# Geometric Asian Options Pricing under the Double Heston Stochastic Volatility Model with Stochastic Interest Rate

Authors: Yanhong Zhong; Guohe Deng
Journal: Complexity
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

Asian option is a special type of option contract in which the payoff depends on the average of the underlying asset price over some predetermined time interval. The averaging feature allows Asian options to reduce the volatility inherent in the option. There are some advantages to trading Asian options in a financial market. One is that these decrease the risk of market manipulation of the financial derivative at expiry. Another is that Asian options have lower relative charge than European or American options. In general, the average considered can be a arithmetic or geometric one and it can be calculated either discretely, for which the average is taken over the underlying asset prices at discrete monitoring time points, or continuously, for which the average is calculated via the integration of the underlying asset price over the monitoring time period. Asian options can be differentiated into two main classes according to their payoff: fixed strike price options (sometimes called “average price”) and floating strike price options (sometimes called “average strike”). All these details are specified by the contracts stipulated by two counterparts, as Asian options are traded actively on the OTC market among investors or traders for hedging the average price of a commodity. For a brief introduction to the development of Asian options, see Boyle and Boyle [1].As the probability distribution of the average prices of the underlying asset generally does not have a simple analytical expression, it is difficult to obtain the analytical pricing formula for Asian option. Since the best-known closed-form pricing formula for the European vanilla option derived by Black and Scholes [2]), many researchers have devoted themselves to developing the Asian options pricing based on the Black-Scholes assumptions; see, e.g., Kemna and Vorst (1990), Turnbull and Wakeman [3], Ritchken et al. [4], Geman and Yor [5], Rogers and Shi [6], Boyle et al. [7], Angus [8], Linetsky [9], Cui et al. [10], and the references therein. For a recent review, one can refer to Fusai and Roncoroni [11] and Sun et al. [12].In practice, the Black-Scholes assumptions are hardly satisfied, especially the constant volatility and constant interest rate hypothesis. As the empirical behaviors of the implied volatility smile and heavy tailed in the distribution of log-returns are commonly observed in financial markets. For this reason, stochastic volatility (hereafter SV) models have been proposed in finance (see Hull and White [13], Stein and Stein [14], Heston [15], and others). These models have been applied to value the Asian options (see, e.g., Wong and Cheung [16], Hubalek and Sgarra [17], Kim and Wee [18], and Shi and Yang [19]). In addition, interest rates are stochastic and stock returns are negatively correlated with interest rate changes, which have been examined in previous research.Although these models mentioned above are able to account for the empirical behaviors, they are still based on a single-factor for volatility dynamics that is inconsistent with the long range memory characteristic of the volatility corrections 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, multifactor SV models have recently generated attention in the 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 volatility processes are driven by two mean-reverting diffusion processes. Gourieroux [23] proposed a multivariate model in which the volatility-covolatility matrix follows a Wishart process.On the basis of the findings of Christoffersen et al. [24], a double Heston (dbH) model, which consists of two independent variance processes, has recently been reported better than the plain Heston [15] model in the performances of hedging (see Sun [29]) and has also been applied to arithmetic Asian option under discrete monitoring (Mehrdoust and Saber [30]) and forward starting option (Zhang and Sun [31]). However, its extension to continuously monitored geometric Asian option is yet to be considered. On the other hand, many of Asian options often have long-dated maturities since they are used as part of the structured notes which has a long maturity. The movement of interest rates becomes an issue in such cases and constant interest rate assumption should be replaced by an appropriate dynamic interest rate model. Several results are available on the Asian option in the stochastic interest rate framework; see, e.g., Nielsen and Sandmann [32, 33], Zhang et al. [34], Donnelly et al. [35], and He and Zhu [36]. In the above stochastic interest rate framework, the short-term interest rate is assumed to follow a specific parametric one-factor model (see, e.g., Cox et al. [37], Hull and White [13], and Vasicek [38]), which tends to 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 structure for the interest rate should involve several sources of uncertainty, and introducing additional state variables (such as the rate of inflation, GDP, etc.) significantly improves the fit.In this paper, we study the pricing of the continuously monitored geometric Asian options under dbH stochastic volatility model with stochastic interest rate framework (hereafter, dbH-SI model). The contribution of the present paper is twofold. Firstly, this paper extends the dbH model by introducing stochastic interest rate, which is assumed to follow two-factor model with two state variables. Secondly, this paper provides a semiexplicit valuation formula for the geometric Asian options with fixed or floating strike price, which is extremely useful also for the arithmetic average option valuation via Monte Carlo methods with control variables.The rest of the paper is organized as follows. Section2 develops the underlying pricing model and describes the geometric Asian option. Section 3 derives the joint characteristic function of a log-return of the underlying asset and its geometric average. Section 4 obtains the analytic expressions for the prices of the fixed strike geometric Asian call option and the floating strike Asian call option under continuous monitoring. Section 5 provides some numerical examples for the proposed approach. Section 6 concludes the paper.


