# Option Pricing under Double Heston Model with Approximative Fractional Stochastic Volatility

Authors: Ying Chang; Yiming Wang; Sumei Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We establish double Heston model with approximative fractional stochastic volatility in this article. Since approximative fractional Brownian motion is a better choice compared with Brownian motion in financial studies, we introduce it to double Heston model by modeling the dynamics of the stock price and one factor of the variance with approximative fractional process and it is our contribution to the article. We use the technique of Radon–Nikodym derivative to obtain the semianalytical pricing formula for the call options and derive the characteristic functions. We do the calibration to estimate the parameters. The calibration demonstrates that the model provides the best performance among the three models. The numerical result demonstrates that the model has better performance than the double Heston model in fitting with the market implied volatilities for different maturities. The model has a better fit to the market implied volatilities for long-term options than for short-term options. We also examine the impact of the positive approximation factor and the long-memory parameter on the call option prices.


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## Body

## 1. Introduction

We propose a double Heston model with approximative fractional stochastic volatility in this article. Though the Black and Scholes [1] model has been a benchmark in European option pricing studies, the existence of the volatility smile made the stochastic volatility model more proper for application. The Hull and White [2] model, the Scott [3] model, the Wiggins [4] model, the Stein and Stein [5] model, and the Lewis [6] model are developed, but most of these stochastic volatility models are not proper for application. The Heston [7] model which is determined by a Cox–Ingersoll–Ross (CIR) [8] process displays the property of mean reversion effect and the character of stationarity. It is the pioneering work in the history of stochastic volatility model studies since Heston technically derived the closed-form formula for the option prices by using characteristic functions. Bakshi et al. [9] empirically showed that stochastic volatility model has better performance than the Black–Scholes [1] model. However, the one-factor stochastic volatility model has some restrictions on exploring the correlation between the volatility and the smile. For instance, one-factor stochastic volatility model cannot explain some specific phenomena that the figure of the smile is unconnected with the volatility. Christoffersen et al. [10] developed a double Heston model for exploring the correlation between the volatility and the smile. Their study showed that a two-factor model can explain the correlation between the volatility and the figure of the smile and has more flexibleness to present the volatility term structure. One factor having greater mean-reverting level explains the correlation between short-period returns and the square of volatility. The other one having smaller mean-reverting level explains the correlation between long-period returns and the square of volatility.The financial market owns the properties of self-similarity and long-range dependence. One should consider it in financial studies. Fractional Brownian motion is centered Gaussian process and it also owns the two properties. Therefore, compared with Brownian motion, it is a better financial tool. Hu and Øksendal [11] developed Wick products to analyze it because it is not a Markov process, or semimartingale, and therefore, the traditional Itô theory is not appropriate for the application of it. Its biggest disadvantage is that it cannot be economically explained very well [12]. If some specific conditions hold, Brownian motion can be expressed equivalently by using the mixed fractional Brownian motion [13], and therefore, one can use the latter for financial application [14–16]. Approximative fractional Brownian motion can also be used to replace fractional Brownian motion [17]. Two terms multiplied by a function can be used to express fractional Brownian motion; one can only concern the second term since it has long-range dependence and the first term has continuous trajectories [17]. Approximative fractional Brownian motion is the approximation of the second term and it is a semimartingale [17]. Thao [17] has proved that when the dynamics of the stock price is driven by approximative fractional Brownian motion, it has no arbitrage opportunity. Several papers have used it as an efficient tool in financial modeling [18, 19]. Several papers presented empirical studies on option pricing models with approximative fractional Brownian motion. Mrázek et al. [20] developed an approximative fractional stochastic volatility model for option pricing application. They applied the Lewis formula [6] for obtaining the semi-closed-form pricing