing Asian options require multiple computing times, as the maturity grows multiply.

However, in applying our analytic solution, we require only negligible CPU time, regardless the length of the maturity. Consequently, as the maturity of options increases, the dominance of the proposed analytic solution becomes even more striking.

The results of this table thus confirm that our analytic solution for Asian options is not

only efficient but also accurate, which means that it can satisfy the rapid option

computing needs of trading practitioners.

3.3 Comparisons of delta between Asian options with stochastic and constant volatilities

To form a risk-neutral portfolio, an option seller must buy in to the underlying asset according to the hedge ratio, delta. With an improper model, the option seller may over- or under-hedge in response to the wrong delta.

To compare the hedge ratios for Asian options with and without stochastic volatility, we use the hedge ratio formulae from equations (11) and (A11). From columns 1 and 2 in Table 3, we note that both the hedge ratios of Asian options with and without stochastic volatilities increase as the initial asset volatility and maturity increase. According to column 3 of Table 3, as the initial volatility and maturity increase, the difference in the delta between the Asian options with stochastic versus constant volatilities also becomes quite remarkable. In the scenario represented by Table 3, the deltas of the Asian option with stochastic volatility are smaller than those of the Asian option with constant volatility. These results indicate that the issuer of an Asian option will over-hedge, because it uses the hedge ratio derived from the constant volatility option pricing model.

【Insert Table 3 about here】

4. Conclusions

This study derives an analytic solution for pricing and hedging Asian options with stochastic volatility and demonstrates its accuracy and efficiency in comparison with the benchmark of large sample simulation trials.

Numerical experiments also show that a stochastic volatility Asian option model is

necessary, because the constant volatility Asian option pricing model results in

significant model errors. We also reveal that our analytic solution provides satisfactory

pricing performance for Asian options with stochastic volatility. The relative pricing

errors in the different scenarios all are less than a penny—much less than those of the benchmark large sample simulations. In addition, the CPU time required by the analytic solution is significantly less than that demanded by simulations. Finally, with longer trading maturity, the computing performance superiority of our analytic solution compared with a simulation becomes more pronounced. These results confirm the need to establish analytic solutions for stochastic volatility Asian options.

Thus, this research contributes to both literature and practice. First, we provide an analytic solution for Asian options that considers the stochastic volatility features inherent in their underlying assets. Second, our analytic solution is very efficient and accurate. Third, the analytic solution provides a good control variate that can enhance the computational accuracy and efficiency of using numerical methods for pricing other exotic Asian options.

Overall, our analytic solution provides an imminently practical tool for valuing Asian

options with stochastic volatility that is not only easy to implement but also more

desirable because of its significant benefits with regard to computing speed.

Appendix. Derivation of an analytic solution for Asian options with stochastic volatility

We first show how to derive an analytic solution for Asian options with constant volatility, then demonstrate the construction of an analytic solution for pure call options with stochastic volatility. Finally, we extend Hull and White’s (1987) research into deriving analytic solution of Asian options with stochastic volatility.

A.1 Analytic solution for Asian options with constant volatility

The price of the underlying asset is assumed to follow a geometric Brownian motion in a risk-neutral scenario, that is,

t t t

dS S = (r-d)dt+σdZ , (A1)

where S is the underlying asset price, r is the annualized risk-free interest rate, d is the continuous dividend yield, ^σ denotes the constant instantaneous volatility of the underlying asset, and Z _t is a Wiener process. By Ito’s lemma,

2

T 0 T

S =S exp[(r-d-σ2)T+σZ ] . (A2)

We assume the option maintains an arithmetic average strike price K over the period [0, T], where T is the maturity of the option. It therefore follows that the strike price is

T 0 t

K= 1 S dt

T  . (A3)

The exact price of a European-style Asian option with constant volatility ^σ at time 0 thus

is given by

-rT +

T 0

C(σ)=e E[(S -K) I ] , (A4)

where E[ I ]  0 is the conditional expectation operator with respect to the risk-neutral probability measure, and I 0 is a filtration.

