# Extremum Options Pricing of Two Assets under a Double Nonaffine Stochastic Volatility Model

Authors: Shubin Ou; Guohe Deng
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

In this paper, we consider the pricing problem for the extremum options by constructing a double nonaffine stochastic volatility model. The joint characteristic function of the logarithm of two asset prices is derived by using the Feynman–Kac theorem and one-order Taylor approximation expansion. The semiclosed analytical pricing formulas of the European extremum options including option on maximum and option on minimum of two underlying assets are derived by using measure change technique and Fourier transform approach. Some numerical examples are provided to analyze the pricing results of extremum options under affine model, nonaffine model, Black–Scholes model, and the influences of some model parameters on the option. Numerical results show that the analytical pricing formulas have higher computational efficiency and accuracy than those of Monte Carlo simulation method. Also, results of sensitivity analysis report that the nonaffine models are more effective than other existing models in capturing the effect of volatility on option pricing.


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## Body

## 1. Introduction

Extremum options are a class of multiasset derivatives whose payoff depends on the maximum or minimum value of multiple underlying assets at maturity. Essentially, according to the payoff function, there are four categories of extremum option: (1) call option on maximum, (2) call option on minimum, (3) put option on maximum, and (4) put option on minimum. This kind of options is convenient hedging instruments when firms want to hedge away price uncertainties of at least two kinds of commodities. Extremum options are widely traded both on organized exchanges and over the counter in equity, fixed income, foreign currency, and commodity markets. They are extensively used to speculate, hedge correlation risks, and even evaluate real assets in corporate finance and bond pricing models (see, e.g., Kwok [1], Dempster et al. [2], and Luciano [3]). .European extremum options of two risky assets are first considered by Stulz [4] and the explicit pricing formula is obtained under the classical Black–Scholes model (BS model, see Black and Scholes [5]. Johnson and Stulz [6] extended the works of Stulz [4] to the European extremum options of several risky underlying assets. Wu and Zhang [7] investigated extremum options of two average prices, and working with two geometric averages, they obtain explicit pricing formulas for these options when underlying assets prices are described by correlated geometric Brownian motions. Ouwehand and West [8] considered the prices of options on the maximum or the minimum of two or several underlying assets and they also apply these to value exchange options. Later, Xu [9], Chen et al. [10], and Liu et al. [11] among others, investigated multiasset options including the extremum option under BS model.Although the BS model is still widely praised for its simplicity, it is a common knowledge that it has important shortcomings (for example, it fails to reflect the stochastic variability in the market parameters and to capture the “volatility smile,” sudden asset jumps, and the leverage effect) which has been proved by many empirical studies. In fact, there are many factors to affect option pricing, such as the return rates, volatility, and other factors of uncertainty. Therefore, there have been a considerable number of studies in the literature to improve the BS model. Among the most popular extensions on valuation model of the extremum option options include stochastic volatility (SV) and jump-diffusions (JD). Deng and Huang [12] and Wang [13] investigated the pricing issue of options on the maximum or the minimum of multiassets by incorporating correlated jump risk (i.e., JD model). Wang [14, 15] considered the extremum options pricing of two assets under Heston SV model (see Heston [16]) and GARCH model (see Wang [14]).Actually, Heston SV model has been considered one of the most popular substitutions in pricing financial derivatives (see e.g., Ma et al. [17], Kim et al. [18], Levendis and Eben [19], He and Lin [20], and He and Chen [21]). The Heston model assumes that the volatility of an underlying asset is not constant and allowed to follow a square root process, which can be correlated with the Brownian motion driving the asset price itself. So this model due to its specific