## 2. Model Formulation

We consider an arbitrage-free, frictionless financial market where only riskless asset and risky asset are traded continuously up to a fixed horizon dateT. Let (Ω,F,{Ft}0≤t≤T,Q) be a complete probability space equipped with a filtration {Ft} satisfying the usual conditions, where Q is a risk-neutral probability measure. Suppose Wtj and Ztj(j=1,2) are all standard Brownian motions defined on the probability space, and the filtration {Ft}t≥0 is generated by these Brownian motions. Moreover, dWt1dZt1=ρ1dt, dWt2dZt2=ρ2dt, and any other Brownian motions are pairwisely independent. Assume that the asset price process St, without paying any dividend, satisfies the following stochastic differential equation under Q:(1)dStSt=rtdt+v1tdWt1+v2tdWt2,dv1t=κ1θ1-v1tdt+σ1v1tdZt1,dv2t=κ2θ2-v2tdt+σ2v2tdZt2,where κj,θj,σj(j=1,2) are all nonnegative constants, which represent the mean-reverting rates, long-term mean levels, and volatilities of variance processes vjt, respectively. We suppose that 2κjθj≥σj2. The instantaneous interest rate, rt, is assumed to be a linear combination of v1t and v2t, i.e., rt=v1t+v2t, which designates the interest rate as an affine function of two-factor economic variables v1 and v2 and offers the analytic tractability (see Duffie et al. [27]).In financial market, there are four types of European style continuously monitoring geometric Asian options: fixed strike geometric Asian calls, fixed strike geometric Asian puts, floating strike geometric Asian calls, and floating strike geometric Asian puts. The payoffs at the expiration dateT for these options are as follows: (2)hST,G0,T=maxG0,T-K,0,a  fixed  strike  geometric  Asian  calls,maxK-G0,T,0,a  fixed  strike  geometric  Asian  puts,maxG0,T-ST,0,a  floating  strike  geometric  Asian  calls,maxST-G0,T,0,a  floating  strike  geometric  Asian  puts,where K is a fixed strike price and G[0,T] is the geometric average of the underlying asset price St until time T; i.e., G[0,T]=exp((1/T)∫0TlnSudu). In the following, we consider only the pricing problem of the geometric Asian call options (hereafter, GAC), while the put options can be dealt with similarly.For the instantaneous interest ratert=v1t+v2t, one can express the price at time t of a zero-coupon bond with maturate T as follows (see Cox et al. [37]):(3)Pt,T=exp⁡at,T-bt,Trt,where(4)at,T=∑j=12κjθjκj-γjT-tσj2+2κjθjσj2ln2γj2γj+κj-γj1-e-γjT-t,bt,T=∑j=1221-e-γjT-t2γj+κj-γj1-e-γjT-t,γj=κj2+2σj2,forj=1,2.


## 3. The Joint Characteristic Function

Given the dynamic of the underlying asset price, it is possible to obtain the discounted joint characteristic function for the log-asset valueST and the log-geometric mean value of the asset price over a certain time period.Letψt(s,w)=EQ[e-∫tTrτdτ+slnG[t,T]+wlnST∣Ft] be the discounted joint characteristic function of two-dimensional random variable, (lnG[t,T],lnST), conditioned on Ft under Q, where G[t,T]=exp{(1/T)∫tTlnSudu} and EQ[·∣Ft] is the conditional 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 thatSt,v1t, and v2t follow the dynamics in (1). If (s,w)∈D and t∈[0,T], then EQ[e-∫tTrτdτ+slnGt,T+wlnST]<∞ and(5)ψts,w=eg0EQexp∑j=12gj1∫tTT-τ2vjτdτ+gj2∫tTT-τvjτdτ+gj3∫tTvjτdτ+gj4vjT∣v1t,v2t,where (6)g0=g0s,w=sT-tTlnSt-1T∑j=12ρjκjθj2σjT-t2+ρjT-tσjvjt+wlnSt-∑j=12ρjκjθjσjT-t+ρjσjvjt,gj1=gj1s,w=s21-ρj22T2,gj2=gj2s,w=s2ρjκj+σj2σjT+sw1-ρj2T,gj3=gj3s,w=sρjTσj+w2ρjκj+σj2σj+w21-ρj22-1,gj4=gj4s,w=wρjσj.Proof.
