Research ArticleGeometric Asian Options Pricing under the Double HestonStochastic Volatility Model with Stochastic Interest Rate

Yanhong Zhong and Guohe Deng

College of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Correspondence should be addressed to Guohe Deng; dengguohe@mailbox.gxnu.edu.cn

Received 16 August 2018; Revised 6 December 2018; Accepted 24 December 2018; Published 10 January 2019

Academic Editor: Hassan Zargarzadeh

Copyright © 2019 Yanhong Zhong and Guohe Deng. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper presents an extension of double Heston stochastic volatility model by incorporating stochastic interest rates and derivesexplicit solutions for the prices of the continuously monitored fixed and floating strike geometric Asian options. The discountedjoint characteristic function of the log-asset price and its log-geometric mean value is computed by using the change of numeraireand the Fourier inversion transform technique. We also provide efficient approximated approach and analyze several effects onoption prices under the proposed model. Numerical examples show that both stochastic volatility and stochastic interest rate havea significant impact on option values, particularly on the values of longer term options.The proposedmodel is suitable formodelingthe longer time real-market changes and managing the credit risks.

1. Introduction

Asian option is a special type of option contract in which thepayoff depends on the average of the underlying asset priceover somepredetermined time interval.The averaging featureallows Asian options to reduce the volatility inherent in theoption. There are some advantages to trading Asian optionsin a financial market. One is that these decrease the riskof market manipulation of the financial derivative at expiry.Another is that Asian options have lower relative chargethan European or American options. In general, the averageconsidered can be a arithmetic or geometric one and it canbe calculated either discretely, for which the average is takenover the underlying asset prices at discrete monitoring timepoints, or continuously, for which the average is calculatedvia the integration of the underlying asset price over themonitoring time period. Asian options can be differentiatedinto two main classes according to their payoff: fixed strikeprice options (sometimes called “average price”) and floatingstrike price options (sometimes called “average strike”). Allthese details are specified by the contracts stipulated bytwo counterparts, as Asian options are traded actively on

the OTC market among investors or traders for hedgingthe average price of a commodity. For a brief introductionto the development of Asian options, see Boyle and Boyle[1].

As the probability distribution of the average prices of theunderlying asset generally does not have a simple analyticalexpression, it is difficult to obtain the analytical pricingformula for Asian option. Since the best-known closed-formpricing formula for the European vanilla option derivedby Black and Scholes [2]), many researchers have devotedthemselves to developing the Asian options pricing based onthe Black-Scholes assumptions; see, e.g., Kemna and Vorst(1990), Turnbull andWakeman [3], Ritchken et al. [4], Gemanand Yor [5], Rogers and Shi [6], Boyle et al. [7], Angus [8],Linetsky [9], Cui et al. [10], and the references therein. For arecent review, one can refer to Fusai and Roncoroni [11] andSun et al. [12].

In practice, the Black-Scholes assumptions are hardly sat-isfied, especially the constant volatility and constant interestrate hypothesis. As the empirical behaviors of the impliedvolatility smile and heavy tailed in the distribution of log-returns are commonly observed in financial markets. For

HindawiComplexityVolume 2019, Article ID 4316272, 13 pageshttps://doi.org/10.1155/2019/4316272

http://orcid.org/0000-0002-9344-5193https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/4316272

2 Complexity

this reason, stochastic volatility (hereafter SV) models havebeen proposed in finance (see Hull and White [13], Steinand Stein [14], Heston [15], and others). These models havebeen applied to value the Asian options (see, e.g., Wongand Cheung [16], Hubalek and Sgarra [17], Kim and Wee[18], and Shi and Yang [19]). In addition, interest rates arestochastic and stock returns are negatively correlated withinterest rate changes, which have been examined in previousresearch.

Although these models mentioned above are able toaccount for the empirical behaviors, they are still based on asingle-factor for volatility dynamics that is inconsistent withthe long range memory characteristic of the volatility correc-tions and the stiff volatility skews. See Alizadeh et al. [20],Fiorentini et al. [21], Chernov et al. [22], Gourieroux [23],Christoffersen et al. [24], Romo [25], and Nagashima et al.[26] for the empirical results. To address this issue, mul-tifactor SV models have recently generated attention inthe option pricing literature. For instance, Duffie et al.[27] proposed multifactor affine stochastic volatility models.Based upon the Black-Scholes framework, Fouque et al. [28]introduced a multiscale SV model, in which the volatilityprocesses are driven by two mean-reverting diffusion pro-cesses. Gourieroux [23] proposed a multivariate model inwhich the volatility-covolatility matrix follows a Wishartprocess.

On the basis of the findings of Christoffersen et al. [24],a double Heston (dbH) model, which consists of two inde-pendent variance processes, has recently been reported betterthan the plain Heston [15] model in the performances ofhedging (see Sun [29]) and has also been applied to arithmeticAsian option under discrete monitoring (Mehrdoust andSaber [30]) and forward starting option (Zhang and Sun [31]).However, its extension to continuously monitored geometricAsian option is yet to be considered. On the other hand,many of Asian options often have long-dated maturities sincethey are used as part of the structured notes which has along maturity. The movement of interest rates becomes anissue in such cases and constant interest rate assumptionshould be replaced by an appropriate dynamic interest ratemodel. Several results are available on the Asian option inthe stochastic interest rate framework; see, e.g., Nielsen andSandmann [32, 33], Zhang et al. [34], Donnelly et al. [35],and He and Zhu [36]. In the above stochastic interest rateframework, the short-term interest rate is assumed to followa specific parametric one-factor model (see, e.g., Cox et al.[37], Hull and White [13], and Vasicek [38]), which tendsto oversimplify the true behavior of interest rate movement.However, empirical tests reported in Lonstaff and Schwartz[39] and Pearson and Sun [40] show that the term struc-ture for the interest rate should involve several sources ofuncertainty, and introducing additional state variables (suchas the rate of inflation, GDP, etc.) significantly improves thefit.

In this paper, we study the pricing of the continuouslymonitored geometric Asian options under dbH stochasticvolatility model with stochastic interest rate framework(hereafter, dbH-SI model). The contribution of the present

paper is twofold. Firstly, this paper extends the dbH modelby introducing stochastic interest rate, which is assumed tofollow two-factor model with two state variables. Secondly,this paper provides a semiexplicit valuation formula for thegeometric Asian options with fixed or floating strike price,which is extremely useful also for the arithmetic averageoption valuation via Monte Carlo methods with controlvariables.

The rest of the paper is organized as follows. Section 2develops the underlying pricing model and describes thegeometric Asian option. Section 3 derives the joint char-acteristic function of a log-return of the underlying assetand its geometric average. Section 4 obtains the analyticexpressions for the prices of the fixed strike geometric Asiancall option and the floating strike Asian call option undercontinuous monitoring. Section 5 provides some numericalexamples for the proposed approach. Section 6 concludes thepaper.