assumption is one of the most significant option pricing approaches which can be applicable to model the behavior of financial derivatives (see Christoffersen et al. [22] and He and Chen [23]). Moreover, one of the most important long-observed features named the volatility smile can be presented under this model. However, it still has several drawbacks. One typical problem is that it does not provide sufficient flexibility for implied volatility surfaces when both short- and long-term options are calibrated simultaneously (see Chernov et al. [24] and Fiorentini et al. [25]). Another well-documented drawback in the Heston model is that it provides incorrect pricing formulas for far in- and out-of-the-money European options (see e.g., Da Fonseca et al. [26] and Kang and Kang [27]).These shortcomings of the Heston model implied that the dynamics of its volatility are not sufficiently described by a single factor. In order to devise a more realistic description of volatility dynamics, the number of factors that drives the volatility levels must be increased. Consequently, multifactor extensions of the Heston model have been introduced in several studies. Duffie et al. [28] and Christoffersen et al. [29] proposed two-factor or multifactor Heston model to solve the problems exposed by the single-factor Heston model. Zhang and Feng [30] derived American option pricing formula under the two-factor Heston model by using the asymptotic expansion method, and the results show that two-factor Heston model is superior to the Heston model in pricing short-term American options. Zhong and Deng [31] explored geometric Asian option pricing under double Heston model and stochastic interest rate, the numerical results show that the long-term stochastic volatility and short-term stochastic volatility have significant effects on option pricing. Deng [32] explored European option under two-factor stochastic volatility model and the numerical results show that the model has more flexibility in capturing the term structure of implied volatility. He and Lin [33] proposed a new three-factor stochastic volatility model, a closed solution for European option pricing was derived, and effective results were achieved.Although the multifactor stochastic volatility models offer several improvements over the single-factor Heston model in capturing the change characteristics of volatility, it still belongs to the class of affine structure. The fact that financial time series exhibit nonlinear characteristics (see Andersen et al. [34] and Chacko and Viceira [35]). By using S&P 500 stock index options, Kaeck and Alexander [36] found that the nonaffine SV model performs better than other affine model in describing the underlying asset price movement process. Huang and He [37] investigated European options pricing under a new two-factor nonaffine stochastic volatility model, they used Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices, empirical analysis revealed that the pricing performance of our proposed model has a significant effect. Recently, many researchers considered option pricing in the nonaffine SV model, for instance, He and Wei [38], Sun et al. [39], Liu et al. [40], and Huang and Guo [41]among others.In this paper, inspired by two-factor Heston model, we usedviγ/2i=1,2γ>0 to describe volatilities of volatility processes, the volatilities are divided into long-term and short-term volatility. In addition, we assume that the short interest rate is modeled by a linear sum of two fluctuation factors. Therefore, we extend the two-factor Heston model to double nonaffine stochastic volatility model within stochastic interest rates framework. To our knowledge, the literature on semiclosed analytical formulas for the extremum options under this model is rather scarce. Our study offers two main contributions. First, the price of the extremum options can be derived analytically in terms of inverse Fourier transforms. Specifically, first-order Taylor expansion is used to obtain the characteristic function of the underlying asset. Second, we contribute to the numerical aspect by providing the accuracy of the price method and comparison of some existing models including the BS model, the Heston model, and our proposed model.The remainder of this paper is organized as follows. Section2 describes the formulation of the double nonaffine stochastic volatility model. In Section 3, we obtain the closed-form solutions of European extremum options. Some numerical experiments are provided in Section 4. The conclusion is given in Section 5.