Substituting equations (A2) and (A3) into equation (A4), we obtain

T 2

2 (r-d-σ2)t+σZ

(r-d-σ2)T+σZ

-rT T t +

0 0 0

0

C(σ)=e E[(S e - 1 S e dt) I ]

T  . (A5)

Let

T 2

2 (r-d-σ2)t+σZ

(r-d-σ2)T+σZ T t

0

X=e - 1 e dt

T  . (A6)

Given the parameters of r, d, σ, and T, by using a Taylor series expansion up to the second order for the exponential function, we can obtain

2 2 2 2 2 2 2

T T

T 2 2 2 2 2 2 2

t t

0

1 1

X Y=1+(r-d-σ2 )T+σ(1+(r-d-σ2 )T)Z + (r-d-σ2 ) T +σZ

2 2

1 1 1

- [1+(r-d-σ2 )t+σ(1+(r-d-σ2 )t )Z + (r-d-σ2 ) t +σZ ]dt

T 2 2





. (A7)

We assume Y is normal distributed; in turn, the mean and variance of Y, conditional on I 0 , can be expressed as

2 4 2 2 2 2

0

1 1 1 1

E[Y I ]= m Tσ- (r-d)Tσ+ (r-d) T + (r-d)T

12 3 2 2

 , (A8)

and

3 6 3 2 4 3 2 2 2

0

7 7 1 7 3 1

Var[Y I ]= = Tσ-[ T (r-d)+ T ]σ+[ T (r-d) + T (r-d) + T]σ

60 15 8 15 4 3

 . (A9)

We use a normal distribution to approximate the real distribution of Y and thereby obtain the analytic solution for the present value of a floating strike price European-style Asian option: ⁷

-rT -m 2 2

C(σ)=S e [m N( m 0   )+e

^v

 2π] , (A10)

where N(.) is the cumulative density function of standard normal distribution.

The value of delta of an Asian option with constant volatility thus can be obtained as follows:

-rT -m 2 2

0

C(σ) = e [m N( m )+e 2π]

S

 





v

 . (A11)

A.2 Analytic solution for options with stochastic volatility

To integrate the stochastic volatility features that are inherent in underlying assets into an analytic solution, we extend the work by Hull and White (1987).

Consider an option f(S ,σ,t)

t t²

with an underlying security price S

t

and its instantaneous variance σ

t²

. The present (time = t) value of this option f(S ,σ,t)

t t²

must be the expected terminal value f(S ,σ,T)

T T²

, discounted at the risk-free rate in a risk-neutral scenario,

2 -r(T-t) 2 2

t t T T T t t T

f(S ,σ,t)=e f(S ,σ,T)p(S S ,σ)dS  , (A12)

where S

t

is the underlying asset price at time t, σ

t

is the instantaneous standard

derivation of the underlying asset at time t, and p(S S ,σ)

T t ²t

is the conditional distribution of S

T

, given the underlying asset price and variance at time t.

7

See Tsao et al. (2003).

The conditional distribution of S

T

therefore depends on two stochastic variables, S

t

and

2

σ

t

. Because for any three related random variables x, y, and z, the conditional density functions are related by

p(x y )= g(x z )h(z y )dz  , (A13)

we let _V be the mean variance over the life [0, T] of the derivative security, defined as

T 2

0 u

V=σdu 1

T  . (A14)

Hence, the distribution of S T can be written as

2 2

T 0 T 0

p(Sσ)= g(S V )h(Vσ)dV  . (A15)

Substituting equation (A14) into equation (A11), we obtain

2 -rT 2

0 0 T T 0 T

f(S ,σ,t=0)=e  f(S )g(S V )h(Vσ)dS dV . (A16)

Equation (A15) also can be rewritten as

2 -rT 2

0 0 T T T 0

f(S ,σ,t=0) = [e  f(S )g(S V )dS ]h(Vσ)dV  . (A17)

The inner term in equation (A17) represents an analytic solution formula for a call option

on a underlying asset with a mean variance _V , which be denoted C(V) . Following Hull

and White (1987), because volatility is tradable or uncorrelated with aggregate

consumption, we can apply Cox and Ross’s (1976) risk-neutral valuation. That is, assume

a risk-neutral world in which a stock price and its instantaneous variance follow the

dynamic stochastic processes,

t t t t

dS S = (r-d)dt+σdZ and (A18)

2 2

t t t

dσσ=μdt+ξdW , (A19)

where ^S _t is the underlying asset price at time t, r is the annualized risk-free interest rate, d is the continuous dividend yield, σ t denotes the instantaneous volatility of the underlying asset at time t, ^μ is the drift term, and ^ξ is the volatility term of stochastic volatility (both of which are constant over time). Also, dZ _t and dW _t are independent Wiener processes.