2. Model Formulation

We consider an arbitrage-free, frictionless financial marketwhere only riskless asset and risky asset are traded continu-ously up to a fixed horizon date 𝑇. Let (Ω,F, {F𝑡}0≤𝑡≤𝑇, 𝑄)be a complete probability space equipped with a filtration{F𝑡} satisfying the usual conditions, where𝑄 is a risk-neutralprobability measure. Suppose 𝑊𝑗𝑡 and 𝑍𝑗𝑡 (𝑗 = 1, 2) are allstandard Brownian motions defined on the probability space,and the filtration {F𝑡}𝑡≥0 is generated by these Brownianmotions. Moreover, 𝑑𝑊1𝑡 𝑑𝑍1𝑡 = 𝜌1𝑑𝑡, 𝑑𝑊2𝑡 𝑑𝑍2𝑡 = 𝜌2𝑑𝑡,and any other Brownian motions are pairwisely independent.Assume that the asset price process 𝑆𝑡, without payingany dividend, satisfies the following stochastic differentialequation under 𝑄:

𝑑𝑆𝑡𝑆𝑡 = 𝑟𝑡𝑑𝑡 + √V1𝑡𝑑𝑊1𝑡 + √V2𝑡𝑑𝑊2𝑡 ,𝑑V1𝑡 = 𝜅1 (𝜃1 − V1𝑡) 𝑑𝑡 + 𝜎1√V1𝑡𝑑𝑍1𝑡 ,𝑑V2𝑡 = 𝜅2 (𝜃2 − V2𝑡) 𝑑𝑡 + 𝜎2√V2𝑡𝑑𝑍2𝑡 ,

(1)

where 𝜅𝑗, 𝜃𝑗, 𝜎𝑗 (𝑗 = 1, 2) are all nonnegative constants,which represent the mean-reverting rates, long-term meanlevels, and volatilities of variance processes V𝑗𝑡, respectively.We suppose that 2𝜅𝑗𝜃𝑗 ≥ 𝜎2𝑗 . The instantaneous interestrate, 𝑟𝑡, is assumed to be a linear combination of V1𝑡 andV2𝑡, i.e., 𝑟𝑡 = V1𝑡 + V2𝑡, which designates the interest rateas an affine function of two-factor economic variables V1and V2 and offers the analytic tractability (see Duffie et al.[27]).

In financial market, there are four types of Europeanstyle continuously monitoring geometric Asian options: fixedstrike geometric Asian calls, fixed strike geometric Asianputs, floating strike geometric Asian calls, and floating strikegeometric Asian puts.The payoffs at the expiration date 𝑇 forthese options are as follows:

Complexity 3

ℎ (𝑆𝑇, 𝐺[0,𝑇]) ={{{{{{{{{{{{{{{{{

max {𝐺[0,𝑇] − 𝐾, 0} , a fixed strike geometric Asian calls,max {𝐾 − 𝐺[0,𝑇], 0} , a fixed strike geometric Asian puts,max {𝐺[0,𝑇] − 𝑆𝑇, 0} , a floating strike geometric Asian calls,max {𝑆𝑇 − 𝐺[0,𝑇], 0} , a floating strike geometric Asian puts,

(2)

where 𝐾 is a fixed strike price and 𝐺[0,𝑇] is the geometricaverage of the underlying asset price 𝑆𝑡 until time 𝑇; i.e.,𝐺[0,𝑇] = exp((1/𝑇) ∫𝑇0 ln 𝑆𝑢𝑑𝑢). In the following, we consideronly the pricing problem of the geometric Asian call options(hereafter, GAC), while the put options can be dealt withsimilarly.

For the instantaneous interest rate 𝑟𝑡 = V1𝑡 + V2𝑡, onecan express the price at time 𝑡 of a zero-coupon bond withmaturate 𝑇 as follows (see Cox et al. [37]):

𝑃 (𝑡, 𝑇) = exp {𝑎 (𝑡, 𝑇) − 𝑏 (𝑡, 𝑇) 𝑟𝑡} , (3)where

𝑎 (𝑡, 𝑇) = 2∑𝑗=1

[𝜅𝑗𝜃𝑗 (𝜅𝑗 − 𝛾𝑗) (𝑇 − 𝑡)𝜎2𝑗+ 2𝜅𝑗𝜃𝑗𝜎2𝑗 ln

2𝛾𝑗2𝛾𝑗 + (𝜅𝑗 − 𝛾𝑗) (1 − 𝑒−𝛾𝑗(𝑇−𝑡))] ,

𝑏 (𝑡, 𝑇) = 2∑𝑗=1

[ 2 (1 − 𝑒−𝛾𝑗(𝑇−𝑡))2𝛾𝑗 + (𝜅𝑗 − 𝛾𝑗) (1 − 𝑒−𝛾𝑗(𝑇−𝑡))] ,𝛾𝑗 = √𝜅2𝑗 + 2𝜎2𝑗 , for 𝑗 = 1, 2.

(4)

3. The Joint Characteristic Function

Given the dynamic of the underlying asset price, it is possibleto obtain the discounted joint characteristic function for thelog-asset value 𝑆𝑇 and the log-geometric mean value of theasset price over a certain time period.

Let 𝜓𝑡(𝑠, 𝑤) = 𝐸𝑄[𝑒− ∫𝑇𝑡 𝑟𝜏𝑑𝜏+𝑠 ln𝐺[𝑡,𝑇]+𝑤 ln 𝑆𝑇 | F𝑡] be thediscounted joint characteristic function of two-dimensionalrandom variable, (ln𝐺[𝑡,𝑇], ln 𝑆𝑇), conditioned on F𝑡 under𝑄, where𝐺[𝑡,𝑇] = exp{(1/𝑇) ∫𝑇𝑡 ln 𝑆𝑢𝑑𝑢} and 𝐸𝑄[⋅ | F𝑡] is theconditional expectation under 𝑄 for 𝑡 ∈ [0, 𝑇]. Denote 𝐷 ={(𝑠, 𝑤) ∈ C2 : R(𝑠) ≥ 0,R(𝑤) ≥ 0, 0 ≤ R(𝑠) +R(𝑤) ≤ 1}.Proposition 1. Suppose that 𝑆𝑡, V1𝑡, and V2𝑡 follow thedynamics in (1). If (𝑠, 𝑤) ∈ 𝐷 and 𝑡 ∈ [0, 𝑇], then𝐸𝑄[|𝑒− ∫𝑇𝑡 𝑟𝜏𝑑𝜏+𝑠 ln𝐺[𝑡,𝑇]+𝑤 ln 𝑆𝑇 |] < ∞ and