## 2. Model Formulation

Suppose that the financial market is a continuously traded, arbitrage-free, and friction-free economy, which consists of a risk-free bond and two risky assets (e.g., stocks) with a terminal horizonT=0,T, where T<∞. Let Ω,F,Ftt∈0,T,Q be a complete probability space with a filtration Ft and Q is a risk-neutral measure, where Ftt∈T is a filtration satisfying the usual conditions of right continuity and completeness. Uncertainties in this system are assumed to be in this probability space and the underlying assets S1t and S2t are expressed by the following dynamic systems:(1)dS1tS1t=rtdt+σs1v1tdWt1+σs1v2tdWt2,dS2tS2t=rtdt+σs2v1tdYt1+σs2v2tdYt2,dv1t=k1θ1−v1tdt+σv1v1tγ/2dZt1,dv2t=k2θ2−v2tdt+σv2v2tγ/2dZt2,where Wt1,Wt2,Yt1,Yt2,Zt1,andZt2 are the standard Brownian motions defined in the probability space. We assume that correlation coefficients between these Brownian motions are given by the following equation:(2)corr<dWt1,dYt1>=ρ1,corr<dWt2,dYt2>=ρ2,corr<dWt1,dZt1>=ρ3,corr<dWt2,dZt2>=ρ4,corr<dYt1,dZt1>=ρ5,corr<dYt2,dZt2>=ρ6.Other Brownian motions are assumed to be independent mutually. In model (1), v1t and v2t denote long term and short-term volatility processes, respectively. rt denotes risk-free interest rate such that rt=v1t+v2t, which designates the interest rate as a linear function of two-factor economic variables v1t and v2t and can offers the analytic tractability. θ1 and θ2 denote long-term mean and short-term mean of the two volatility processes, respectively. σs1,σs2,σv1, and σv2 denote volatilities of the variances, respectively. k1 and k2 denote the mean-reverting rates. ki,θi,σs1,σs2,σv1, and σv2 are assumed to be positive constants. Let Ftt∈T be generated by S1t,S2t,v1t, and v2t.In addition,γ≥0 is a nonaffine parameter. If γ≠1, model (1) is a nonaffine model. Model (1) contains some extant models as its special cases. If γ=1, the model is Heston [16] model. If γ=2, model (1) is the GARCH diffusion model by Nelson [42]. If γ=3, the model is the 3/2 nonaffine stochastic volatility model (see Drimus [43]), it is used to simulate the evolution of short-term interest rates by Ahn and Gao [44]. If γ=4, this model is the 4/2 stochastic volatility model proposed by Grasselli [45]. If γ=0, the model degenerates into an diffusion model (see Cox and Ross [46]).The payoff of European-style call option on the maximum (or minimum) of two assets with maturityT and strike price K is maxmaxS1T,S2T−K,0=maxS1T,S2T−K+ (or minS1T,S2T−K+). In stochastic volatility models in general, options can be valued using analytical formulas via the characteristic function of the underlying log-price. In the following, we obtain the joint characteristic function of the underlying log-prices X1t=lnS1t and X2t=lnS2t at time t. Let Ψt,x1,x2,v1,v2;u1,u2,u3,u4,T be the joint characteristic function given by the following equation:(3)Ψt,x1,x2,v1,v2;u1,u2,u3,u4,T=EQe−∫tTrsds+iu1X1T+iu2X2T+iu3v1T+iu4v2T|Ft=EQe−∫tTrsds+iu1X1T+iu2X2T+iu3v1T+iu4v2TX1t=x1,X2t=x2,v1t=v1,v2t=v2=EtQe−∫tTrsds+iu1X1T+iu2X2T+iu3v1T+iu4v2T,where i is an imaginary unit, u1,u2,u3,u4∈C,t∈0,T, and EtQ⋅ denotes expectation conditional on the filtration Ft under measure Q.Lemma 1.
Suppose that the dynamics ofSit and viti=1,2 follow model (1). The joint characteristic function defined in equation (3) has the following expression:(4)Ψt,x1,x2,v1,v2;u1,u2,u3,u4,T=expiu1x1+iu2x2+Aτ,u+Bτ,uv1+Cτ,uv2,where u=u1,u2,u3,u4∈C,τ=T−t.(5)Aτ,u=d34d12f1+h12τ−4f1lneτh1−q11−q1+4h1q1eτh1−1eτh1−q11−q1+d44d22f2+h22τ−4f2lneτh2−q21−q2+4h2q2eτh2−1eτh2−q21−q2+f32d1f1−h1τ−2ln1−q1e−τh11−q1+f42d2f2−h2τ−2ln1−q2e−τh21−q2,Bτ,u=12d1f1+h1−2h11−q1e−τh1,Cτ,u=12d2f2+h2−2h21−q2e−τh2,where(6)d1=12σv12γθ1γ−1,d2=12σv22γθ2γ−1,d3=1−γσv12θ1γ2,d4=1−γσv22θ2γ2,f1=f1u1,u2,0,0=1+γ2σs1σv1θ11−γ/2ρ3iu1+1+γ2σs2σv1θ11−γ/2ρ5iu2−k1,f2=f2u1,u2,0,0=1+γ2σs1σv2θ21−γ2ρ4iu1+1+γ2σs2σv2θ21−γ/2ρ6iu2−k2,f3=f3u1,u2,0,0=1−γ2σs1σv1θ11+γ/2ρ3iu1+1−γ2σs2σv1θ1γ+12ρ5iu2+k1θ1,f4=f4u1,u2,0,0=1−γ2σs1σv2θ2γ+12ρ4iu1+1−γ2σs2σv2θ2γ+12ρ6iu2+k2θ2,g1=g1u1,u2,0,0=−12σs12u1u1+i+12σs22u2u2+i+iu1+u2−σs1σs2ρ1u1u2−1,g2=g2u1,u2,0,0=−12σs12u1u1+i+12σs22u2u2+i+iu1+u2−σs1σs2ρ2u1u2−1,with(7)h1=h1u1,u2,0,0=f12u1,u2,0,0−4d1g1u1,u2,0,0,h2=h2u1,u2,0,0=f22u1,u2,0,0−4d2g2u1,u2,0,0,q1=q1u1,u2,u3,0=2d1iu3−f1+h12d1iu3−f1−h1,q2=q2u1,u2,0,u4=2d2iu4−f2+h22d2iu4−f2−h2.Proof.
See Appendix A.