In addition, let _V be the mean variance over the time interval [0, T], defined as in equation (A14). Then, the first three moments of _V with μ=0 can be derived as

E(V)=V

0

, (A20)

ξT 2 2

2 2

4 2 0

2(e -ξT-1)

E(V )= V

ξT , and (A21)

2 2

3ξTξT 2

3 3

6 3 0

e -9e +6ξT+8

E(V )= V

3ξT . (A22)

Furthermore, it becomes straightforward to derive the variance and skewness of _V :

4 k

0 4 2 0

2σ(e -k-1)

Var(V) = -σ

k and (A23)

3k k 2 3

6

0 3

e -(9+18k)e +(8+24k+18k +6k )

Skew(V)=σ[ ]

3k , (A24)

where k=ξT

²

.

If we accept equations (A14), (A18), and (A19), the distribution of log(S S )

T 0

,

conditional on _V , is normal with a mean rT- VT 2 and variance _VT . To approximate

2

0 0

f(S ,σ) in a Taylor series by expanding C(V) around its expected value _V , we consider

8

2 3

2

0 0 2 3

V V

1 C 1 C

f(S ,σ) C(V) + Var(V) + Skew(V)

2 V 6 V

 

   , (A25)

where Var(V) and Skew(V) are the second and third central moments of _V .

We thus obtain an analytic approximate solution for a plain vanilla European call option with stochastic volatility from equation (A25). Following the same arguments, we substitute an analytic Asian option formula with variance _V into equation (A23) and thus derive an analytic solution for an Asian option with stochastic volatility.

A.3 Analytic approximation of Asian option with stochastic volatility

To extend the model proposed by Hull and White (1987) for pricing Asian options with

stochastic volatility, we must derive the components in the equation _C(V) ,

2 2

C(V) V



 and

3 3

C(V) V



 , where C(V) is the analytic formula of Asian options, according to equation

(A10), except that the underlying stock has an expected mean variance _V .

From equation (10), we can derive C(V) straightforwardly:

-rT -m 2 2

C(V)=S e [m N( m 0      )+e ^

^v

^   2π] , (A26)

where

2 2 2 2 2

1 1 1 1

m T V - (r-d)T V+ (r-d) T + (r-d)T

12 3 2 2

  , and (A27)

3 3 3 2 2 3 2 2

7 7 1 7 3 1

= T V -[ T (r-d)+ T ]V +[ T (r-d) + T (r-d)+ T]V

60 15 8 15 4 3

  . (A28)

The analytic solutions for

2 2

C(V) V



 and

3 3

C(V) V



 in turn are derived as follows:

m 2 1

2v

rT 2

0 1 1

m

C(V) 1

S e M N v e V

V v 2 2π

 

 



    

                   

, (A29)

2 2

m m

3 2

2v 2v

rT 2 2

0 2 1 1

2

2 2

m 1 m 1

2v 2 2v 2

1 2

m

C(V) 1 1

S e [M N e M A e v V

V v 2π32π

1 1

e v V B e v V ] 8π8π

 

    



 

     

  

            

 

 

, and (A30)

 

2 2 2

m m m

3

2v 2v 2v

rT

0 3 1 2 3 4

3

m

C(V) 1 1 1

S e M N eαeαeαα

V v 2π32π8π

  

  

  



    

                     

, (A31)

where

2 2 2 2 2

1 1 1 1

m T V - (r-d)T V+ (r-d) T + (r-d)T

12 3 2 2

  ,

3 3 3 2 2 3 2 2

7 7 1 7 3 1

= T V -[ T (r-d)+ T ]V +[ T (r-d) + T (r-d)+ T]V

60 15 8 15 4 3

  ,

1 3

2 2

1 1

m 1

A M v m v V

V v 2

 