𝜓𝑡 (𝑠, 𝑤) = 𝑒𝑔0𝐸𝑄[[exp{{{2∑𝑗=1

[𝑔𝑗1 ∫𝑇𝑡(𝑇 − 𝜏)2 V𝑗𝜏𝑑𝜏 + 𝑔𝑗2 ∫𝑇

𝑡(𝑇 − 𝜏) V𝑗𝜏𝑑𝜏 + 𝑔𝑗3 ∫𝑇

𝑡V𝑗𝜏𝑑𝜏 + 𝑔𝑗4V𝑗𝑇]}}} | V1𝑡, V2𝑡

]] , (5)

where

𝑔0 = 𝑔0 (𝑠, 𝑤) = 𝑠[[𝑇 − 𝑡𝑇 ln 𝑆𝑡

− 1𝑇2∑𝑗=1

(𝜌𝑗𝜅𝑗𝜃𝑗2𝜎𝑗 (𝑇 − 𝑡)2 +𝜌𝑗 (𝑇 − 𝑡)𝜎𝑗 V𝑗𝑡)]]

+ 𝑤[[ln 𝑆𝑡 −2∑𝑗=1

(𝜌𝑗𝜅𝑗𝜃𝑗𝜎𝑗 (𝑇 − 𝑡) +𝜌𝑗𝜎𝑗 V𝑗𝑡)]] ,

𝑔𝑗1 = 𝑔𝑗1 (𝑠, 𝑤) = 𝑠2 (1 − 𝜌2𝑗 )2𝑇2 ,

𝑔𝑗2 = 𝑔𝑗2 (𝑠, 𝑤) = 𝑠 (2𝜌𝑗𝜅𝑗 + 𝜎𝑗)2𝜎𝑗𝑇 +𝑠𝑤 (1 − 𝜌2𝑗 )𝑇 ,

𝑔𝑗3 = 𝑔𝑗3 (𝑠, 𝑤) = 𝑠𝜌𝑗𝑇𝜎𝑗 +𝑤 (2𝜌𝑗𝜅𝑗 + 𝜎𝑗)2𝜎𝑗

+ 𝑤2 (1 − 𝜌2𝑗 )2 − 1,𝑔𝑗4 = 𝑔𝑗4 (𝑠, 𝑤) = 𝑤𝜌𝑗𝜎𝑗 .

(6)

4 Complexity

Proof. (i) We first prove that the integrability conditionguarantees the existence of the cumulant function 𝜓𝑡(𝑠, 𝑤) in𝐷. If (𝑠, 𝑤) ∈ 𝐷 and 𝑡 ∈ [0, 𝑇), then

𝐸𝑄 [𝑒−∫𝑇

𝑡𝑟𝜏𝑑𝜏+𝑠ln𝐺[𝑡,𝑇]+𝑤ln𝑆𝑇

]= 𝑃 (𝑡, 𝑇) 𝐸𝑄𝑇 [𝑒𝑠ln𝐺[𝑡,𝑇]+𝑤ln𝑆𝑇 ]≤ 𝑃 (𝑡, 𝑇) 𝐸𝑄𝑇 [ 1𝑇 − 𝑡 ∫

𝑇

𝑡𝑆𝑢𝑑𝑢 + 𝑆𝑇]

= [ 1𝑇 − 𝑡 ∫𝑇

𝑡

𝑃 (𝑡, 𝑇)𝑃 (𝑡, 𝑢) 𝑑𝑢 + 1] 𝑆𝑡 < ∞,

(7)

where 𝑄𝑇 is the 𝑇− forward measure given by the Radon-Nikodym derivative: (𝑑𝑄𝑇/𝑑𝑄)|F𝑇 = exp{− ∫𝑇0 𝑟𝑢𝑑𝑢}/𝑃(0, 𝑇), and 𝑃(𝑡, 𝑇) is given above in (3). In the case of 𝑡 = 𝑇,it is triviality.

(ii) In order to determine (5), we start from model(1) and develop ln 𝑆𝑇 with the Brownian motions 𝑊1𝑡 and𝑊2𝑡 expressed as 𝑊1𝑡 = 𝜌1𝑍1𝑡 + √1 − 𝜌12𝑍1𝑡 and 𝑊2𝑡 =𝜌1𝑍2𝑡 + √1 − 𝜌22𝑍2𝑡 , respectively, where (𝑍1𝑡 , 𝑍2𝑡 , 𝑍1𝑡 , 𝑍2𝑡 ) are4-dimensional Brownian motion defined on the probabilityspace. For the process, ln 𝑆𝑇, we have

ln 𝑆𝑇 = ln 𝑆𝑡 + 2∑𝑗=1

[−𝜌𝑗𝜅𝑗𝜃𝑗𝜎𝑗 (𝑇 − 𝑡) −𝜌𝑗𝜎𝑗 V𝑗𝑡

+ (𝜌𝑗𝜅𝑗𝜎𝑗 +12)∫𝑇

𝑡V𝑗𝜏𝑑𝜏 + 𝜌𝑗𝜎𝑗 V𝑗𝑇

+ √1 − 𝜌2𝑗 ∫𝑇𝑡√V𝑗𝜏𝑑𝑍𝑗𝜏] = ln 𝑆𝑡 + 2∑

𝑗=1[𝜃𝑗2 (𝑇 − 𝑡)

+ V𝑗𝑡2𝜅𝑗 −V𝑗𝑇2𝜅𝑗 + (𝜌𝑗 +

𝜎𝑗2𝜅𝑗)∫𝑇

𝑡√V𝑗𝜏𝑑𝑍𝑗𝜏

+ √1 − 𝜌2𝑗 ∫𝑇𝑡√V𝑗𝜏𝑑𝑍𝑗𝜏] .

(8)

On the other hand, we have

ln𝐺[𝑡,𝑇] = 𝑇 − 𝑡𝑇 ln 𝑆𝑡 + 1𝑇2∑𝑗=1

[𝜃𝑗4 (𝑇 − 𝑡)2

+ V𝑗𝑡2𝜅𝑗 (𝑇 − 𝑡) −12𝜅𝑗 ∫𝑇

𝑡V𝑗𝜏𝑑𝜏

+ (𝜌𝑗 + 𝜎𝑗2𝜅𝑗)∫𝑇

𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏

+ √1 − 𝜌2𝑗 ∫𝑇𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏] .

(9)

Using the fact

∫𝑇𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏 = 1𝜎𝑗 [𝜅𝑗 ∫

𝑇

𝑡(𝑇 − 𝜏) V𝑗𝜏𝑑𝜏

+ ∫𝑇𝑡V𝑗𝜏𝑑𝜏 − (𝑇 − 𝑡) V𝑗𝑡 − 𝜅𝑗𝜃𝑗 (𝑇 − 𝑡)22 ] ,

(10)

for 𝑗 = 1, 2, thenln𝐺[𝑡,𝑇] = 𝑇 − 𝑡𝑇 ln 𝑆𝑡 + 1𝑇

2∑𝑗=1

[−𝜌𝑗𝜅𝑗𝜃𝑗2𝜎𝑗 (𝑇 − 𝑡)2

− 𝜌𝑗V𝑗𝑡𝜎𝑗 (𝑇 − 𝑡) +𝜌𝑗𝜎𝑗 ∫𝑇

𝑡V𝑗𝜏𝑑𝜏

+ (𝜌𝑗𝜅𝑗𝜎𝑗 +12)∫𝑇

𝑡(𝑇 − 𝜏) V𝑗𝜏𝑑𝜏

+ √1 − 𝜌2𝑗 ∫𝑇𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏] .

(11)

Let G be the 𝜎−field generated byF𝑡 and {(𝑍1𝑢, 𝑍2𝑢) : 𝑡

Complexity 5

+ 2∑𝑗=1

[𝑤(𝜌𝑗V𝑗𝑇𝜎𝑗 +2𝜌𝑗𝜅𝑗 + 𝜎𝑗2𝜎𝑗 ∫

𝑇

𝑡V𝑗𝜏𝑑𝜏)

− ∫𝑇𝑡V𝑗𝜏𝑑𝜏]}}} ,

𝐴3 = exp{{{2∑𝑗=1

√1 − 𝜌2𝑗 ∫𝑇𝑡[ 𝑠𝑇 (𝑇 − 𝜏) + 𝑤]

⋅ √V𝑗𝜏𝑑𝑍𝑗𝜏}}} .(13)

Since

𝐸𝑄 [𝐴3 | G] = exp{{{122∑𝑗=1

(1 − 𝜌2𝑗 )

⋅ ∫𝑇𝑡[ 𝑠𝑇 (𝑇 − 𝜏) + 𝑤]

2V𝑗𝜏𝑑𝜏}}}

= exp{{{2∑𝑗=1

(1 − 𝜌2𝑗 )2⋅ ∫𝑇𝑡[ 𝑠2𝑇2 (𝑇 − 𝜏)2 + 2𝑠𝑤𝑇 (𝑇 − 𝜏) + 𝑤2] V𝑗𝜏𝑑𝜏}}} ,

(14)

substituting (14) into (12) and applying the Markov propertyof {V𝑗𝑢 : 0 ≤ 𝑢 ≤ 𝑇}, 𝑗 = 1, 2, lead to (5), which completes theproof.

FromProposition 1, it is clear thatweneed to search for anexact formula for the discounted joint characteristic functionof V1𝑡 and V2𝑡 and the three different integrals of V𝑗𝑡 (𝑗 =1, 2) appearing in (5). We use the same approach introducedby Kim and Wee [18] to obtain an explicit formula for thediscounted joint characteristic function 𝜓𝑡(𝑠, 𝑤). Therefore,

two series of functions are introduced as follows. Define 𝐹𝑗𝜏and 𝐹𝑗𝜏 : C4 → C as

𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) = ∞∑𝑛=0

𝑓𝑗𝑛 , (15)𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) = ∞∑

𝑛=1

𝑛𝜏𝑓𝑗𝑛 , (16)where𝑓𝑗𝑛 , 𝑛 = 0, 1, 2, ⋅ ⋅ ⋅ , are functions of 𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, and 𝑔𝑗4defined as

𝑓𝑗−2 = 𝑓𝑗−1 = 0,𝑓𝑗0 = 1,𝑓𝑗1 = (𝜅𝑗 − 𝑔𝑗4𝜎

2𝑗 ) 𝜏2 ,

𝑓𝑗𝑛 = − 𝜎2𝑗𝜏22𝑛 (𝑛 − 1) (𝑔𝑗1𝜏2𝑓𝑗𝑛−4 + 𝑔𝑗2𝜏𝑓𝑗𝑛−3

+ (𝑔𝑗3 − 𝜅2𝑗2𝜎2𝑗 )𝑓

𝑗𝑛−2) , 𝑛 ≥ 2,

(17)

for 𝑗 = 1, 2.For 𝑗 = 1, 2, denote𝐷𝑗𝜏 = {(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ C4 :

𝐸𝑄 [exp{R (𝑔𝑗1) ∫𝜏0(𝜏 − 𝑡)2 V𝑗𝑡𝑑𝑡

+R (𝑔𝑗2)∫𝜏0(𝜏 − 𝑡) V𝑗𝑡𝑑𝑡 +R (𝑔𝑗3)∫𝜏

0V𝑗𝑡𝑑𝑡

+R (𝑔𝑗4) V𝑗𝜏}] < ∞} .

(18)

Proposition 2. (1) �e argument of 𝐹𝑗𝜏 : arg𝐹𝑗𝜏(𝑔𝑗1,𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ̸= 0 for every (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ 𝐷𝑗𝜏 (𝑗 = 1, 2).In particular, arg𝐹𝑗𝜏 is continuous on 𝐷𝑗𝜏, andarg𝐹𝑗𝜏(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) = 0 for 𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, and 𝑔𝑗4 areall real numbers in𝐷𝑗𝜏 for 𝑗 = 1, 2.

(2) For (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ 𝐷𝑗𝜏, 𝑗 = 1, 2, we have

𝐸𝑄[[exp{{{2∑𝑗=1

[𝑔𝑗1 ∫𝜏0(𝜏 − 𝑡)2 V𝑗𝑡𝑑𝑡 + 𝑔𝑗2 ∫𝜏

0(𝜏 − 𝑡) V𝑗𝑡𝑑𝑡 + 𝑔𝑗3 ∫𝜏

0V𝑗𝑡𝑑𝑡 + 𝑔𝑗4V𝑗𝜏]}}}

]]

= exp{{{2∑𝑗=1

(𝜅𝑗V𝑗0 + 𝜅2𝑗𝜃𝑗𝜏𝜎𝑗 −2V𝑗0𝜎2𝑗

𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4)𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) −2𝜅𝑗𝜃𝑗𝜎2𝑗 ln𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4))

}}} .(19)

6 Complexity

�e proof of Proposition 2 is similar to that of Kim andWee[18].