(i) We first prove that the integrability condition guarantees the existence of the cumulant functionψt(s,w) in D. If (s,w)∈D and t∈[0,T), then (7)EQe-∫tTrτdτ+slnGt,T+wlnST=Pt,TEQTeslnGt,T+wlnST≤Pt,TEQT1T-t∫tTSudu+ST=1T-t∫tTPt,TPt,udu+1St<∞, where QT is the T- forward measure given by the Radon-Nikodym derivative: (dQT/dQ)FT=exp{-∫0Trudu}/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 lnST with the Brownian motions Wt1 and Wt2 expressed as Wt1=ρ1Zt1+1-ρ12Z~t1 and Wt2=ρ1Zt2+1-ρ22Z~t2, respectively, where (Zt1,Zt2,Z~t1,Z~t2) are 4-dimensional Brownian motion defined on the probability space. For the process, lnST, we have(8)lnST=lnSt+∑j=12-ρjκjθjσjT-t-ρjσjvjt+ρjκjσj+12∫tTvjτdτ+ρjσjvjT+1-ρj2∫tTvjτdZ~τj=lnSt+∑j=12θj2T-t+vjt2κj-vjT2κj+ρj+σj2κj∫tTvjτdZτj+1-ρj2∫tTvjτdZ~τj.On the other hand, we have (9)lnGt,T=T-tTlnSt+1T∑j=12θj4T-t2+vjt2κjT-t-12κj∫tTvjτdτ+ρj+σj2κj∫tTT-τvjτdZτj+1-ρj2∫tTT-τvjτdZ~τj.
Using the fact(10)∫tTT-τvjτdZτj=1σjκj∫tTT-τvjτdτ+∫tTvjτdτ-T-tvjt-κjθjT-t22, for j=1,2, then(11)lnGt,T=T-tTlnSt+1T∑j=12-ρjκjθj2σjT-t2-ρjvjtσjT-t+ρjσj∫tTvjτdτ+ρjκjσj+12∫tTT-τvjτdτ+1-ρj2∫tTT-τvjτdZ~τj.
LetG be the σ-field generated by Ft and {(Zu1,Zu2):t<u≤T}. By (8) and (11), for (s,w)∈D, we have(12)ψts,w=EQexp-∫tTrτdτ+slnGt,T+wlnST∣Ft=EQEQexp-∫tTrτdτ+slnGt,T+wlnST∣G∣Ft=A1EQA2EQA3∣G∣Ft,where(13)A1=expsT-tTlnSt-1T∑j=12ρjκjθj2σjT-t2+ρjT-tσjvjt+wlnSt-∑j=12ρjκjθjσjT-t+ρjσjvjt,A2=expsT∑j=122ρjκj+σj2σj∫tTT-τvjτdτ+ρjσj∫tTvjτdτ+∑j=12wρjvjTσj+2ρjκj+σj2σj∫tTvjτdτ-∫tTvjτdτ,A3=exp∑j=121-ρj2∫tTsTT-τ+wvjτdZ~τj.Since(14)EQA3∣G=exp12∑j=121-ρj2∫tTsTT-τ+w2vjτdτ=exp∑j=121-ρj22∫tTs2T2T-τ2+2swTT-τ+w2vjτdτ,substituting (14) into (12) and applying the Markov property of {vju:0≤u≤T},j=1,2, lead to (5), which completes the proof.From Proposition1, it is clear that we need to search for an exact formula for the discounted joint characteristic function of v1t and v2t and the three different integrals of vjt(j=1,2) appearing in (5). We use the same approach introduced by Kim and Wee [18] to obtain an explicit formula for the discounted joint characteristic function ψt(s,w). Therefore, two series of functions are introduced as follows. Define Fτj and F~τj:C4→C as(15)Fτjgj1,gj2,gj3,gj4=∑n=0∞fnj,(16)F~τjgj1,gj2,gj3,gj4=∑n=1∞nτfnj,where fnj,n=0,1,2,⋯, are functions of gj1, gj2, gj3, and gj4 defined as (17)f-2j=f-1j=0,f0j=1,f1j=κj-gj4σj2τ2,fnj=-σj2τ22nn-1gj1τ2fn-4j+gj2τfn-3j+gj3-κj22σj2fn-2j,n≥2,for j=1,2.Forj=1,2, denote(18)Djτ=gj1,gj2,gj3,gj4∈C4:EQexpRgj1∫0ττ-t2vjtdt+Rgj2∫0ττ-tvjtdt+Rgj3∫0τvjtdt+Rgj4vjτ<∞.Proposition 2.