formula and the fundamental transform and used the optimization approach for calibration. They considered two errors and five weights for calibration. Pospíšil and Sobotka [21] also developed an approximative fractional stochastic volatility model. They used the Heston approach [7] for obtaining the semi-closed-form pricing formula and the characteristic functions and used the optimization approach for calibration. They considered three errors and three weights for calibration. Several authors also developed an option pricing model with approximative fractional Brownian motion from a different point of view. Kang et al. [22] developed an approximative fractional stochastic volatility model to study foreign exchange (FX) options. They derived the pricing formula and used some numerical approach to examine the impact of the long-memory parameter and the positive approximation factor on the option prices.There are two typical approaches to derive the formulae for the characteristic functions. The first approach is developed by Duffie et al. [23]. They expended the Heston [7] model on the multifactor models which are called the affine jump-diffusion (AJD) models. Using the Feynman–Kac theorem, the closed form of characteristic functions can be derived. It is generally used to derive the characteristic functions for obtaining the pricing formula for the options. The advantage of the approach is that the solutions for the characteristic functions can be obtained when some specific conditions are met. The second one is introduced by Schöbel and Zhu [24] and developed by Zhu [25]. Schöbel and Zhu [24] used the expectation approach to derive the characteristic functions, and Zhu [25] extended it on the multifactor models and used the modular approach to derive the characteristic functions. Specifically, the formulae for the characteristic functions can be expressed as a structure consisting of several modules, and the formulae for the modules are derived by the expectation approach. The advantage of this approach is that when the option pricing models are determined by several stochastic factors, one can obtain the characteristic functions by integrating the corresponding parts of the formulae for the characteristic functions to be a unified structure. Ahlip et al. [26] proposed a double Heston hybrid model with jumps. They used expectation approach and martingale approach to obtain the pricing formula for the options. Meanwhile, another approach can also be used in several papers. For instance, He and Zhu [27] developed an option pricing model; they expressed the pricing formula for the options with the forward characteristic function. To obtain the formula for the options and corresponding characteristic function, they changed the dynamics of their model to the forward measure.The loss function approach is generally used for the estimation of the parameters and there are two classes of the loss functions [28]. The first class is minimizing the error between the quoted and model prices. There are two approaches generally used in the first class. The first one is to minimize the absolute value of the mean-squared error (MSE) between the quoted and model prices. Since this approach puts too much weight on long-term and in-the-money options, some authors prefer the second approach for estimating parameters [29]. The second one is to minimize the relative value of the mean-squared error (RVMSE) between the quoted and model prices. The second class is minimizing the error between the quoted and model implied volatilities. If one considers the denominator to be the weight attached to the loss function, there are several choices one can decide to be the weight. Christoffersen et al. [10] developed a loss function that they used the Black–Scholes sensitivity as the weight. The difference between the market prices of the bid and ask also can be taken as the weight [21]. Though there is no concern that which approach is better, the approach used for estimation and evaluation should be the same [29].We develop double Heston model with approximative fractional stochastic volatility in this article. It is an extensive study based on the study of Ahlip et al. [26] and Kang et al. [22]. Our contribution is that we introduce approximative fractional Brownian motion to double Heston model by modeling the dynamics of the stock price and one factor of the variance with approximative fractional process and estimate the parameters under this model. We do some pioneering work to study this innovative model. We derive the semianalytical pricing formula. We do calibration to estimate the parameters and examine the impact of the long-memory parameter and the positive approximation factor on the call option prices.We form the article as follows. We develop the model and then derive the semianalytical pricing formula in Section2. We derive the characteristic functions in Section 3. We estimate the parameters and give some numerical study in Section 4. The conclusion is presented in Section 5.