      

         ,

2 1 2 2

1 1

m 1

B m v M m v V

V 2v 2

 



 

    

           ,

 

2 2

1

m 1 1

M T V r d T

V 6 3

 

   

 ,

2

2

2 2

m 1

M T

V 6

 

 

 ,

3

3 3

M m 0

V

 

 

 ,

    ²  

2 3 2 3 3 2

1

v 1 3 7 1 14 7

V [ T T r d T r d ] [ T T r d ]V T V

V 3 4 15 4 15 20

 

          

 ,

 

2

2 3 3

2 2

v 1 14 7

V [ T T r d ] T V

V 4 15 10

 

     

 ,

3

3

3 3

v 7

V T

V 10

 

 

 ,

1 3 3 5

2 2 2 2 2

1 2 1 1 2 1 1 2 1

1 3

α2M A M AB M v M v M V m v V m v V

2 4

   

 

     

       

  ,

5 3 3

3 2

2 2 2

2 1 1 1 2

αv V 3 v V B 2v V V

2

  

  

    ,

3 1 1

2 2

2 2 2

3 1 1 2

1

1 2 2 1 2 3 2 2

2 1 1 1 1 2 1 2

αv V B v V B 1 v V B

2

v V v M 2m v M V m v M m v V 1 m v V

2

  

     

  

   

 

         

         

, and

3 1 1

2 2 2

4 1 2 2 3

αv V V 1 v V B v V

2

  

  

    .

Substituting equations (A23), (A24), (A26), (A30), and (A31) into equation (A25), we can derive the following analytic approximation formula for Asian options with stochastic volatility:

2 -rT -m22

0 0 0

2 2

3

2 2 2 2

0 2 1 1

2 2

1 1 4 k

2 2 2 2 0

1 2

f(S ,σ) S e [m N( m )+e 2π]

1 1 1

+ { [ ( )

2 2 32

2σ(e -k-1)

1 1

]} (

8 8 k

 

 

    



 

     

   

 

   



 

 

v

m m

v v

rT

m m

v v

S e M N m e M A e v V

v

e v V B e v V

 

 

 

 

4 2 0

2 2

2 2

0 3 1 2

2

3k k 2 3

2 6

3 4 0 3

-σ)

1 1 1

+ { [ ( )

6 2 32

1 e -(9+18k)e +(8+24k+18k +6k )

] }(σ[ ])

8 3k

 

 

 



 



  



 

m m

v v

rT

m v

S e M N m e e

v e

 

 

  

(A32)

Q.E.D.

References

Black, F., and M.J. Scholes (1973), “The Pricing of Options and Corporate Liability,”

Journal of Political Economy, 81, 637-659.

Blattberg, R.C., and N.J. Gonedes (1974), “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices,” Journal of Business, 47, 244-280.

Castanias, R.P. (1979), “Macroinformation and the Variability of Stock Market Prices,”

Journal of Finance, 34, 439-450.

Christie, A.A. (1982), “The Stochastic Behavior of Common Stock Variance: Value, Leverage and Interest Rate Effects,” Journal of Financial Economics, 6, 213-234.

Cox, J., and M. Ross (1976), “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, 3, 145-166.

Curran, M. (1992), “Beyond Average Intelligence,” Risk, 5(10), 60.

Hull, J., and A. White (1987), “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance, 42, 281-300.

Hull, J., and A. White (1993), “Efficient Procedures for Valuing European and American Path-Dependent Options,” Journal of Derivatives, 1(1), 21-31.

Johnson, H. and D. Shanno (1987), “Option Pricing When the Variance Is Changing,”

Journal of Financial and Quantitative Analysis, 22, 143-151.

Ju, N. (2002), “Pricing Asian and Basket Options via Taylor Expansion,” Journal of Computational Finance, 5, 79-103.

Kemna, A.G. Z. and A.C.F. Vorst (1990), “A Pricing Method for Options Based on

Average Asset Values,” Journal of Banking and Finance, 14, 113-129.