(  1) The argument of Fτj: argFτj(gj1,gj2,gj3,gj4)≠0 for every (gj1,gj2,gj3,gj4)∈Djτ(j=1,2). In particular, argFτj is continuous on Djτ, and argFτj(gj1,gj2,gj3,gj4)=0 for gj1,gj2,gj3, and gj4 are all real numbers in Djτ for j=1,2.
(  2) For (gj1,gj2,gj3,gj4)∈Djτ,j=1,2, we have(19)EQexp∑j=12gj1∫0ττ-t2vjtdt+gj2∫0ττ-tvjtdt+gj3∫0τvjtdt+gj4vjτ=exp∑j=12κjvj0+κj2θjτσj-2vj0σj2F~τjgj1,gj2,gj3,gj4Fτjgj1,gj2,gj3,gj4-2κjθjσj2lnFτjgj1,gj2,gj3,gj4.
The proof of Proposition2 is similar to that of Kim and Wee [18].
Using Proposition2 to Proposition 1 leads to the explicit expression of ψt(s,w) in Proposition 3. To describe the simplicity of the result, we need to introduce new functions. Define Ht,Tj and H~t,Tj: C2→C as(20)Ht,Tjs,w=FT-tjgj1s,w,gj2s,w,gj3s,w,gj4s,w,(21)H~t,Tjs,w=F~T-tjgj1s,w,gj2s,w,gj3s,w,gj4s,w,with j=1,2.Proposition 3.
(  1) If (s,w)∈D, then Ht,Tj(s,w)≠0 and argHt,Tj is continuous on D. In particular, argHt,Tj(s,w)=0 if s and w are real numbers for j=1,2.
(  2) For ψt(s,w)∈D with argHt,Tj defined as above, then(22)ψts,w=exp∑j=12-cj1H~t,Tjs,wHt,Tjs,w-cj2lnHt,Tjs,w+cj3s+cj4w+cj5.Here cj1=2vjt/σj2,cj2=2κjθj/σj2, (23)cj3=T-t2TlnSt-ρjκjθjT-t22σjT-ρjT-tσjTvjt,cj4=lnSt2-ρjσjvjt-ρjκjθjT-tσj,cj5=κjvjt+κj2θjT-tσj2for 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)∈Dτj⊂DT-tj for every 0≤t≤T and j=1,2. Therefore, (19) remains valid for any vj0>0. The time homogeneous Markov property of vjt implies that (22) holds when Propositions 1 and 2 are satisfied.
Now Substituting the expressions ofgj1,gj2,gj3,gj4,j=1,2 into (15) and (16), Ht,Tj and H~t,Tj are expressed as follows, i.e., for (s,w)∈C2,(24)Ht,Tjs,w=∑n=0∞hnjs,w,(25)H~t,Tjs,w=∑n=1∞nT-thnjs,w,where(26)h-2js,w=h-1js,w=0,h0js,w=1,h1js,w=T-tκj-wρjσj2,hnjs,w=T-t24nn-1T2-s2σj21-ρj2T-t2hn-4js,w-sσjTσj+2ρjκj+2swσj2T1-ρj2T-thn-3js,w+Tκj2T-2sρjσj-w2ρjκj+σjσjT-w21-ρj2σj2T+2σj2Thn-2js,w,n≥2, for j=1,2.