## 2. The Double Heston Model with Approximative Fractional Stochastic Volatility

LetΩ,ℱ,ℙ be a probability space; F=ℱtt∈0,T is the filtration and ℙ is the risk-neutral probability measure. The stock price process S=Stt∈0,T and the variance processes v=vtt∈0,T and v^=v^tt∈0,T are given by the following dynamic system:(1)dStSt=rdt+σsdWtS+ξvtdB˜tε,H+v^tdW^tS,dvt=θ−κvtdt+σvvtdB˜tε,H,dvt=θ^−κ^v^tdt+σv^v^tdW^tv.First, we give some preliminaries about fractional Brownian motionBtHt≥0 and the approximation of it. It is a Gaussian process including Hurst index H∈0,1. It has zero mean and the covariance equation EBtHBsH=1/2t2H+s2H−t−s2H. For H=1/2, it is a standard Brownian motion. We usually assume H∈1/2,1 for financial application since it displays long-range dependence in that case [19]. It can be decomposed as(2)BtH=1ΓH+1/2Zt+∫0tt−sH−1/2dWs,where Zt=∫−∞0t−sH−1/2−−sH−1/2dWs, Wt denotes a standard Brownian motion, and Γ denotes the gamma function. Since Zt has continuous trajectories, we can just consider the second term in the bracket which is B˜t=∫0tt−sH−1/2dWs and it has long-range dependence property [17, 19–21]. B˜t can be approximated by approximative fractional Brownian motion B˜tε,H which is given by [17](3)B˜tε,H=∫0tt−s+εH−1/2dWs.B˜tε,H⟶L2ΩB˜t as ε⟶0+. B˜tε,H is semimartingale [17]. The dynamics of it can be given by [17](4)dB˜tε,H=H−12ψtdt+εH−1/2dWtv,where H represents a long-memory parameter, we consider H∈1/2,1 since we only concern the long-range dependence property [20, 21], ε is an approximation factor, and ψt is a stochastic process given by(5)ψt=∫0tt−s+εH−3/2dWsψ.Some assumptions are given by the following statements:(1)
W^S=W^tSt∈0,T and W^v=W^tvt∈0,T are correlated Brownian motions that dW^S,W^vt=ρ^dt, ρ^∈−1,1(2)