## 3. Option Prices for Extremum Options

In this section, we solve the pricing of options on the maximum or the minimum of two asset prices. We first provide a closed-form solution for a call option on the maximum of two asset prices and then use this analytical solution with two explicit formulas for plain vanilla call option written on single asset to obtain the price of a closed-form solution for a call option on the minimum of two asset prices. To calculate the price of these call options in model (1), we apply the change of numeraire technique. We now state the main results for the Extremum call options of two risky assets.


### 3.1. Option on the Maximum of Two Asset Prices

Theorem 1.
Assume that the underlying assetS1t and S2t satisfy model (1), the price at time t∈0,T of call option on the maximum of two asset prices is as follows:(8)Cmax2S1t,S2t,t=S1tI1+S2tI2−KPt,TI˜1−KPt,TI˜2,where Pt,T denotes the price at time t∈0,T of a risk-free zero-coupon bond with maturity date T such that Pt,T=Ψt,x1,x2,v1,v2;0,0,0,0,T and R⋅ means taking the real part,(9)I1=14+12π∫0∞Re−iu1lnKϕ1u1,0iu1du1+12π∫0∞Rϕ10,u2iu2du2−12π2∫0∞∫0∞Re−iu1lnKϕ1u1,u2−eiu1lnKϕ1u1,−u2u1u2du1du2,(10)I˜1=14+12π∫0∞Re−iu1lnKϕ3u1,0iu1du1+12π∫0∞Rϕ30,u2iu2du2−12π2∫0∞∫0∞Re−iu1lnKϕ3u1,u2−eiu1lnKϕ3u1,−u2u1u2du1du2,(11)I2=14+12π∫0∞Rϕ2u1,0iu1du1+12π∫0∞Re−iu2lnKϕ20,u2iu2du2−12π2∫0∞∫0∞Re−iu2lnKϕ2u1,u2−eiu2lnKϕ2u1,−u2u1u2du1du2,and(12)I˜2=14+12π∫0∞Rϕ4u1,0iu1du1+12π∫0∞Re−iu2lnKϕ40,u2iu2du2−12π2∫0∞∫0∞Re−iu2lnKϕ4u1,u2−eiu2lnKϕ4u1,−u2u1u2du1du2,with the above characteristic functions defined by the following equations:(13)ϕ1u1,u2=Ψt,x1,x2,v1,v2;u1+u2−i,−u2,0,0,TΨt,x1,x2,v1,v2;−i,0,0,0,T,(14)ϕ2u1,u2=Ψt,x1,x2,v1,v2;−u1,u1+u2−i,0,0,TΨt,x1,x2,v1,v2;0,−i,0,0,T,(15)ϕ3u1,u2=Ψt,x1,x2,v1,v2;u1+u2,−u2,0,0,TΨt,x1,x2,v1,v2;0,0,0,0,T,(16)ϕ4u1,u2=Ψt,x1,x2,v1,v2;−u1,u1+u2,0,0,TΨt,x1,x2,v1,v2;0,0,0,0,T.Proof.
The proof can be found in Appendix B.


### 3.2. Option on the Minimum of Two Asset Prices

Theorem 2.
Assume that the underlying assetS1t and S2t satisfy model (1), the price at time t∈0,T of call option on the minimum of two asset prices is as follows:(17)Cmin2S1t,S2t,t=S1tΠ1−I1+S2tΠ2−I2−KPt,TI˜1+I˜2−Π˜1−Π˜2,where(18)Π1=12+1π∫0∞Rψ1u1,0e−iu1lnKiu1du1,Π˜1=12+1π∫0∞Rψ3u1,0e−iu1lnKiu1du1,Π2=12+1π∫0∞Rψ20,u2e−iu2lnKiu2du2,Π˜2=12+1π∫0∞Rψ40,u2e−iu2lnKiu2du2,with the characteristic function given by the following equation:(19)ψ1u1,0=Ψt,x1,0,v1,v2;u1−i,0,0,0,TΨt,x1,0,v1,v2;−i,0,0,0,T,ψ20,u2=Ψt,0,x2,v1,v2;0,u2−i,0,0,TΨt,0,x2,v1,v2;0,−i,0,0,T,ψ3u1,0=Ψt,0,x2,v1,v2;u1,0,0,0,TΨt,0,x2,v1,v2;0,0,0,0,T,ψ40,u2=Ψt,0,x2,v1,v2;0,u2,0,0,TΨt,0,x2,v1,v2;0,0,0,0,T.Proof.
The proof can be found in Appendix C.


