Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model

Shou-de Huang; Xin-Jiang He; Shou-de Huang; Xin-Jiang He

doi:10.3934/math.2023243

AIMS Mathematics

2023, Volume 8, Issue 2: 4875-4891. doi: 10.3934/math.2023243

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Research article

Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model

Shou-de Huang ¹,
Xin-Jiang He ^{2
,
,}

1.
School of Mathematics and Computer Science, Anshun University, Guizhou, China
2.
School of Economics, Zhejiang University of Technology, Hangzhou, China

Received: 22 August 2022 Revised: 06 November 2022 Accepted: 16 November 2022 Published: 09 December 2022
MSC : 91G20

In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.

Keywords:

Citation: Shou-de Huang, Xin-Jiang He. Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model[J]. AIMS Mathematics, 2023, 8(2): 4875-4891. doi: 10.3934/math.2023243

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Abstract

1. Introduction

Although the Black-Scholes-Merton (BS) model ^[1,2] proposed in 1973 has attracted a lot of attention from theoretical researchers and financial practitioners because of its simplicity and tractability, the assumptions of this model are far away from reality. For example, the model assumes that the volatility of the underlying asset price is constant, which is inconsistent with the fact that the implied volatility extracted from real trading data often presents a "smile" or "smirk". Therefore, a number of researchers have improved the BS model by relaxing the model assumption that the volatility is constant. Among them, the stochastic volatility model has received a lot of attention.

Hull and White ^[3] first proposed the concept of stochastic volatility and used the Taylor expansion to obtain the pricing formula of European options. However, this model assumes a zero correlation between the asset price and volatility process, which contradicts the "leverage effect" demonstrated by Bakshi et al. ^[4]. Stein and Stein ^[5] derived an analytical pricing formula for European options by assuming that the underlying asset price follows the Ornstein-Uhlenbeck process. However, this model produces negative volatility values, which is clearly contrary to reality. Heston ^[6] made a breakthrough by proposing to use the Cox-Ingersoll-Ross (CIR) process to describe the dynamic process of the volatility since it satisfies a series of properties, including the non-negativity and mean reversion. Under this model, the analytical solution of European options can also be derived, so that model calibration can be carried out at a reasonable speed.

Furthermore, Christoffersen et al. ^[7] proposed the double Heston model and their empirical results demonstrated that the double Heston model was more flexible than the Heston model in establishing the volatility term structure and could provide a better fit to market option data than the Heston model. Moreover, the double Heston model, which consists of two unrelated processes, keeps the characteristics of the Heston model that it is easy to calculate, and it is possible to get analytical or semi-analytical solutions when pricing path-dependent options under the double Heston model. However, it should be remarked that the square root process of the Heston model ignores the nonlinear properties of option prices observed in the real market, which has prompted many researchers to improve the Heston stochastic volatility model in recent years ^[8,9,10]. In particular, Christoffersen et al. ^[11] and Chourdakis ^[12] found that the non-affine stochastic volatility model was better than other stochastic volatility models (including the Heston model) in describing the nonlinear characteristics observed from the trading data of the options market.

Motivated by the advantages of the non-affine stochastic volatility model as well as the double Heston model, we propose a two-factor non-affine stochastic volatility model for the price process of the underlying asset, and consider the pricing problem of European options under the newly proposed model. Due to the complicated model dynamics, it is not possible to derive an analytical solution to the characteristic function of the underlying log price, and we adopt the perturbation method to approximate the characteristic function, so that European options can be analytically evaluated with a Fourier cosine series through the COS method ^{[13,14,15,16,17]}. The economic implication of our proposed two-factor non-affine stochastic volatility model can be illustrated from two aspects; a) volatility smile or smirk can be better captured by multi-factor stochastic volatility models ^[7], and (b) non-affine stochastic volatility is able to not only describe the mean reversion characteristics of the time series of option price volatility, but also describe the nonlinear characteristics observed from the trading data of the options market.

The rest of this paper is organized as follows. A two-factor non-affine stochastic volatility pricing model is proposed in Section 2. In Section 3, an analytical pricing formula is derived using the Taylor expansion and COS method which can reduce the computational complexity. In Section 4, Some numerical examples and calibration analysis are provided to test our results, after which we conclude the paper.