Using Proposition 2 to Proposition 1 leads to the explicitexpression of 𝜓𝑡(𝑠, 𝑤) in Proposition 3. To describe the simplic-ity of the result, we need to introduce new functions. Define𝐻𝑗𝑡,𝑇and �̃�𝑗𝑡,𝑇: C2 → C as𝐻𝑗𝑡,𝑇 (𝑠, 𝑤)= 𝐹𝑗𝑇−𝑡 (𝑔𝑗1 (𝑠, 𝑤) , 𝑔𝑗2 (𝑠, 𝑤) , 𝑔𝑗3 (𝑠, 𝑤) , 𝑔𝑗4 (𝑠, 𝑤)) ,

(20)

�̃�𝑗𝑡,𝑇 (𝑠, 𝑤)= 𝐹𝑗𝑇−𝑡 (𝑔𝑗1 (𝑠, 𝑤) , 𝑔𝑗2 (𝑠, 𝑤) , 𝑔𝑗3 (𝑠, 𝑤) , 𝑔𝑗4 (𝑠, 𝑤)) ,

(21)

with 𝑗 = 1, 2.Proposition 3. (1) If (𝑠, 𝑤) ∈ 𝐷, then 𝐻𝑗𝑡,𝑇(𝑠, 𝑤) ̸= 0 andarg𝐻𝑗𝑡,𝑇 is continuous on𝐷. In particular, arg𝐻𝑗𝑡,𝑇(𝑠, 𝑤) = 0 if𝑠 and 𝑤 are real numbers for 𝑗 = 1, 2.

(2) For 𝜓𝑡(𝑠, 𝑤) ∈ 𝐷 with arg𝐻𝑗𝑡,𝑇 defined as above, then

𝜓𝑡 (𝑠, 𝑤) = exp{{{2∑𝑗=1

(−𝑐𝑗1 �̃�𝑗𝑡,𝑇 (𝑠, 𝑤)𝐻𝑗𝑡,𝑇 (𝑠, 𝑤)

− 𝑐𝑗2ln𝐻𝑗𝑡,𝑇 (𝑠, 𝑤) + 𝑐𝑗3𝑠 + 𝑐𝑗4𝑤 + 𝑐𝑗5)}}} .(22)

Here 𝑐𝑗1 = 2V𝑗𝑡/𝜎2𝑗 , 𝑐𝑗2 = 2𝜅𝑗𝜃𝑗/𝜎2𝑗 ,

𝑐𝑗3 = 𝑇 − 𝑡2𝑇 ln 𝑆𝑡 −𝜌𝑗𝜅𝑗𝜃𝑗 (𝑇 − 𝑡)22𝜎𝑗𝑇 −

𝜌𝑗 (𝑇 − 𝑡)𝜎𝑗𝑇 V𝑗𝑡,𝑐𝑗4 = ln 𝑆𝑡2 −

𝜌𝑗𝜎𝑗 V𝑗𝑡 −𝜌𝑗𝜅𝑗𝜃𝑗 (𝑇 − 𝑡)𝜎𝑗 ,

𝑐𝑗5 = 𝜅𝑗V𝑗𝑡 + 𝜅2𝑗𝜃𝑗 (𝑇 − 𝑡)𝜎2𝑗

(23)

for 𝑗 = 1, 2.Proof. Assume that (𝑠, 𝑤) ∈ 𝐷, using the definitions of 𝑔𝑗𝑘 =𝑔𝑗𝑘(𝑠, 𝑤) and 𝐷𝑗𝜏 for 𝑘 = 1, 2, 3, 4 and 𝑗 = 1, 2. Note that(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ 𝐷𝑗𝜏 ⊂ 𝐷𝑗𝑇−𝑡 for every 0 ≤ 𝑡 ≤ 𝑇 and𝑗 = 1, 2. Therefore, (19) remains valid for any V𝑗0 > 0. Thetime homogeneous Markov property of V𝑗𝑡 implies that (22)holds when Propositions 1 and 2 are satisfied.

Now Substituting the expressions of 𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4, 𝑗 =1, 2 into (15) and (16),𝐻𝑗𝑡,𝑇 and �̃�𝑗𝑡,𝑇 are expressed as follows,i.e., for (𝑠, 𝑤) ∈ C2,

𝐻𝑗𝑡,𝑇 (𝑠, 𝑤) = ∞∑𝑛=0

ℎ𝑗𝑛 (𝑠, 𝑤) , (24)�̃�𝑗𝑡,𝑇 (𝑠, 𝑤) = ∞∑

𝑛=1

𝑛𝑇 − 𝑡ℎ𝑗𝑛 (𝑠, 𝑤) , (25)where

ℎ𝑗−2 (𝑠, 𝑤) = ℎ𝑗−1 (𝑠, 𝑤) = 0,ℎ𝑗0 (𝑠, 𝑤) = 1,ℎ𝑗1 (𝑠, 𝑤) = (𝑇 − 𝑡) (𝜅𝑗 − 𝑤𝜌𝑗𝜎𝑗)2 ,ℎ𝑗𝑛 (𝑠, 𝑤) = (𝑇 − 𝑡)24𝑛 (𝑛 − 1) 𝑇2 {−𝑠2𝜎2𝑗 (1 − 𝜌2𝑗 ) (𝑇 − 𝑡)2

⋅ ℎ𝑗𝑛−4 (𝑠, 𝑤) − [𝑠𝜎𝑗𝑇(𝜎𝑗 + 2𝜌𝑗𝜅𝑗)+ 2𝑠𝑤𝜎2𝑗𝑇 (1 − 𝜌2𝑗 )] (𝑇 − 𝑡) ℎ𝑗𝑛−3 (𝑠, 𝑤) + 𝑇 [𝜅2𝑗𝑇− 2𝑠𝜌𝑗𝜎𝑗 − 𝑤 (2𝜌𝑗𝜅𝑗 + 𝜎𝑗) 𝜎𝑗𝑇 − 𝑤2 (1 − 𝜌2𝑗 ) 𝜎2𝑗𝑇+ 2𝜎2𝑗𝑇] ℎ𝑗𝑛−2 (𝑠, 𝑤)} , 𝑛 ≥ 2,

(26)

for 𝑗 = 1, 2.4. Pricing Geometric Asian Option

Once the discounted joint characteristic function is found,the European continuously monitored geometric Asianoption can be valued using numeraire change technique andthe inverse Fourier transform approach which are appliedin many research works (see, e.g., Geman et al. [41] andDeng [42]). This section derives the pricing formulas for thecontinuously monitored fixed and floating strike geometricAsian call options.