## 3.1. Option on the Maximum of Two Asset Prices

Theorem 1.
Assume that the underlying assetS1t and S2t satisfy model (1), the price at time t∈0,T of call option on the maximum of two asset prices is as follows:(8)Cmax2S1t,S2t,t=S1tI1+S2tI2−KPt,TI˜1−KPt,TI˜2,where Pt,T denotes the price at time t∈0,T of a risk-free zero-coupon bond with maturity date T such that Pt,T=Ψt,x1,x2,v1,v2;0,0,0,0,T and R⋅ means taking the real part,(9)I1=14+12π∫0∞Re−iu1lnKϕ1u1,0iu1du1+12π∫0∞Rϕ10,u2iu2du2−12π2∫0∞∫0∞Re−iu1lnKϕ1u1,u2−eiu1lnKϕ1u1,−u2u1u2du1du2,(10)I˜1=14+12π∫0∞Re−iu1lnKϕ3u1,0iu1du1+12π∫0∞Rϕ30,u2iu2du2−12π2∫0∞∫0∞Re−iu1lnKϕ3u1,u2−eiu1lnKϕ3u1,−u2u1u2du1du2,(11)I2=14+12π∫0∞Rϕ2u1,0iu1du1+12π∫0∞Re−iu2lnKϕ20,u2iu2du2−12π2∫0∞∫0∞Re−iu2lnKϕ2u1,u2−eiu2lnKϕ2u1,−u2u1u2du1du2,and(12)I˜2=14+12π∫0∞Rϕ4u1,0iu1du1+12π∫0∞Re−iu2lnKϕ40,u2iu2du2−12π2∫0∞∫0∞Re−iu2lnKϕ4u1,u2−eiu2lnKϕ4u1,−u2u1u2du1du2,with the above characteristic functions defined by the following equations:(13)ϕ1u1,u2=Ψt,x1,x2,v1,v2;u1+u2−i,−u2,0,0,TΨt,x1,x2,v1,v2;−i,0,0,0,T,(14)ϕ2u1,u2=Ψt,x1,x2,v1,v2;−u1,u1+u2−i,0,0,TΨt,x1,x2,v1,v2;0,−i,0,0,T,(15)ϕ3u1,u2=Ψt,x1,x2,v1,v2;u1+u2,−u2,0,0,TΨt,x1,x2,v1,v2;0,0,0,0,T,(16)ϕ4u1,u2=Ψt,x1,x2,v1,v2;−u1,u1+u2,0,0,TΨt,x1,x2,v1,v2;0,0,0,0,T.Proof.
The proof can be found in Appendix B.


## 3.2. Option on the Minimum of Two Asset Prices

Theorem 2.
Assume that the underlying assetS1t and S2t satisfy model (1), the price at time t∈0,T of call option on the minimum of two asset prices is as follows:(17)Cmin2S1t,S2t,t=S1tΠ1−I1+S2tΠ2−I2−KPt,TI˜1+I˜2−Π˜1−Π˜2,where(18)Π1=12+1π∫0∞Rψ1u1,0e−iu1lnKiu1du1,Π˜1=12+1π∫0∞Rψ3u1,0e−iu1lnKiu1du1,Π2=12+1π∫0∞Rψ20,u2e−iu2lnKiu2du2,Π˜2=12+1π∫0∞Rψ40,u2e−iu2lnKiu2du2,with the characteristic function given by the following equation:(19)ψ1u1,0=Ψt,x1,0,v1,v2;u1−i,0,0,0,TΨt,x1,0,v1,v2;−i,0,0,0,T,ψ20,u2=Ψt,0,x2,v1,v2;0,u2−i,0,0,TΨt,0,x2,v1,v2;0,−i,0,0,T,ψ3u1,0=Ψt,0,x2,v1,v2;u1,0,0,0,TΨt,0,x2,v1,v2;0,0,0,0,T,ψ40,u2=Ψt,0,x2,v1,v2;0,u2,0,0,TΨt,0,x2,v1,v2;0,0,0,0,T.Proof.
The proof can be found in Appendix C.