2. Model specification

Let $\left\{\Omega, \mathcal{F}_{t}, \mathbb Q\right\}$ be a complete probability space with a filtration continuous on the right, where $\mathbb Q$ is a risk-neutral probability measure. The price process of the underlying asset, $S_{t}$ , and the processes of the volatility, $v_{1t}$ and $v_{2t}$ , are specified under $\mathbb Q$ as follows

$\begin{eqnarray} \frac {{\mathrm{d}}S_t}{S_t} & = &r {\mathrm{d}} t+\sqrt{v_{1t}} {\mathrm{d}} W_{1s}(t)+\sqrt{v_{2t}} {\mathrm{d}} W_{2s}(t), \end{eqnarray}$

(2.1)

$\begin{eqnarray} {\mathrm{d}} v_{1t} & = &\kappa_{1}\left(\theta_{1}-v_{1t}\right) {\mathrm{d}} t+\sigma_{1} {v_{1t}} {\mathrm{d}} W_{1v}(t), \end{eqnarray}$

(2.2)

$\begin{eqnarray} {\mathrm{d}} v_{2t} & = &\kappa_{2}\left(\theta_{2}-v_{2t}\right) {\mathrm{d}} t+\sigma_{2} {v_{2t}} {\mathrm{d}} W_{2v}(t), \end{eqnarray}$

(2.3)

where $\operatorname{Cov}\left({\mathrm{d}}W_{1s}(t), {\mathrm{d}}W_{1v}(t)\right) = \rho_1{\mathrm{d}} t$ and $\operatorname{Cov}\left({\mathrm{d}}W_{2s}(t), {\mathrm{d}}W_{2v}(t)\right) = \rho_2{\mathrm{d}} t$ . The pairs of $W_{1s}(t)$ and $W_{2s}(t)$ , $W_{1s}(t)$ and $W_{2v}(t)$ , as well as $W_{2s}(t)$ and $W_{1v}(t)$ , are uncorrelated. $k_1, \theta_1, \sigma_1$ and $k_2, \theta_2, \sigma_2$ represent the mean-reversion speed, long-term mean and instantaneous volatility of the volatility processes $v_{1t}$ and $v_{2t}$ , respectively.

If we make the transformation of $x_{t} = \ln({S_{t}/K})$ , Eqs (2.1)–(2.3) can be reformulated as

$\begin{eqnarray} {\mathrm{d}} x_t & = &(r-\frac{v_{1t}+v_{2t}}{2} ){\mathrm{d}} t+\sqrt{v_{1t}} {\mathrm{d}} W_{1s}(t)+\sqrt{v_{2t}} {\mathrm{d}} W_{2s}(t), \end{eqnarray}$

(2.4)

$\begin{eqnarray} {\mathrm{d}} v_{1t} & = &\kappa_{1}\left(\theta_{1}-v_{1t}\right) {\mathrm{d}} t+\sigma_1 v_{1t}\left(\rho_1 d W_{1s}(t)+\sqrt{\left(1-\rho_1^{2}\right)} d W^{\bot}_{1v}(t)\right), \end{eqnarray}$

(2.5)

$\begin{eqnarray} {\mathrm{d}} v_{2t} & = &\kappa_{2}\left(\theta_{2}-v_{2t}\right) {\mathrm{d}} t+\sigma_{2} {v_{2t}}\left(\rho_2 d W_{2s}(t)+\sqrt{\left(1-\rho_2^{2}\right)} d W^{\bot}_{2v}(t)\right). \end{eqnarray}$

(2.6)

3. The pricing of European options

In this section, we derive an approximation to the characteristic function of the underlying log-price, based on which an analytical formula for European option prices with the COS method is obtained.

3.1. An approximate characteristic function

It is well-known that if we are able to derive the characteristic function of the underlying log price, then it would be fairly straightforward to derive the European option pricing formula. With the definition of the characteristic function as

$\Phi(x, v_1, v_2, \tau; u) = E^{\mathbb Q}\left[e^{i u x_T} | x_{t} = x, v_{1t} = v_1,v_{2t} = v_2\right],$

where $T \geq t, \tau = T-t, i = \sqrt{-1}$ , its analytical approximation is presented in the following theorem.