## 4. Pricing Geometric Asian Option

Once the discounted joint characteristic function is found, the European continuously monitored geometric Asian option can be valued using numeraire change technique and the inverse Fourier transform approach which are applied in many research works (see, e.g., Geman et al. [41] and Deng [42]). This section derives the pricing formulas for the continuously monitored fixed and floating strike geometric Asian call options.Theorem 4.
Suppose thatSt,v1t, and v2t follow the dynamics in (1), then the price at time t∈[0,T] of the continuously monitored fixed strike geometric Asian call option with maturity T and the strike price K is given by(27)GACfit,S,v1,v2,K,T=e1/T∫0tlnSuduψt1,0Π1-KPt,TΠ2,where P(t,T) is given above in (3) and (28)Πj=12+1π∫0+∞Re-iulnKt,Tϕjuiudu,j=1,2,Kt,T=Ke-1/T∫0tlnSudu,ϕ1u=ψtiu+1,0ψt1,0,ϕ2u=ψtiu,0ψt0,0,where i is the imaginary unit (i2=-1).Proof.
Since(29)GACfit,S,v1,v2,K,T=EQe-∫tTruduG0,T-K+∣Ft=e1/T∫0tlnSuduEQe-∫tTruduGt,T1lnGt,T≥lnKt,T∣Ft-KEQe-∫tTrudu1lnGt,T≥lnKt,T∣Ft=e1/T∫0tlnSuduψt1,0Q1lnGt,T≥lnKt,T-KPt,TQTlnGt,T≥lnKt,T,where Q1 is defined by the following Radon-Nikodym derivative: (30)dQ1dQFT=e-∫tTruduGt,TEQe-∫tTruduGt,T∣Ft, and QT is the T- forward measure given above, it is well known that the probability distribution functions can be calculated by using the Fourier inversion transform, and then the above two probabilities Q1 and QT in (29) are given by(31)Q1lnGt,T≥lnKt,T=12+1π∫0+∞Re-iulnKt,Tϕ1uiudu≔Π1,(32)QTlnGt,T≥lnKt,T=12+1π∫0+∞Re-iulnKt,Tϕ2uiudu≔Π2,where(33)ϕ1u=EQ1eiulnGt,T∣Ft=EQe-∫tTrudu+iulnGt,TGt,T∣FtEQe-∫tTruduGt,T∣Ft=EQe-∫tTrudu+iu+1lnGt,T∣FtEQe-∫tTrudu+lnGt,T∣Ft=ψtiu+1,0ψt1,0,(34)ϕ2u=EQTeiulnGt,T∣Ft=EQe-∫tTrudu+iulnGt,T∣FtEQe-∫tTrudu∣Ft=ψtiu,0Pt,T. From (29), (31), and (32), we can obtain the required Theorem 4.Theorem 5.
Suppose thatSt,v1t, and v2t follow the dynamics in (1), then the price at time t∈[0,T] of the continuously monitored floating strike geometric Asian call option with maturity T is given by(35)GACflt,S,v1,v2,T=StΠ^1-e1/T∫0tlnSuduψt1,0Π^2,where (36)Π^1=12-1π∫0+∞Rψtiu,1-iueiu/T∫0tlnSuduiuStdu,Π^2=12-1π∫0+∞Rψtiu+1,-iueiu/T∫0tlnSuduiuψt1,0du.Proof.