## 4. Numerical Experiments

In this section, we provide numerical experiments to show the performance of the new pricing formulas (8) and (17) for European-style call option on maximum or minimum of two assets under our proposed model (1). We first tests the accuracy and efficiency of the analytical formulas mentioned above for this call option, after which the option prices produced by our formulas will be compared with those obtained by some other well-known models. Finally, the impacts of some main parameters in model (1) on the option prices are also presented. The basic parameters of the model are shown in Table 1.Table 1
Basic parameter values for the numerical experiments.
Model parametersValueS10100S20120K90σs10.25σs20.25σv10.15σv20.1ρ10.4ρ20.4ρ30.5ρ40.5ρ50.5ρ60.5v100.01v200.01k14k24θ10.15θ20.15T1γ3


### 4.1. Comparison of Numerical Integral Prices with Monte Carlo Results

Since the analytical formulas (8) and (17) have 2-fold integrals with infinite integration ranges, we apply the numerical integration method to approximate it. In order to test the accuracy and efficiency of the approximation for our analytical formulas, the functions “quad()” and “dblquad()” in MATLAB are used for 2-dimensional numerical integration based on automatic adaptive rule. The article takes Monte Carlo (MC) simulations with reducing variance technique as benchmark. The results are shown in Table 2.Table 2
Comparison of analytical integration results with Monte Carlo results.
StrikeMaximum optionMinimum optionKIntegrationMCRE (%)CISDIntegrationMCRE (%)CISD9046.388146.38160.01(46.2987 and 46.4646)0.042326.458126.57290.43(26.5055 and 26.6402)0.03449542.598842.63330.08(42.5503 and 42.7164)0.042422.844722.91130.29(22.8441 and 22.9785)0.034310038.769338.90330.34(38.8202 and 38.9863)0.042419.406519.21441.00(19.1475 and 19.2812)0.034110534.958735.22370.75(35.1412 and 35.3061)0.042115.736815.60550.84(15.5412 and 15.6698)0.032811031.229831.53500.97(31.4523 and 31.6177)0.042212.597912.39230.94(12.3308 and 12.4538)0.031411527.599127.79950.72(27.7169 and 27.8823)0.04229.20869.33781.38(9.2812 and 9.3944)0.028912024.599124.09620.13(24.0141 and 24.1783)0.04196.78646.70131.27(6.6512 and 6.7513)0.025512520.617820.51910.22(20.4384 and 20.5998)0.04124.52324.60441.76(4.5615 and 4.6474)0.021913017.225017.06530.14(16.9872 and 17.1433)0.03982.99333.02040.90(2.9853 and 3.0554)0.017913513.860913.91240.37(13.8380 and 13.9868)0.03801.84331.81851.36(1.7911 and 1.8458)0.0140CPU(s)55.37485.75———57.45463.86———Note.RE=CInt−CMC/CMC∗100%.For MC simulation for calculating the option prices, we simulate 50000 sampling paths and take the time steps 252. The comparisons of accuracy and speed between the numerical integration and MC methods are reported in Table2, where the Monte Carlo results are also accompanied with a 95% confidence interval (CI) and standard deviation (SD).Table2 shows that the numerical integration results and Monte Carlo (MC) stimulation results are very close (i.e., RE is less than 1%), and there is no significant difference between the two methods, which implies the integral approximation method being the accurate one for pricing this options. In addition, it is observed that integral approximation method takes much less time to calculate each option price than that by the MC method. Consequently, the results using our analytical formulas are correct.


### 4.2. Comparison of Heston Model, Nonaffine Model, and the BS Model

We compare the option pricing difference between our model (1) and other models such as the BS model (see Johnson and Stulz [6]), and Heston model in Figures 1(a) and 1(b). The numerical results illustrate that call option prices on both maximum and minimum of two assets under the BS model are lower than those under the affine and nonaffine stochastic volatility models, which is due to the fact that the two models have stochastic volatility. On the other hand, it is observed that the option prices under double nonaffine stochastic volatility model (1) are lower than those of two-factor Heston stochastic volatility model. This is due to the nonaffine parameter γ>1 which has bigger volatility on the underlying asset.Figure 1
Comparison of Heston model, nonaffine model, and the BS model. (a) Maximum case. (b) Minimum case.
(a)(b)


### 4.3. Comparison between Single-Asset Call Option and Extremum Options

Figure2, we compare the results of single-asset call options corresponding to S1 and S2 in model (1) with call options on maximum and minimum of two assets. From Figure 2, it shows that the option prices of the call option on maximum of two assets are close to those of the single asset S1, while the option prices of the call option on minimum of two assets are close to those of the single asset S2. This is because option on maximum with maximum value of initial value of assets and option on minimum with minimum value of initial value of assets.Figure 2
Single-asset call option and extremum call options.