Theorem 4. Suppose that 𝑆𝑡, V1𝑡, and V2𝑡 follow the dynamicsin (1), then the price at time 𝑡 ∈ [0, 𝑇] of the continuously mon-itored fixed strike geometric Asian call option with maturity 𝑇and the strike price 𝐾 is given by

𝐺𝐴𝐶𝑓𝑖 (𝑡, 𝑆, V1, V2, 𝐾, 𝑇)= 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0) Π1 − 𝐾𝑃 (𝑡, 𝑇)Π2, (27)

where 𝑃(𝑡, 𝑇) is given above in (3) andΠ𝑗 = 12 + 1𝜋 ∫

+∞

0R[𝑒−𝑖𝑢ln𝐾𝑡,𝑇𝜙𝑗 (𝑢)𝑖𝑢 ]𝑑𝑢,

𝑗 = 1, 2,

Complexity 7

𝐾𝑡,𝑇 = 𝐾𝑒−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢,𝜙1 (𝑢) = 𝜓𝑡 (𝑖𝑢 + 1, 0)𝜓𝑡 (1, 0) ,𝜙2 (𝑢) = 𝜓𝑡 (𝑖𝑢, 0)𝜓𝑡 (0, 0) ,

(28)

where 𝑖 is the imaginary unit (𝑖2 = −1).Proof. Since

𝐺𝐴𝐶𝑓𝑖 (𝑡, 𝑆, V1, V2, 𝐾, 𝑇)= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 (𝐺[0,𝑇] − 𝐾)+ | F𝑡]= 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝐸𝑄 [𝑒− ∫T𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇]1{ln𝐺[𝑡,𝑇]≥ln𝐾𝑡,𝑇} |F𝑡] − 𝐾𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢1{ln𝐺[𝑡,𝑇]≥ln𝐾𝑡,𝑇} | F𝑡]= 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0)𝑄1 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇)− 𝐾𝑃 (𝑡, 𝑇)𝑄𝑇 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇) ,

(29)

where𝑄1 is defined by the following Radon-Nikodymderiva-tive:

𝑑𝑄1𝑑𝑄F𝑇 = 𝑒− ∫

𝑇

𝑡𝑟𝑢𝑑𝑢

𝐺[𝑡,𝑇]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇] | F𝑡] , (30)

and 𝑄𝑇 is the 𝑇− forward measure given above, it is wellknown that the probability distribution functions can becalculated by using the Fourier inversion transform, and thenthe above two probabilities 𝑄1 and 𝑄𝑇 in (29) are given by

𝑄1 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇)= 12 + 1𝜋 ∫

+∞

0R [𝑒−𝑖𝑢ln𝐾𝑡,𝑇𝜙1 (𝑢)𝑖𝑢 ] 𝑑𝑢 fl Π1,

(31)

𝑄𝑇 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇)= 12 + 1𝜋 ∫

+∞

0R [𝑒−𝑖𝑢ln𝐾𝑡,𝑇𝜙2 (𝑢)𝑖𝑢 ] 𝑑𝑢 fl Π2,

(32)

where

𝜙1 (𝑢) = 𝐸𝑄1 [𝑒𝑖𝑢ln𝐺[𝑡,𝑇] | F𝑡]= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+𝑖𝑢ln𝐺[𝑡,𝑇]𝐺[𝑡,𝑇] | F𝑡]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇] | F𝑡]

= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+(𝑖𝑢+1)ln𝐺[𝑡,𝑇] | F𝑡]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+ln𝐺[𝑡,𝑇] | F𝑡]

= 𝜓𝑡 (𝑖𝑢 + 1, 0)𝜓𝑡 (1, 0) ,(33)

𝜙2 (𝑢) = 𝐸𝑄𝑇 [𝑒𝑖𝑢ln𝐺[𝑡,𝑇] | F𝑡]= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+𝑖𝑢ln𝐺[𝑡,𝑇] | F𝑡]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 | F𝑡] =

𝜓𝑡 (𝑖𝑢, 0)𝑃 (𝑡, 𝑇) .(34)

From (29), (31), and (32), we can obtain the requiredTheorem 4.

Theorem 5. Suppose that 𝑆𝑡, V1𝑡, and V2𝑡 follow the dynamicsin (1), then the price at time 𝑡 ∈ [0, 𝑇] of the continuouslymonitored floating strike geometric Asian call option withmaturity 𝑇 is given by

𝐺𝐴𝐶𝑓𝑙 (𝑡, 𝑆, V1, V2, 𝑇) = 𝑆𝑡Π̂1− 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0) Π̂2, (35)

where

Π̂1= 12

− 1𝜋 ∫+∞

0R[[

𝜓𝑡 (𝑖𝑢, 1 − 𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝑖𝑢𝑆𝑡 ]]𝑑𝑢,Π̂2= 12

− 1𝜋 ∫+∞

0R[[

𝜓𝑡 (𝑖𝑢 + 1, −𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝑖𝑢𝜓𝑡 (1, 0) ]]𝑑𝑢.

(36)

Proof. Since

𝐺𝐴𝐶𝑓𝑙 (𝑡, 𝑆, V1, V2, 𝑇) = 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 (𝑆𝑇 − 𝐺[0,𝑇])+ | F𝑡]= 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 (𝑆𝑇𝑒−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢 − 𝐺[𝑡,𝑇])+ |F𝑡] = 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝑆𝑇1{ln(𝐺[𝑡,𝑇]/𝑆𝑇)≤−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢} | F𝑡]− 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇]1{ln(𝐺𝑡,𝑇/𝑆𝑇)≤−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢} |

8 Complexity

F𝑡] = 𝑆𝑡𝑄2 (ln(𝐺[𝑡,𝑇]𝑆𝑇 ) ≤ −1𝑇 ∫𝑡

0ln 𝑆𝑢𝑑𝑢)

− 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0)𝑄1 (ln(𝐺[𝑡,𝑇]𝑆𝑇 ) ≤ −1𝑇 ∫𝑡

0ln 𝑆𝑢𝑑𝑢) ,

(37)

where𝑄2 is defined by the following Radon-Nikodymderiva-tive:

𝑑𝑄2𝑑QF𝑇 = 𝑒− ∫

𝑇

𝑡𝑟𝑢𝑑𝑢 𝑆𝑇𝑆𝑡 , (38)

under the two probability measures 𝑄1 and 𝑄2, the condi-tional characteristic functions of ln(𝐺[𝑡,𝑇]/𝑆𝑇) are given by

𝐸𝑄1 [𝑒𝑖𝑢ln(𝐺[𝑡,𝑇]/𝑆𝑇) | F𝑡]= 𝐸𝑄 [𝑒−∫𝑇𝑡 𝑟𝑢𝑑𝑢+𝑖𝑢ln(𝐺[𝑡,𝑇]/S𝑇)𝐺[𝑡,𝑇] | F𝑡]

𝐸𝑄 [𝑒−∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇] | F𝑡]= 𝜓𝑡 (𝑖𝑢 + 1, −𝑖𝑢)𝜓𝑡 (1, 0) ,

(39)

𝐸𝑄2 [𝑒𝑖𝑢ln(𝐺[𝑡,𝑇]/𝑆𝑇) | F𝑡]= 𝐸𝑄 [𝑒−∫𝑇𝑡 𝑟𝑢𝑑𝑢+𝑖𝑢ln(𝐺[𝑡,𝑇]/𝑆𝑇)+ln𝑆𝑇 | F𝑡]

𝑆𝑡= 𝜓𝑡 (𝑖𝑢, 1 − 𝑖𝑢)𝑆𝑡 .