Since(37)GACflt,S,v1,v2,T=EQe-∫tTruduST-G0,T+∣Ft=e1/T∫0tlnSuduEQe-∫tTruduSTe-1/T∫0tlnSudu-Gt,T+∣Ft=EQe-∫tTruduST1ln⁡Gt,T/ST≤-1/T∫0tlnSudu∣Ft-e1/T∫0tlnSuduEQe-∫tTruduGt,T1ln⁡Gt,T/ST≤-1/T∫0tlnSudu∣Ft=StQ2lnGt,TST≤-1T∫0tlnSudu-e1/T∫0tlnSuduψt1,0Q1lnGt,TST≤-1T∫0tlnSudu,where Q2 is defined by the following Radon-Nikodym derivative:(38)dQ2dQFT=e-∫tTruduSTSt,under the two probability measures Q1 and Q2, the conditional characteristic functions of ln(G[t,T]/ST) are given by(39)EQ1eiulnGt,T/ST∣Ft=EQe-∫tTrudu+iulnGt,T/STGt,T∣FtEQe-∫tTruduGt,T∣Ft=ψtiu+1,-iuψt1,0,(40)EQ2eiulnGt,T/ST∣Ft=EQe-∫tTrudu+iulnGt,T/ST+lnST∣FtSt=ψtiu,1-iuSt.Therefore (41)Q1lnGt,TST≤-1T∫0tlnSudu=12-1π∫0+∞Rψtiu+1,-iueiu/T∫0tlnSuduiuψt1,0du,(42)Q2lnGt,TST≤-1T∫0tlnSudu=12-1π∫0+∞Rψtiu,1-iueiu/T∫0tlnSuduSt·iudu.Plugging (41) and (42) in (37) yields (35).In addition to option prices, one can compute derivatives to hedge against changes in the underlying asset priceS and volatilities 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 valuation of the continuously monitored fixed strike geometric Asian call option using some numerical examples. To implement the analytic formula given in (27) numerically, we first determine the number of terms taken for the computation of the infinite series expansions for Ht,Tj and H~t,Tj for j=1,2 in (24) and (25). Second, we investigate the accuracy and efficiency of the approximated analytic formula given in (27). In the end, we compare the option prices varying by S0 and T under different models including the dbH-SI, H-SI (i.e., Heston stochastic volatility model with stochastic interest rate), dbH, and Heston models. Furthermore, we use this proposed model (1), i.e., the dbH-SI model, to examine the impacts of the model parameters on option prices. Here we implement the integral formulas given in (27) without truncation, modification, and approximation in Mathematica and Matlab software. Numerical integration was performed using the “NIntegrate()” or “quadgk()” commands which can handle infinite ranges, oscillatory integrands, and singularities in their default version, which employs automatic adaptive 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. The results are shown in Table 1.Table 1
G A C f i 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 0.294 15 17.7849 0.294   15 25.2254 0.299 20 17.7849 0.296   20 25.2251 0.302 25 17.7849 0.301   25 25.2251 0.307 30 17.7849 0.312   30 25.2251 0.315 100 5 9.7733 0.279  100 5 19.8642 0.276 10 9.8624 0.288   10 19.1846 0.285 15 9.8623 0.290   15 19.1758 0.290 20 9.8623 0.295   20 19.1755 0.302 25 9.8623 0.298   25 19.1755 0.309 30 9.8623 0.302   30 19.1755 0.311 110 5 4.2061 0.283  110 5 14.1405 0.283 10 4.5395 0.291   10 14.0976 0.291 15 4.5415 0.304   15 14.1003 0.297 20 4.5415 0.306   20 14.1002 0.304 25 4.5415 0.312   25 14.1002 0.310 30 4.5415 0.315   30 14.1002 0.315  1 90 5 22.3192 0.268 3 90 5 30.5831 0.289 10 22.0988 0.283   10 29.6250 0.293 15 22.0938 0.287   15 29.2870 0.299 20 22.0938 0.296   20 29.2835 0.305 25 22.0938 0.313   25 29.2835 0.314 30 22.0938 0.319   30 29.2835 0.317 100 5 15.0971 0.281  100 5 26.7286 0.268 10 15.1874 0.287   10 25.6231 0.284 15 15.1850 0.294   15 25.3192 0.291 20 15.1849 0.298   20 25.3151 0.295 25 15.1849 0.301   25 25.3151 0.298 30 15.1849 0.307   30 25.3151 0.307 110 5 9.2791 0.275  110 5 22.5318 0.281 10 9.7604 0.278   10 21.9684 0.284 15 9.7650 0.281   15 21.7105 0.297 20 9.7651 0.292   20 21.7064 0.301 25 9.7651 0.294   25 21.7064 0.305 30 9.7651 0.301   30 21.7064 0.310Table1 investigates the influence of the number of terms n taken in (24) and (25). It is shown that the numerical values tend to be quite stable as the number of terms increases. In particular, the option prices stay unchanged after taking n=20 and n=30 terms, at most. In addition, Table 1 provides option prices with various maturities T, strike prices K, and the required CPU times (in seconds). From Table 1, we can see that some characteristics of the GACfi option are similar to the plain European call option. For example, the values of the GACfi option are decreasing functions of the strike price. In addition, under the same parameter values setting in dbH-SI model, longer maturating date T will result in higher option values.It would be interesting to see the performance of our approximated analytic formula approach implemented to the fixed strike GAC option under dbH-SI model. Our numerical example uses the number of termsn=20 taken in (24) and (25); other parameter values are similar to those settings of Table 1. We compute the 0.5-,1.5-, and 3-year maturity GAC options by our approximated analytic formula approach and compare the results with Monte Carlo simulation with 10000 sample paths and 100 points for time axis. The numerical results are displayed in Table 2.Table 2