Theorem 1. If we assume that the price process of the underlying asset and the volatility processes satisfy Eqs (2.1)–(2.3), the characteristic function of $x_{T}$ can be approximated by

$\begin{eqnarray} \Phi(x, v_1,v_2 ,\tau; u) = exp\left\{iux+A(u,\tau)+B_1(u,\tau)v_1+B_2(u,\tau)v_2\right\}, \end{eqnarray}$

(3.1)

where

$\begin{split} A\left(u,\tau\right)& = riu\tau-\frac{1}{4}(iu+u^2)(\theta_1+\theta_2)\tau-\frac{\alpha_{3}}{\alpha_{2}}\left[\beta_{1} \tau+\ln \left(\frac{-\beta_{2}+\beta_{1} e^{-\alpha \tau}}{\alpha}\right)\right] \\ &-\frac{\xi_{3}}{\xi_{2}}\left[\eta_{1} \tau+\ln \left(\frac{-\eta_{2}+\eta_{1} e^{-\xi \tau}}{\xi}\right)\right]-\frac{1}{2}\left( B_1(u, \tau)\theta_1+B_2(u, \tau)\theta_2\right),\\ B_1\left(u,\tau\right)& = \alpha_{0} \frac{1-e^{-\alpha \tau}}{-\beta_{2}+\beta_{1} e^{-\alpha \tau}}, B_2\left(u,\tau\right) = \alpha_{0} \frac{1-e^{-\xi \tau}}{-\eta_{2}+\eta_{1} e^{-\xi \tau}}, \end{split}$

with

$\begin{split} \alpha_{0}& = -\frac{1}{2}(iu+u^2), \; \alpha_{1} = \frac{3}{2} \theta_1^{\frac{1}{2}}\sigma_1\rho_1 iu-\kappa_1,\\ \alpha_{2}& = \sigma_1^{2}\theta_1,\beta_{1} = \frac{\alpha_{1} +\alpha}{2}, \; \beta_{2} = \frac{\alpha_{1} -\alpha}{2}, \; \eta_{2} = \frac{\xi_{1} -\xi}{2} ,\\ \alpha& = \sqrt{\alpha_{1}^{2}-4 \alpha_{0} \alpha_{2}}, \; \xi_{1} = \frac{3}{2} \theta_2^{\frac{1}{2}}\sigma_2\rho_2iu-\kappa_2, \; \xi_{2} = \sigma_2^{2}\theta_2,\eta_{1} = \frac{\xi_{1} +\xi}{2} ,\\ \xi& = \sqrt{\xi_{1}^{2}-4 \alpha_{0} \xi_{2}}, \; \alpha_{3} = \frac{1}{2} \theta_1 \kappa_1+\frac{1}{4}\sigma_1 \rho_1 iu\theta_1^{\frac{3}{2}}, \; \xi_{3} = \frac{1}{2} \theta_2 \kappa_2+\frac{1}{4}\sigma_2 \rho_2 iu\theta_2^{\frac{3}{2}}. \end{split}$

Proof. Applying the Feynman-Kac theorem yields the partial differential equation (PDE) governing $\Phi(x, v_1, v_2, \tau; u)$ as

$\begin{align} &-\frac{\partial \Phi}{\partial \tau}+\left(r-\frac{v_1+v_2}{2} \right) \frac{\partial \Phi}{\partial x}+ \frac{v_1+v_2}{2}\frac{\partial^{2}\Phi}{\partial x^{2}}+\sum\limits_{j = 1}^{2}\left(\kappa_j (\theta_{j} -v_{j} ) \frac{\partial \Phi}{\partial v_{j} } \right. \\&\left.+\frac{1}{2} \sigma_{j} ^{2} v_{j} ^{2} \frac{\partial \Phi^{2} }{\partial v_{j} ^{2}}+v_{j} ^{\frac{3}{2}} \sigma_{j} \rho_{j}\frac{\partial^{2} \Phi}{\partial x \partial v_{j} }\right) = 0, \end{align}$

(3.2)

with the boundary condition given by

$\Phi\left(x, v_1,v_2, 0 ; u\right) = e^{i u x}.$

As the above PDE is clearly nonlinear, it does not admit a closed-form solution, and thus we try to first linearize it. The idea is to approximate $v^{\frac{3}{2}}$ and $v^{2}$ in the PDE using the Taylor expansion around the long-term mean of the volatility as follows:

$\begin{eqnarray} v_j^{2}& {\approx} &2 \theta_j v_j-\theta_j^{2}, \end{eqnarray}$

(3.3)