(40)

Therefore

𝑄1 (ln(𝐺𝑡,𝑇𝑆𝑇 ) ≤ −1𝑇 ∫𝑡

0ln 𝑆𝑢𝑑𝑢)

= 12− 1𝜋 ∫

+∞

0R[[

𝜓𝑡 (𝑖𝑢 + 1, −𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln 𝑆𝑢𝑑𝑢𝑖𝑢𝜓𝑡 (1, 0) ]]𝑑𝑢,(41)

𝑄2 (ln(𝐺𝑡,𝑇𝑆𝑇 ) ≤ −1𝑇 ∫𝑡

0ln 𝑆𝑢𝑑𝑢)

= 12− 1𝜋 ∫

+∞

0R[[

𝜓𝑡 (𝑖𝑢, 1 − 𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝑆𝑡 ⋅ 𝑖𝑢 ]] 𝑑𝑢.(42)

Plugging (41) and (42) in (37) yields (35).

In addition to option prices, one can compute derivativesto hedge against changes in the underlying asset price 𝑆 andvolatilities V1 and V2. We omit it due to its triviality.

5. Numerical Examples

In this section, we use the dbH-SI model to analyze the val-uation of the continuously monitored fixed strike geometricAsian call option using some numerical examples. To imple-ment the analytic formula given in (27) numerically, we firstdetermine the number of terms taken for the computationof the infinite series expansions for 𝐻𝑗𝑡,𝑇 and �̃�𝑗𝑡,𝑇 for 𝑗 =1, 2 in (24) and (25). Second, we investigate the accuracyand efficiency of the approximated analytic formula givenin (27). In the end, we compare the option prices varyingby 𝑆0 and 𝑇 under different models including the dbH-SI,H-SI (i.e., Heston stochastic volatility model with stochasticinterest rate), dbH, and Heston models. Furthermore, we usethis proposed model (1), i.e., the dbH-SI model, to examinethe impacts of the model parameters on option prices. Herewe implement the integral formulas given in (27) withouttruncation,modification, and approximation inMathematicaand Matlab software. Numerical integration was performedusing the “NIntegrate()” or “quadgk()” commands whichcan handle infinite ranges, oscillatory integrands, and sin-gularities in their default version, which employs automaticadaptive integration.

Model (1) parameters are set as follows: 𝜅1 = 1.15, 𝜅2 =2.2, 𝜃1 = 0.2, 𝜃2 = 0.15, 𝜎1 = 0.5, 𝜎2 = 0.8, 𝜌1 = −0.4,𝜌2 = −0.64, V10 = 0.09, V20 = 0.04, 𝑆0 = 100, and 𝑡 = 0. Theresults are shown in Table 1.

Table 1 investigates the influence of the number of terms𝑛 taken in (24) and (25). It is shown that the numerical valuestend to be quite stable as the number of terms increases. Inparticular, the option prices stay unchanged after taking 𝑛 =20 and 𝑛 = 30 terms, at most. In addition, Table 1 providesoption prices with various maturities 𝑇, strike prices 𝐾, andthe requiredCPU times (in seconds). FromTable 1, we can seethat some characteristics of the GAC𝑓𝑖 option are similar tothe plain European call option. For example, the values of theGAC𝑓𝑖 option are decreasing functions of the strike price. Inaddition, under the same parameter values setting in dbH-SImodel, longer maturating date 𝑇 will result in higher optionvalues.

It would be interesting to see the performance of ourapproximated analytic formula approach implemented to thefixed strike GAC option under dbH-SImodel. Our numericalexample uses the number of terms 𝑛 = 20 taken in (24) and(25); other parameter values are similar to those settings ofTable 1. We compute the 0.5-,1.5-, and 3-year maturity GACoptions by our approximated analytic formula approach andcompare the results with Monte Carlo simulation with 10000sample paths and 100 points for time axis. The numericalresults are displayed in Table 2.

Table 2 shows that the approximated analytic formulaapproach is considerably faster than those of the MonteCarlo simulation (MC). For a given set of parameters,the approximated analytic formula approach calculates theoption prices for 5 different strikes and 3 different maturatesin approximately 5 seconds. The Monte Carlo simulationtakes around 110 seconds for each option price. Table 2 alsocompares their pricing accuracy. It can be seen that therelative error (RE) in prices is less than 0.4% for all cases. If

Complexity 9

Table 1: GAC𝑓𝑖 option prices with different maturities, strike prices, and varying n.

𝑇 𝐾 𝑛 Option price CPU (sec.) 𝑇 𝐾 𝑛 Option price CPU (sec.)0.5 90 5 17.8597 0.287 1.5 90 5 26.4805 0.291

10 17.7859 0.291 10 25.2393 0.29415 17.7849 0.294 15 25.2254 0.29920 17.7849 0.296 20 25.2251 0.30225 17.7849 0.301 25 25.2251 0.30730 17.7849 0.312 30 25.2251 0.315

100 5 9.7733 0.279 100 5 19.8642 0.27610 9.8624 0.288 10 19.1846 0.28515 9.8623 0.290 15 19.1758 0.29020 9.8623 0.295 20 19.1755 0.30225 9.8623 0.298 25 19.1755 0.30930 9.8623 0.302 30 19.1755 0.311

110 5 4.2061 0.283 110 5 14.1405 0.28310 4.5395 0.291 10 14.0976 0.29115 4.5415 0.304 15 14.1003 0.29720 4.5415 0.306 20 14.1002 0.30425 4.5415 0.312 25 14.1002 0.31030 4.5415 0.315 30 14.1002 0.315

1 90 5 22.3192 0.268 3 90 5 30.5831 0.28910 22.0988 0.283 10 29.6250 0.29315 22.0938 0.287 15 29.2870 0.29920 22.0938 0.296 20 29.2835 0.30525 22.0938 0.313 25 29.2835 0.31430 22.0938 0.319 30 29.2835 0.317

100 5 15.0971 0.281 100 5 26.7286 0.26810 15.1874 0.287 10 25.6231 0.28415 15.1850 0.294 15 25.3192 0.29120 15.1849 0.298 20 25.3151 0.29525 15.1849 0.301 25 25.3151 0.29830 15.1849 0.307 30 25.3151 0.307

110 5 9.2791 0.275 110 5 22.5318 0.28110 9.7604 0.278 10 21.9684 0.28415 9.7650 0.281 15 21.7105 0.29720 9.7651 0.292 20 21.7064 0.30125 9.7651 0.294 25 21.7064 0.30530 9.7651 0.301 30 21.7064 0.310

we regard the Monte Carlo price as the benchmark, then thisnumerical example confirms that the approximated analyticformula approach is accurate and efficient.