Levy, E. (1992), “Pricing European Average Rate Currency Options,” Journal of International Money and Finance, 14, 474-491.

Scott, L. (1987), “Option Pricing When the Variance Changes Randomly: Theory, Estimation and an Application,” Journal of Financial and Quantitative Analysis, 26, 419- 438.

Stein, E., and J. Stein (1991), “Stock Price Distributions with Stochastic Volatility: An Analytic Approach,” Review of Financial Studies, 4, 727-752.

Tsao, C.Y., C.C. Chang, and C.G. Lin (2002), “Analytic Approximation Formulae for Pricing Forward-Starting Asian Options,” Journal of Futures Markets, 23, 487-516.

Turnbull, S. M., and L. M. Wakeman (1991), “A Quick Algorithm for Pricing European Average Options,” Journal of Financial and Quantitative Analysis, 26, 377–389.

Wiggins, J. (1987), “Options Values under Stochastic Volatility: Theory and Empirical

Estimates,” Journal of Financial Economic, 19, 351-372.

Table 1

Comparisons of Asian Call Options with Stochastic and Constant Volatilities.

T

(in years) μ ξ

Asian Option with Stochastic Volatility

(1)

Asian Option with Constant Volatility

(2)

Relative Error (3) = (2) – (1)/(1)

0.25

0.00

0.40 4.0682(0.0241)

4.0774(0.0244)

0.2261%

0.80 4.0385(0.0237) 0.9632%

1.20 3.9891(0.0231) 2.2135%

0.20

0.40 4.1423(0.0246) -1.5668%

0.80 4.1119(0.0242) -0.8390%

1.20 4.0614(0.0236) 0.3940%

0.40

0.40 4.2182(0.0250) -3.3379%

0.80 4.1871(0.0246) -2.6200%

1.20 4.1356(0.0240) -1.4073%

0.5

0.00

0.40 6.1048(0.0154)

6.1338(0.0158)

0.4750%

0.80 6.0209(0.0140) 1.8751%

1.20 5.8871(0.0128) 4.1905%

0.20

0.40 6.3307(0.0159) -3.1102%

0.80 6.2432(0.0145) -1.7523%

1.20 6.1037(0.0132) 0.4931%

0.40

0.40 6.5684(0.0165) -6.6165%

0.80 6.4771(0.0149) -5.3002%

1.20 6.3318(0.0137) -3.1271%

1

0.00

0.40 9.3027(0.0198)

9.3791(0.0139)

0.8213%

0.80 9.0922(0.0262) 3.1555%

1.20 8.7823(0.0356) 6.7955%

0.20

0.40 10.0156(0.0229) -6.3551%

0.80 9.7883(0.0303) -4.1805%

1.20 9.4582(0.0411) -0.8363%

0.40

0.40 10.8124(0.0266) -13.2561%

0.80 10.5685(0.0349) -11.2542%

1.20 10.2200(0.0477) -8.2280%

Notes: The base parameters are set as follows: S

₀

= 100, r = 0.1, d = 0, and σ

₀

= 0.3. We assume the

option has an arithmetic average strike price and one year has 252 trading days. Option prices in

columns 1 and 2 are derived by applying a Monte Carlo simulation with 100,000 paths, repeated 50

times. Standard errors appear in parenthesis. Column 3 indicates the relative error between the Asian

call options with constant volatility and the benchmark stochastic volatility Asian options.

Table 2

Comparisons of Asian Options in Analytic Solution and Monte Carlo Method

S

0

T

(in years) Simulation Values (1)

Analytic Solution Values

(2)

Relative Error (3) = [(2) – (1)]/(1)

Simulation CPU Time (in seconds)

(4)

Analytic Solution CPU

Time (in seconds)

(5)

Comparative Efficiency (6) = (4)/(5)