$\begin{eqnarray} v_j^{\frac{3}{2}}&{\approx}&\frac{3}{2} \theta_j^{\frac{1}{2}} v_j-\frac{1}{2} \theta_j^{\frac{3}{2}} \end{eqnarray}$

(3.4)

where $j = 1, 2$ . Substituting Eqs (3.3) and (3.4) into (3.2) leads to

$\begin{align} &-\frac{\partial \Phi}{\partial \tau}+\left(r-\frac{v_1+v_2}{2} \right) \frac{\partial \Phi}{\partial x}+ \frac{v_1+v_2}{2}\frac{\partial^{2}\Phi}{\partial x^{2}}+\sum\limits_{j = 1}^{2}\left(\kappa_{j} (\theta_{j} -v_{j} ) \frac{\partial \Phi}{\partial v_{j} } \right. \\&\left.+\frac{1}{2} \sigma_{j} ^{2}(2\theta_j v_j-\theta_j^{2})\frac{\partial \Phi^{2} }{\partial v_{j} ^{2}}+\frac{\partial^{2} \Phi}{\partial x \partial v_{j} } (\frac{3}{2} \theta_j^{\frac{1}{2}} v_j-\frac{1}{2} \theta_j^{\frac{3}{2}} )\sigma_{j} \rho_{j}\right) = 0. \end{align}$

(3.5)

Following Duffie et al. ^[18], we now assume that the solution to PDE (3.5) takes the form of

$\begin{eqnarray} \Phi(x, v_1,v_2 ,\tau; u) = exp\left\{iux+A(u,\tau)+B_1(u,\tau)v_1+B_2(u,\tau)v_2\right\}, \end{eqnarray}$

(3.6)

with the boundary conditions

$A(u,0 ) = B_1(u, 0) = B_2(u, 0) = 0.$

The substitution of Eq (3.6) into (3.5) yields

$\begin{align} &-\left(\frac{\partial A}{\partial \tau}+\frac{\partial B_1}{\partial \tau}v_1+\frac{\partial B_2}{\partial \tau}v_2\right)+\left(r-\frac{v_1+v_2}{2} \right) iu+\frac{v_1+v_2}{2}{(iu)}^2 \\&+\sum\limits_{j = 1}^{2}\left(\kappa_{j} (\theta_{j} -v_{j} ) B_{j}+\frac{1}{2} \sigma_{j} ^{2} (2 \theta_j v_j-\theta_j^{2}) B^{2}_{j}+ (\frac{3}{2} \theta_j^{\frac{1}{2}} v_j-\frac{1}{2} \theta_j^{\frac{3}{2}}) \sigma_{j} \rho_{j}iuB_{j}\right) = 0. \end{align}$

(3.7)

By matching the coefficients, we can derive the following three ordinary differential equations (ODEs)

$\begin{eqnarray} \frac{\partial B_1}{\partial \tau}& = &\sigma_1^{2}\theta_1 B_1^2+\left(\frac{3}{2}\sigma_1\rho_1\theta_1^{\frac{1}{2}}iu-\kappa_1\right)B_1-\frac{1}{2}(iu+u^2), \end{eqnarray}$

(3.8)

$\begin{eqnarray} \frac{\partial B_2}{\partial \tau}& = &\sigma_2^{2}\theta_2 B_1^2+\left(\frac{3}{2}\sigma_2\rho_2 \theta_2^{\frac{1}{2}}iu-\kappa_2\right)B_2-\frac{1}{2}(iu+u^2), \end{eqnarray}$

(3.9)

$\begin{eqnarray} \frac{\partial A}{\partial \tau}& = &riu+\sum\limits_{j = 1}^{2}\left[\kappa_j\theta_j B_j -\frac{1}{2}\sigma_j^2\theta_j^2 B_j^2-\frac{1}{2} \sigma_j\rho_j \theta_j^{\frac{3}{2}}iuB_j\right]. \end{eqnarray}$

(3.10)

ODEs (3.8) and (3.9) are clearly Riccati equations, whose solutions can be respectively written as

$\begin{split} &B_1\left(u,\tau\right) = \alpha_{0} \frac{1-e^{-\alpha \tau}}{-\beta_{2}+\beta_{1} e^{-\alpha \tau}},\\ &B_2\left(u,\tau\right) = \alpha_{0} \frac{1-e^{-\xi \tau}}{-\eta_{2}+\eta_{1} e^{-\xi \tau}}. \end{split}$