After examining accuracy and efficiency. We shall turnto investigate the comparison of the option prices underdifferent models. Figure 1 compares the dbH-SI model withHeston, dbH, and H-SI models in pricing the fixed strikeGAC option against the initial asset price 𝑆0 and maturatingdate 𝑇 with the strike price 𝐾 = 100. From Figure 1, it isinteresting to notice that the option prices of the fixed strikeGAC option calculated from the dbH-SI model are muchhigher than those of the dbHmodel, H-SImodel, and Hestonmodel. One possible reason for this phenomena is that thevolatilities are always bigger for the dbH-SI model than those

for other models. This implies that the proposed model, i.e.,the dbH-SI model, has a more significant influence than thedbHmodel, H-SImodel, and Hestonmodel on option prices.We can also clearly observe that the bigger the value of 𝑆0(or𝑇) is, the higher the GAC𝑓𝑖 option price is.

In Figure 2, we use the dbH-SI model to examine theeffects of the mean-reverting rates (𝜅1, 𝜅2) and volatilities(𝜎1, 𝜎2) of variance processes (V1𝑡, V2𝑡) on the GAC𝑓𝑖 optionprices (see Figures 2(a)–2(d)). We also examine the impactsof the correlation coefficients (𝜌1, 𝜌2) (see Figures 2(e)and 2(f)), which determine both the correlation betweenthe underlying asset and its volatilities, and the correla-tion between the underlying asset and the short interestrates.

10 Complexity

Table 2: Comparison of the approximated approach and MC for GAC𝑓𝑖 options.

𝑇 𝐾 Approximated approach CPU(sec.) Monte Carlo CPU(sec.) RE(%)0.5 90 17.7849 0.284 17.7892 110.302 -0.0242

95 13.5305 0.291 13.5246 110.384 0.0436100 9.8623 0.278 9.8602 110.403 0.0213105 6.8598 0.296 6.8624 110.329 -0.3810110 4.5415 0.287 4.5407 110.376 0.0176

1.5 90 25.2251 0.265 25.2206 110.376 0.017895 22.0851 0.281 22.0891 110.391 -0.0181100 19.1755 0.294 19.1787 110.417 -0.0167105 16.5109 0.312 16.5045 110.485 0.3888110 14.1002 0.293 14.1039 110.392 -0.0262

3 90 29.2835 0.271 29.2894 110.428 -0.020195 27.2560 0.289 27.2513 110.397 0.0173100 25.3151 0.307 25.3106 110.405 0.0178105 23.4644 0.295 23.4595 110.471 0.0209110 21.7065 0.292 21.7128 110.396 -0.0290

80 85 90 95 100 105 110 115 120S0

0

5

10

15

20

25

30

35

40

45

optio

n pr

ice

dbH-SI modeldbH modelH-SI modelHeston model

(a) Option price against 𝑆0

0.5 1 1.5 2 2.5 3 3.5 4 4.5T

5

10

15

20

25

30

35

40

optio

n pr

ice

dbH-SI modeldbH modelH-SI modelHeston model

(b) Option price against 𝑇

Figure 1: Comparison of option prices under Heston, dbH, H-SI, and dbH-SI models.

From Figures 2(a) and 2(b), we can observe that theoption prices increase as the mean-reverting rate increases.Particularly, it is quite remarkable that the mean-revertingrates have a significant effect on the longer termoption values.The option price decreases as the volatilities of varianceprocesses increase (see Figures 2(c) and 2(d)).The correlationcoefficients (𝜌1, 𝜌2) have several effects depending on therelation between the strike price and the initial asset price.A negative (𝜌1, 𝜌2) tends to produce higher values for ITM(in-the-money) options and lower values for OTM (out-of-the-money) options (see Figure 2(e)). A similar result holdswith respect to the expiration date: a negative (𝜌1, 𝜌2) tends to

produce higher value for the longer term options and lowervalues for the shorter options (see Figure 2(f)).

6. Conclusions

Theproposedmodel (the dbH-SImodel) incorporates severalimportant features of the underlying asset returns variability.We derive the discounted joint characteristic function ofthe log-asset price and its log-geometric mean value overtime period [0, 𝑇] and obtain approximated analytic solutionsto the continuously monitored fixed and floating strikegeometric Asian call options using the change of numeraire

Complexity 11

S0

5

10

15

20

25

30

35

optio

n pr

ice

80 85 90 95 100 105 110 115 120

(Ｅ1, Ｅ2)=(1.5,1)

(Ｅ1, Ｅ2)=(1.5,2)

(Ｅ1, Ｅ2)=(2,1.5)

(Ｅ1, Ｅ2)=(1.5,1.5)

(Ｅ1, Ｅ2)=(1,1.5)

(a)

T

5

10

15

20

25

30

35

optio

n pr

ice

0.5 1 1.5 2 2.5 3 3.5 4

(Ｅ1, Ｅ2)=(1.5,1)

(Ｅ1, Ｅ2)=(1.5,2)

(Ｅ1, Ｅ2)=(2,1.5)

(Ｅ1, Ｅ2)=(1.5,1.5)

(Ｅ1, Ｅ2)=(1,1.5)

(b)

S0

05

101520253035404550

optio

n pr

ice

80 85 90 95 100 105 110 115 120

(1, 2)=(0.2,0.5)

(1, 2)=(0.2,0.7)

(1, 2)=(0.2,0.9)

(1, 2)=(0.6,0.5)

(1, 2)=(0.9,0.5)

(c)

T

10

15

20

25

30

35

40

optio

n pr

ice

0.5 1 1.5 2 2.5 3 3.5 45

(1, 2)=(0.2,0.5)

(1, 2)=(0.2,0.7)

(1, 2)=(0.2,0.9)

(1, 2)=(0.6,0.5)

(1, 2)=(0.9,0.5)

(d)

S0

0

10

20

30

40

50

60

70

80

optio

n pr

ice

80 85 90 95 100 105 110 115 120

(1, 2)=(0.6,-0.6)

(1, 2)=(0.6,0)

(1, 2)=(0.6,0.6)

(1, 2)=(-0.6,0.6)

(1, 2)=(0,0.6)

(1, 2)=(-0.6,-0.6)

(e)

T

510152025303540455055

optio

n pr

ice

0.5 1 1.5 2 2.5 3 3.5 4

(1, 2)=(0.6,-0.6)

(1, 2)=(0.6,0)

(1, 2)=(0.6,0.6)

(1, 2)=(-0.6,0.6)

(1, 2)=(0,0.6)

(1, 2)=(-0.6,-0.6)

(f)

Figure 2: Option prices with respect to 𝑆0 and 𝑇 in the dbH-SI model.

12 Complexity

technique and the Fourier inversion transform approach.Some numerical examples are provided to examine theeffects of the proposed model, which reveals some additionalfeatures having a significant impact on option values, espe-cially long-term options. The proposed model can be testedempirically by using the option price data from the optionmarket.

Data Availability

All data used to support of the findings for this study areincluded within this article.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

The authors acknowledge the financial support providedby the Natural Sciences Foundation of China (11461008),the Guangxi Natural Science Foundation (2018JJA110001),and the Guangxi Graduate Education Innovation Programproject (XYCSZ2018059).

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