80

0.25 3.2308(0.0190) 3.2567 0.8017% 5.9219 1.61E-05 367249.61

0.50 4.8167(0.0112) 4.8632 0.9654% 12.6188 1.61E-05 782561.25

0.75 6.1231(0.0111) 6.1601 0.6043% 21.0750 1.61E-05 1306976.75

1.00 7.2737(0.0210) 7.2647 -0.1237% 27.1563 1.61E-05 1684111.64

90

0.25 3.6346(0.0214) 3.6638 0.8034% 5.9938 1.61E-05 371708.53

0.50 5.4188(0.0126) 5.4711 0.9652% 12.4875 1.61E-05 774418.61

0.75 6.8885(0.0125) 6.9302 0.6054% 20.4594 1.61E-05 1268800.01

1.00 8.1830(0.0236) 8.1728 -0.1246% 26.9063 1.61E-05 1668607.76

100

0.25 4.0385(0.0237) 4.0709 0.8023% 5.9094 1.61E-05 366474.42

0.50 6.0209(0.0140) 6.0790 0.9650% 12.4563 1.61E-05 772483.73

0.75 7.6539(0.0139) 7.7002 0.6049% 20.4719 1.61E-05 1269575.20

1.00 9.0922(0.0262) 9.0809 -0.1243% 26.9250 1.61E-05 1669767.45

110

0.25 4.4423(0.0261) 4.4780 0.8036% 5.9094 1.61E-05 366474.42

0.50 6.6230(0.0154) 6.6869 0.9648% 12.4563 1.61E-05 772483.73

0.75 8.4193(0.0153) 8.4702 0.6046% 20.4312 1.61E-05 1267051.17

1.00 10.0014(0.0288) 9.9890 -0.1240% 26.8750 1.61E-05 1666666.68

120

0.25 4.8462(0.0285) 4.8851 0.8027% 5.8906 1.61E-05 365308.53

0.50 7.2251(0.0168) 7.2948 0.9647% 12.4500 1.61E-05 772093.03

0.75 9.1847(0.0167) 9.2402 0.6043% 20.4344 1.61E-05 1267249.62

1.00 10.9106(0.0315) 10.8971 -0.1237% 26.8969 1.61E-05 1668024.82

Notes: The base parameters are set as follows: r = 0.1, d = 0, σ

0

= 0.3, μ _{= 0, and} ξ = 0.8. We assume the option

has an arithmetic average strike price. Option prices in column 1 are derived by applying a Monte Carlo simulation

method with 100,000 paths, repeated 50 times; standard errors are in parentheses. Option prices in column 2 are

derived by applying the analytic stochastic volatility Asian option solution described by equation (10). Column 3

indicates relative error between the option prices of both solutions. Columns 4 and 5 report the CPU times of the

simulation and analytic solution, respectively. Column 6 indicates the efficiency of the analytic solution compared

with the simulation.

Table 3

Comparisons of Delta in Asian Options with Stochastic and Constant Volatilities

σ

0

T (in years)

The Delta of Asian Option with Stochastic Volatility

(1)

The Delta of Asian Option with Constant Volatility

(2)

Relative Error (3) = [(2) – (1)]/(1)

0.1

0.25 0.0187 0.0188 0.5348%

0.50 0.0313 0.0314 0.3195%

0.75 0.0431 0.0433 0.4640%

1.00 0.0544 0.0546 0.3676%

0.2

0.25 0.0296 0.0297 0.3378%

0.50 0.0458 0.0462 0.8734%

0.75 0.0599 0.0604 0.8347%

1.00 0.0728 0.0734 0.8242%

0.3

0.25 0.0408 0.0410 0.4902%

0.50 0.0611 0.0616 0.8183%

0.75 0.0779 0.0787 1.0270%

1.00 0.0929 0.0939 1.0764%

0.4

0.25 0.0519 0.0522 0.5780%

0.50 0.0764 0.0771 0.9162%

0.75 0.0961 0.0971 1.0406%

1.00 0.1133 0.1145 1.0591%

Notes: The base parameters are set as follows: S

0

= 100, r = 0.1, d = 0, μ _{=0, and} ξ =0.8. We assume the

option has an arithmetic average strike price and one year has 252 trading days. Deltas in columns 1 and 2

are derived by applying equations (11) and (A11), respectively. Column 3 indicates the relative error of

deltas between the Asian call options with constant and stochastic volatilities.