Comparison of the approximated approach and MC forGACfi 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 0.0436 100 9.8623 0.278 9.8602 110.403 0.0213 105 6.8598 0.296 6.8624 110.329 -0.3810 110 4.5415 0.287 4.5407 110.376 0.0176  1.5 90 25.2251 0.265 25.2206 110.376 0.0178 95 22.0851 0.281 22.0891 110.391 -0.0181 100 19.1755 0.294 19.1787 110.417 -0.0167 105 16.5109 0.312 16.5045 110.485 0.3888 110 14.1002 0.293 14.1039 110.392 -0.0262  3 90 29.2835 0.271 29.2894 110.428 -0.0201 95 27.2560 0.289 27.2513 110.397 0.0173 100 25.3151 0.307 25.3106 110.405 0.0178 105 23.4644 0.295 23.4595 110.471 0.0209 110 21.7065 0.292 21.7128 110.396 -0.0290Table2 shows that the approximated analytic formula approach is considerably faster than those of the Monte Carlo simulation (MC). For a given set of parameters, the approximated analytic formula approach calculates the option prices for 5 different strikes and 3 different maturates in approximately 5 seconds. The Monte Carlo simulation takes around 110 seconds for each option price. Table 2 also compares their pricing accuracy. It can be seen that the relative error (RE) in prices is less than 0.4% for all cases. If we regard the Monte Carlo price as the benchmark, then this numerical example confirms that the approximated analytic formula approach is accurate and efficient.After examining accuracy and efficiency. We shall turn to investigate the comparison of the option prices under different models. Figure1 compares the dbH-SI model with Heston, dbH, and H-SI models in pricing the fixed strike GAC option against the initial asset price S0 and maturating date T with the strike price K=100. From Figure 1, it is interesting to notice that the option prices of the fixed strike GAC option calculated from the dbH-SI model are much higher than those of the dbH model, H-SI model, and Heston model. One possible reason for this phenomena is that the volatilities 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 the dbH model, H-SI model, and Heston model on option prices. We can also clearly observe that the bigger the value of S0(or T) is, the higher the GACfi option price is.Figure 1
Comparison of option prices under Heston, dbH, H-SI, and dbH-SI models.
(a)
Option price againstS0 (b)
Option price againstTIn Figure2, we use the dbH-SI model to examine the effects of the mean-reverting rates (κ1,κ2) and volatilities (σ1,σ2) of variance processes (v1t,v2t) on the GACfi option prices (see Figures 2(a)–2(d)). We also examine the impacts of the correlation coefficients (ρ1,ρ2) (see Figures 2(e) and 2(f)), which determine both the correlation between the underlying asset and its volatilities, and the correlation between the underlying asset and the short interest rates.Figure 2
Option prices with respect toS0 and T in the dbH-SI model.
(a) (b) (c) (d) (e) (f)From Figures2(a) and 2(b), we can observe that the option prices increase as the mean-reverting rate increases. Particularly, it is quite remarkable that the mean-reverting rates have a significant effect on the longer term option values. The option price decreases as the volatilities of variance processes increase (see Figures 2(c) and 2(d)). The correlation coefficients (ρ1,ρ2) have several effects depending on the relation 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 holds with respect to the expiration date: a negative (ρ1,ρ2) tends to produce higher value for the longer term options and lower values for the shorter options (see Figure 2(f)).


## 6. Conclusions

The proposed model (the dbH-SI model) incorporates several important features of the underlying asset returns variability. We derive the discounted joint characteristic function of the log-asset price and its log-geometric mean value over time period[0,T] and obtain approximated analytic solutions to the continuously monitored fixed and floating strike geometric Asian call options using the change of numeraire technique and the Fourier inversion transform approach. Some numerical examples are provided to examine the effects of the proposed model, which reveals some additional features having a significant impact on option values, especially long-term options. The proposed model can be tested empirically by using the option price data from the option market.



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