### 4.4. Sensitivity Analysis of Nonaffine Model

In the following Figures3–6, we investigated the influence of nonaffine parameter γ, volatilities σs1 and σs2, correlation coefficients ρ1 and ρ2, and the long-term means θ1 and θ2 in model (1) on option prices.Figure 3
The influence of nonaffine parameterγ. (a) Maximum case. (b) Minimum case.
(a)(b)Figure 4
The influence of volatilityσs1 and σs2. (a)σs1 case. (b)σs2 case.
(a)(b)Figure 5
The influence of coefficientsρ1 and ρ2. (a)ρ1 case. (b)ρ2 case.
(a)(b)Figure 6
The influence of long-term meansθ1 and θ2. (a)θ1 case. (b)θ2 case.
(a)(b)Figures3(a) and 3(b) illustrate that call option prices on both maximum and minimum of two assets are decreased with nonaffine parameter γ increasing.Figures4(a) and 4(b) show that both σs1 and σs2 have positive impact on option price on maximum of two assets, because the increase of volatility leads to the maximum value of assets increasing. On the contrary, both σs1 and σs2 have negative effect to option price on minimum of two assets. This is due to the fact that the increasing of volatility leads to the minimum value of assets decreasing.Figures5(a) and 5(b) show that call option price on maximum of two assets decrease as ρ1 and ρ2 both increase, while option price on minimum of two assets is opposite. Since the positive correlation between two assets reduces the difference of two assets which results to option prices on maximum or minimum of two assets are closed to single-asset option prices.Figures6(a) and 6(b) show that both θ1 and θ2 have positive impact on option prices on maximum or minimum of two assets. This is because the two volatilities have larger values as θ1 and θ2 both increase which will generate larger appreciation for the underlying assets.


## 4.1. Comparison of Numerical Integral Prices with Monte Carlo Results

Since the analytical formulas (8) and (17) have 2-fold integrals with infinite integration ranges, we apply the numerical integration method to approximate it. In order to test the accuracy and efficiency of the approximation for our analytical formulas, the functions “quad()” and “dblquad()” in MATLAB are used for 2-dimensional numerical integration based on automatic adaptive rule. The article takes Monte Carlo (MC) simulations with reducing variance technique as benchmark. The results are shown in Table 2.Table 2
Comparison of analytical integration results with Monte Carlo results.
StrikeMaximum optionMinimum optionKIntegrationMCRE (%)CISDIntegrationMCRE (%)CISD9046.388146.38160.01(46.2987 and 46.4646)0.042326.458126.57290.43(26.5055 and 26.6402)0.03449542.598842.63330.08(42.5503 and 42.7164)0.042422.844722.91130.29(22.8441 and 22.9785)0.034310038.769338.90330.34(38.8202 and 38.9863)0.042419.406519.21441.00(19.1475 and 19.2812)0.034110534.958735.22370.75(35.1412 and 35.3061)0.042115.736815.60550.84(15.5412 and 15.6698)0.032811031.229831.53500.97(31.4523 and 31.6177)0.042212.597912.39230.94(12.3308 and 12.4538)0.031411527.599127.79950.72(27.7169 and 27.8823)0.04229.20869.33781.38(9.2812 and 9.3944)0.028912024.599124.09620.13(24.0141 and 24.1783)0.04196.78646.70131.27(6.6512 and 6.7513)0.025512520.617820.51910.22(20.4384 and 20.5998)0.04124.52324.60441.76(4.5615 and 4.6474)0.021913017.225017.06530.14(16.9872 and 17.1433)0.03982.99333.02040.90(2.9853 and 3.0554)0.017913513.860913.91240.37(13.8380 and 13.9868)0.03801.84331.81851.36(1.7911 and 1.8458)0.0140CPU(s)55.37485.75———57.45463.86———Note.RE=CInt−CMC/CMC∗100%.For MC simulation for calculating the option prices, we simulate 50000 sampling paths and take the time steps 252. The comparisons of accuracy and speed between the numerical integration and MC methods are reported in Table2, where the Monte Carlo results are also accompanied with a 95% confidence interval (CI) and standard deviation (SD).Table2 shows that the numerical integration results and Monte Carlo (MC) stimulation results are very close (i.e., RE is less than 1%), and there is no significant difference between the two methods, which implies the integral approximation method being the accurate one for pricing this options. In addition, it is observed that integral approximation method takes much less time to calculate each option price than that by the MC method. Consequently, the results using our analytical formulas are correct.