WS=WtSt∈0,T, Wv=Wtvt∈0,T, and Wψ=Wtψt∈0,T are mutually independent Brownian motionsThe call option price formula at timet∈0,T with strike K under ℙ is given by(6)CtT,K=Etℙe−rT−tCTT,K=Etℙe−rT−tST−K+=e−rT−tEtℙST1ST≻K−Ke−rT−tEtℙ1ST≻K.The forward price for the stockFt,T at time t can be defined by Ft,T:=St/e−rT−t, and then, we can have that ST=FT,T, and CtT,K can be expressed as(7)CtT,K=e−rT−tEtℙFT,T1ST≻K−Ke−rT−tEtℙ1ST≻K.Lemma 1.
The dynamics ofFt,T under ℙ can be given by the following equation:(8)FT,T=Ft,Texp∫tTσ˜Fu,TdW˜u−12∫tTσ˜Fu,T2du,where σ˜Fu,Tt∈0,T and W˜=W˜tt∈0,T are ℝ3-valued processes given by(9)σ˜Fu,T=σS,ξεH−1/2vt,v^t,dW˜t=dWtS+1σSξH−12ψtvtdt,dWtv,dW^tS∗.Proof.
We can obtain the following equation using Itô’s lemma:(10)dFt,T=Ft,TσSdWtS+ξH−12ψtvtdt+ξεH−1/2vtdWtv+v^tdW^tS.
We can obtain the expression for the dynamics ofFt,T by applying Itô’s lemma, and the proof is complete.
Therefore, we can obtain the following equation:(11)e−rT−tEtℙFT,T1sT≻K=StEtℙexp∫tTσ˜Fu,TdW˜u−12∫tTσ˜Fu,T2du1sT≻K.Definition 1.
We define the forward measureℙ^T which is equivalent to ℙ on Ω,ℱT:(12)η^t=dℙ^Tdℙ|ℱt=exp∫0tσ˜Fu,T,dW˜u−12∫0tσ˜Fu,T2du.
Therefore, the formula for the call option price can be expressed as follows:(13)CtT,K=StEtℙ^T1ST≻K−Ke−rT−tEtℙ1ST≻K=Stℙ^TST≻K|St,vt,v^t−Ke−rT−tℙST≻K|St,vt,v^t=Stℙ^TxT≻lnK|St,vt,v^t−Ke−rT−tℙxT≻lnK|St,vt,v^t=StP1t,St,vt,v^t,K−Ke−rT−tP2t,St,vt,v^t,K,where xt=lnFt,T.(14)P1t,St,vt,v^t,K=ℙ^TxT≻lnK|St,vt,v^t=12+1π∫0∞Ree−iϕlnKf1ϕiϕdϕ,P2t,St,vt,v^t,K=ℙxT≻lnK|St,vt,v^t=12+1π∫0∞Ree−iϕlnKf2ϕiϕdϕ,where f1ϕ:f1ϕ,t,St,vt,v^t=Etℙ^TexpiϕxT and f2ϕ:f2ϕ,t,St,vt,v^t=EtℙexpiϕxT, and f1ϕ and f1ϕ are the characteristic functions.
Therefore, we only need to derive the formulae for the two characteristic functions for obtaining the formula for the call option price.


## 3. The Derivation of the Characteristic Functions

We derive the formulae for the characteristic functions in this section.Theorem 1.
Suppose that the stock price has the dynamics given by the model, then the formulae for the characteristic functionsf1ϕ and f2ϕ can be expressed as follows:(15)f1ϕ=ctexpαβσS2τ2−αξσvvt+θτ−αρ^σv^v^t+θ^τ×Etℙexp−s1vT−s2∫tTvudu−s3v^T−s4∫tTv^udu,(16)f2ϕ=ctexp−βγσS2τ2−βξσvvt+θτ−βρ^σv^v^t+θ^τ×Etℙexp−q1vT−q2∫tTvudu−q3v^T−q4∫tTv^udu,where τ=T−t, ct=expiϕxt, α=1+iϕ, β=iϕ, and γ=1−iϕ. s1, s2, s3, and s4 are given by s1=−αξ/σv, s2=−αξκ/σv−ξε2H−1/2, s3=−αρ^/σv^, and s4=−α21−ρ^2/2+ακ^ρ^/σv^−α/2. q1, q2, q3, and q4 are given by q1=−βξ/σv, q2=−βξκ/σv−ξε2H−1/2, q3=−βρ^/σv^, and q4=−β21−ρ^2/2+βκ^ρ^/σv^−β/2.Proof.
We can obtain the following equation:(17)f1ϕ=Etℙ^TexpiϕxT=EtℙexpiϕxTexp∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+∫tTv^udW^uS−12∫tTσS2+ξ2ε2H−1vu+v^udu.
Therefore, we can obtain that(18)f1ϕ=Etℙexpiϕxt+∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+∫tTv^udW^uS×exp−iϕ2∫tTσS2+ξ2ε2H−1vu+v^udu×exp∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+∫tTv^udW^uS×exp−12∫tTσS2+ξ2ε2H−1vu+v^udu.
We insertct=expiϕxt, α=1+iϕ, and β=iϕ into the above equation, and then we can obtain that(19)f1ϕ=ctEtℙexpα∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+∫tTv^udW^uS×exp−α2∫tTσS2+ξ2ε2H−1vu+v^udu.
We decomposeW^S into two parts:(20)W^tS=ρ^W^tv+1−ρ^2W^t,where W^=W^tt∈0,T is a Brownian motion under ℙ and not correlated with WS, W^S, Wv, W^v, and Wψ. Consequently, we can obtain that(21)f1ϕ=ctEtℙexpα∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+ρ^∫tTv^udW^uv+1−ρ^2∫tTv^udW^u×exp−α2∫tTσS2+ξ2ε2H−1vu+v^udu.
We denoteτ=T−t and we can obtain the following equations according to the dynamics of the model:(22)∫tTvudWuv=ε1/2−HσvvT−vt−θτ+κ∫tTvudu−∫tTH−12σvψuvudu,∫tTv^udW^uv=1σv^v^T−v^t−θ^τ+κ^∫tTv^udu.
We insert the above equations into the above expression forf1ϕ and rearrange it, and then we can obtain that(23)f1ϕ=ctEtℙexpαβσS2τ2−αξσvvt+θτ−αρ^σv^v^t+θ^τ×expαξσvvT+αξκσv−ξε2H−12∫tTvudu+αρ^σv^v^T+α21−ρ^22+ακ^ρ^σv^−α2∫tTv^udu.
We can rearrange it to obtain that(24)f1ϕ=ctexpαβσS2τ2−αξσvvt+θτ−αρ^σv^v^t+θ^τ×EtℙexpαξσvvT+αξκσv−ξε2H−12∫tTvudu+αρ^σv^v^T+α21−ρ^22+ακ^ρ^σv^−α2∫tTv^udu.
Therefore, we can obtain the following equation:(25)f1ϕ=ctexpαβσS2τ2−αξσvvt+θτ−αρ^σv^v^t+θ^τ×Etℙexp−s1vT−s2∫tTvudu−s3v^T−s4∫tTv^udu,where s1, s2, s3, and s4 are given by equation (15).
We can obtain the formula forf2ϕ by applying the same method used for deriving the formula for f1ϕ:(26)f2ϕ=EtℙexpiϕxT=Etℙexpiϕxt+∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+∫tTv^udW^uS×exp−iϕ2∫tTσS2+ξ2ε2H−1vu+v^udu.