## 4.2. Comparison of Heston Model, Nonaffine Model, and the BS Model

We compare the option pricing difference between our model (1) and other models such as the BS model (see Johnson and Stulz [6]), and Heston model in Figures 1(a) and 1(b). The numerical results illustrate that call option prices on both maximum and minimum of two assets under the BS model are lower than those under the affine and nonaffine stochastic volatility models, which is due to the fact that the two models have stochastic volatility. On the other hand, it is observed that the option prices under double nonaffine stochastic volatility model (1) are lower than those of two-factor Heston stochastic volatility model. This is due to the nonaffine parameter γ>1 which has bigger volatility on the underlying asset.Figure 1
Comparison of Heston model, nonaffine model, and the BS model. (a) Maximum case. (b) Minimum case.
(a)(b)


## 4.3. Comparison between Single-Asset Call Option and Extremum Options

Figure2, we compare the results of single-asset call options corresponding to S1 and S2 in model (1) with call options on maximum and minimum of two assets. From Figure 2, it shows that the option prices of the call option on maximum of two assets are close to those of the single asset S1, while the option prices of the call option on minimum of two assets are close to those of the single asset S2. This is because option on maximum with maximum value of initial value of assets and option on minimum with minimum value of initial value of assets.Figure 2
Single-asset call option and extremum call options.


## 4.4. Sensitivity Analysis of Nonaffine Model

In the following Figures3–6, we investigated the influence of nonaffine parameter γ, volatilities σs1 and σs2, correlation coefficients ρ1 and ρ2, and the long-term means θ1 and θ2 in model (1) on option prices.Figure 3
The influence of nonaffine parameterγ. (a) Maximum case. (b) Minimum case.
(a)(b)Figure 4
The influence of volatilityσs1 and σs2. (a)σs1 case. (b)σs2 case.
(a)(b)Figure 5
The influence of coefficientsρ1 and ρ2. (a)ρ1 case. (b)ρ2 case.
(a)(b)Figure 6
The influence of long-term meansθ1 and θ2. (a)θ1 case. (b)θ2 case.
(a)(b)Figures3(a) and 3(b) illustrate that call option prices on both maximum and minimum of two assets are decreased with nonaffine parameter γ increasing.Figures4(a) and 4(b) show that both σs1 and σs2 have positive impact on option price on maximum of two assets, because the increase of volatility leads to the maximum value of assets increasing. On the contrary, both σs1 and σs2 have negative effect to option price on minimum of two assets. This is due to the fact that the increasing of volatility leads to the minimum value of assets decreasing.Figures5(a) and 5(b) show that call option price on maximum of two assets decrease as ρ1 and ρ2 both increase, while option price on minimum of two assets is opposite. Since the positive correlation between two assets reduces the difference of two assets which results to option prices on maximum or minimum of two assets are closed to single-asset option prices.Figures6(a) and 6(b) show that both θ1 and θ2 have positive impact on option prices on maximum or minimum of two assets. This is because the two volatilities have larger values as θ1 and θ2 both increase which will generate larger appreciation for the underlying assets.


## 5. Conclusion

In this article, we incorporate double nonaffine stochastic volatility factors and stochastic interest rate in the model for valuation of extremum option written on two underlying assets. By employing the technique of changing probability measures and Taylor expansion method, we obtained closed-form solutions for the pricing of extremum options in both maximum and minimum setting. Numerical experiments are performed to analyze the effects of model parameters on option prices. Our numerical results show that some model parameters have significant impacts on option prices.A further subject of research could be that of providing the formulas for the hedging coefficients, the “Greeks;” this can certainly be performed following the approach proposed in this paper. A pricing problem of options on the maximum or minimum of two average prices is also considered in the market model of the present paper. We leave these to the future research.



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*Source: 1165629-2023-02-01.xml*