Insertingβ=iϕ and decomposition of W^S into the above equation, then we can obtain that(27)f2ϕ=ctEtℙexpβ∫tTσSdWuS+∫tTξH−12ψuvudu+∫tTξεH−1/2vudWuv+ρ^∫tTv^udW^uv+1−ρ^2∫tTv^udW^u×exp−β2∫tTσS2+ξ2ε2H−1vu+v^udu.
We denoteγ=1−iϕ and rearrange it to obtain that(28)f2ϕ=ctEtℙexp−βγσS2τ2−βξσvvt+θτ−βρ^σv^v^t+θ^τ×expβξσvvT+βξκσv−ξε2H−12∫tTvudu+βρ^σv^v^T+β21−ρ^22+βκ^ρ^σv^−β2∫tTv^udu.
Moreover,(29)f2ϕ=ctexp−βγσS2τ2−βξσvvt+θτ−βρ^σv^v^t+θ^τ×EtℙβξσvvT+βξκσv−ξε2H−12∫tTvudu+βρ^σv^v^T+β21−ρ^22+βκ^ρ^σv^−β2∫tTv^udu.
Therefore, we can obtain the following equation:(30)f2ϕ=ctexp−βγσS2τ2−βξσvvt+θτ−βρ^σv^v^t+θ^τ×Etℙexp−q1vT−q2∫tTvudu−q3v^T−q4∫tTv^udu.where q1, q2, q3, and q4 are given by equation (16).
To complete the formulae forf1ϕ and f2ϕ, we present the calculation of the expectations in the formulae for f1ϕ and f2ϕ.Lemma 2.
If the processesv and v^ follow the dynamics in the model with independent Brownian motions Wv and W^v, if(31)Fτ,vt,v^t=Etℙexp−λ1vT−μ1∫tTvudu−λ2v^T−μ2∫tTv^udu,then the formula for it is given by(32)Fτ,vt,v^t=exp−G1τ,λ1,μ1vt−G2τ,λ2,μ2v^t−θH1τ,λ1,μ1−θ^H2τ,λ2,μ2,where(33)G1τ,λ1,μ1=λ1γ1+κ+eγ1τγ1−κ+2μ1eγ1τ−1ε2H−1σv2λ1eγ1τ−1+γ1−κ+eγ1τγ1+κ,G2τ,λ2,μ2=λ2γ2+κ^+eγ2τγ2−κ^+2μ2eγ2τ−1σv^2λ2eγ2τ−1+γ2−κ^+eγ2τγ2+κ^,H1τ,λ1,μ1=−2ε2H−1σv2ln2γ1eκ+γ1τ/2ε2H−1σv2λ1eγ1τ−1+γ1−κ+eγ1τγ1+κ,H2τ,λ2,μ2=−2σv^2ln2γ2eκ^+γ2τ/2σv^2λ2eγ2τ−1+γ2−κ^+eγ2τγ2+κ^,γ1=κ2+2ε2H−1σv2μ1,γ2=κ^2+2σv^2μ2.Proof.
Fort∈0,T, we denote(34)Mt=Fτ,vt,v^texp−μ1∫0tvudu−μ2∫0tv^udu.
It can also be expressed as(35)Mt=Etℙexp−λ1vT−μ1∫0Tvudu−λ2v^t−μ2∫0Tv^udu.
Mt is an F-martingale under ℙ [26, 30]. Fτ,v,v^ satisfies the following PIDE by applying Feynman–Kac theorem:(36)−∂F∂τ+θ−κv∂F∂v+θ^−κ^v^∂F∂v^+12ε2H−1σv2v∂2F∂v2+12σv^2v^∂2F∂v^2−μ1v+μ2v^F=0,with the boundary condition F0,v,v^=exp−λ1v−λ2v^; we recall that ψt is a martingale and ψ0=Eℙψt=0 is used in the above equation.
We conjecture that the solution is(37)Fτ,v,v^=exp−G1τ,λ1,μ1v−G2τ,λ2,μ2v^−θH1τ,λ1,μ1−θ^H2τ,λ2,μ2,with the boundary condition G10,λ1,μ1=λ1, G20,λ2,μ2=λ2,andH10,λ1,μ1=H20,λ2,μ2=0.
We rearrange equation (36) by instituting the above expression for conjecture into it(38)0=∂G1∂τv+∂G2∂τv^+θ∂H1∂τ+θ^∂H2∂τ−θ−κvG1−θ^−κ^v^G2+12ε2H−1σv2vG12+12σv^2v^G22−μ1v+μ2v^.
We can obtain the following ODEs:(39)∂G1∂τ=−12ε2H−1σv2G12−κG1+μ1,∂H1∂τ=G1,∂G2∂τ=−12σv^2G22−κ^G2+μ2,∂H2∂τ=G2.
We can obtain the formulae forG1τ,λ1,μ1, G2τ,λ2,μ2, H1τ,λ1,μ1, and H2τ,λ2,μ2 by solving the above ODEs. Therefore, the proof is complete.


## 4. Calibration

In this section, we use the loss function to estimate the parameters in the Heston model, double Heston model, and the model we developed and compare the performance among the three models. The market data offered on May 10, 2012, we use are the implied volatilities of the put options on Dow Jones Industrial Average ETF [28]. Since there are four maturities, we consider 0.001 as the risk-free interest rate. We set ε=1e−5. We need to estimate the following parameters:(40)Θ=κ,θ,σv,v0,κ^,θ^,σv^,v^0,ρ^,ε,H,σs.As we mentioned before, there are two approaches generally used for calibration, to minimize the absolute value of the mean-squared error (MSE) and the relative value of the mean-squared error (RVMSE) between the quoted and model prices. We consider the first approach for calibration. To be specific, first, we choose to use the bisection algorithm [28] to obtain the market prices since we only have the quoted implied volatilities data. Second, we use the market prices and the first approach to estimate the parameters. Suppose that the total number of the quotes is N, for a set number of NT maturities τt (t=1,…,NT), there are a set number of NK strikes Kkk=1,…,NK. For each τt,Kk, the corresponding market price is Ct,k and the model price is Ct,kΘ. We consider the following loss function:(41)MSE=1N∑t,kCt,k−Ct,kΘ2.Table1 shows the calibration results. It demonstrates that the model provides the smallest MSE and IVMSE. The estimation of the parameters also accords with the study based on Christoffersen et al. [10]. For both double Heston model and the model, one factor of the variance offers a higher mean-reverting level and the other factor of the variance offers a lower mean-reverting level. Since double Heston model offers better performance on MSE and IVMSE compared with the Heston model under the loss function we use, we plot the implied volatilities based on the former and the model using the estimated parameters, and the market implied volatilities for different maturities. Figure 1 shows the result. It demonstrates the model gives better performance than double Heston model on fitting with the market implied volatilities for different maturities. Compared with short-term options, the model has a better fit to the market implied volatilities on long-term options.Table 1
Calibration of the parameters.
HestonκHθHσHvH0ρHMSEIVMSE8.98140.04090.29700.0244−0.96210.023955.266e − 5Double HestonκD1θD1σD1vD10ρD110.75260.03300.36130.0252−0.89160.017434.928e − 5κD2θD2σD2vD20ρD20.94910.02570.05170.00030.7009The modelκθσvv0σs11.64901.08980.53460.53440.00370.002771.316e − 5κ^θ^σv^v^0ρ^0.16730.10680.99060.0316−0.4422ξH0.43160.7976Figure 1
Market and model implied volatilities for different maturities.
(a)(b)(c)(d)We also give some numerical results to examine the impact of the approximation factorε and the long-memory parameter H on the call option prices. We set the values of parameters as follows: S = 100, K = 100, r = 0.005, and ξ=0.5. The values of the other parameters are set in Table 2.Table 2
The values of the parameters.
ParametersValuesParametersValuesParametersValuesParametersValuesθ2.2κ1.5σv0.5v00.5θ^1.2κ^0.5σv^0.9v^00.25ρ^−0.5σs0.15Table3 shows the result. It demonstrates the impact of the parameters ε and H on the prices of the call options. When the value of ε is fixed, the call option price decreases as H increases. When the value of H is fixed, the call option price decreases as ε decreases.Table 3
The prices of the call options.
T1M2M3M6M9M1YεH1e−50.57.730411.513414.737923.040030.200836.56320.66.56339.800712.579319.825926.192531.94730.76.43459.611812.341419.473625.755531.44590.86.42159.592712.317419.438025.711331.39531e−60.57.730411.513414.737923.040030.200836.56320.66.51089.723712.482319.682326.014331.74270.76.42589.599012.325419.449825.725931.41200.86.42049.591112.315419.435025.707631.39101e−70.57.730411.513414.737923.040030.200836.56320.66.47759.674812.420819.591125.901131.61290.76.42239.593912.318919.440325.714231.39850.86.42019.590712.314919.434325.706731.3900


## 5. Conclusion

We developed a double Heston model with approximative fractional stochastic volatility in this article. It is an extensive study based on the study of Ahlip et al. [26] and Kang et al. [22]. Since approximative fractional Brownian motion can replace fractional Brownian motion, we introduced it to double Heston model by modeling the dynamics of the stock price and one factor of the variance with approximative fractional process and it is our contribution to this article. We did some pioneering work to study this innovative model. We derived the pricing formula for the options and used the expectation approach to derive the characteristic functions. We did the calibration to estimate the parameters using the market data. The calibration demonstrates that the model we developed provides the smallest MSE and IVMSE. In both double Heston model and the model, one factor of the variance has a higher mean-reverting level and the other factor of the variance has a lower mean-reverting level. The result accords with the study of Christoffersen et al. [10]. Since double Heston model offers better performance on MSE and IVMSE compared with the Heston model under the loss function we use, we plotted the implied volatilities based on the former and the model using the estimated parameters, and the market implied volatilities for different maturities. It demonstrates the model gives better performance than double Heston model on fitting with the market implied volatilities for different maturities. Compared with short-term options, the model has a better fit to the market implied volatilities on long-term options. We also examined the impact of the positive approximation factor and the long-memory parameter on the call option prices. The result demonstrates that when the value of the positive approximation factor is fixed, the call option price decreases as the long-memory parameter increases. When the value of the long-memory parameter is fixed, the call option price decreases as the positive approximation factor decreases.



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