Multiplying ${\theta_1}/{2}$ and ${\theta_2}/{2}$ respectively on both sides of Eqs (3.8) and (3.9), and substituting them into (3.10), we can obtain

$\frac{\partial A}{\partial \tau} = riu+\sum\limits_{j = 1}^{2}\left((\frac{1}{2} \theta_j \kappa_j+\frac{1}{4}\sigma_j \rho_j iu\theta_j^{\frac{3}{2}})B_j-\frac{1}{4}(iu+u^2) \theta_j\right)-\frac{1}{2}\left(\frac{\partial B_1}{\partial \tau}\theta_1+\frac{\partial B_2}{\partial \tau}\theta_2\right),$

integrating on both sides of which leads to the final expression of $A\left(\tau, u\right)$ . This completes the proof.

It should be pointed that once the analytic expression of the characteristic function has been obtained, the cumulants of $lnS_{T}$ can be computed, which will be used in the truncation of the computational domain of option pricing ^[13]. In particular, the $n$ -th cumulant of $lnS_{T}$ is given by

$c_{n} = \left.\frac{1}{i^{n}} \frac{\partial^{n}\left(\ln \Phi(u)\right)}{\partial u^{n}}\right|_{u = 0}.$

3.2. An analytical pricing formula

The risk-neutral pricing rule implies that European options can be evaluated through ^[19]

$\begin{eqnarray} P\left(x,v_1, v_2, t_{0}\right) = e^{-r \Delta t} \int_{-\infty}^{\infty} p(y, T) f(y | x, v_1, v_2) d y, \end{eqnarray}$

(3.11)

where $x = \ln \left(S_{0} / K\right)$ , $y = \ln \left(S_{T} / K\right)$ , $f(y|x, v_1, v_2)$ is the probability function of $y$ given $x, v_1, v_2$ , and $p(y, T)$ is the payoff function of a European option at maturity given by

$p(y, T) = g(y) = \left[\alpha K\left(e^{y}-1\right)\right]^{+}, \quad \alpha = \left\{\begin{array}{ll}{1,} & {\rm { for\; a \;call }} \\ {-1,} & {\rm { for\; a\; put. }}\end{array}\right.$

Although the expression of $f(y|x, v_1, v_2)$ is unknown, we can formulate it with a Fourier cosine series ^[10] as

$\begin{align} f(y|x, v_1, v_2) \approx \frac{2}{b-a} \sideset{}{'}\sum\limits_{k = 0}^{N-1}\operatorname{Re}\left\{\Phi\left(x, v_1,v_2 ,\tau; \frac{k \pi}{b-a} \right) e^{i k \pi \frac{x-a}{b-a}}\right\} \cos \left(k \pi \frac{y-a}{b-a}\right) \end{align}$

(3.12)

where $\sum^{\prime}$ means the first term of the summation is multiplied by ${1}/{2}$ , $\operatorname{Re}\{\cdot\}$ denotes taking the real part of a complex number, and $\Phi(x, v_1, v_2, \tau; u)$ is the characteristic function of $f(y|x, v_1, v_2)$ . $a, b$ are respectively the lower and upper bounds used for the integration interval in the Fourier cosine method.

Substituting Eq (3.12) into (3.11) and interchanging integration and summation, the approximate price of European options $P\left(x, v_1, v_2, t_{0}\right)$ can be derived as

$\hat{P}\left(x, v_1,v_2, t_{0}\right) = e^{-r \Delta t} \sideset{}{'}\sum\limits_{k = 0}^{N-1}\operatorname{Re}\left\{\Phi\left(x, v_1,v_2 ,\tau; \frac{k \pi}{b-a}\right) e^{i k \pi \frac{x-a}{b-a}}\right\}V_{k}$

where

$\begin{align} V_{k} = \frac{2}{b-a} \int_{a}^{b} p(y, T) \cos \left(k \pi \frac{y-a}{b-a}\right) d y. \end{align}$

This means that the remaining task is to calculate the coefficients $V_{k}$ . For a European call option, we denote

$\begin{equation} \begin{split} \chi_{k}\left(x_{1}, x_{2}\right) = &\int_{x_{1}}^{x_{2}} e^{x} \cos \left(k \pi \frac{x-a}{b-a}\right) d x \\ \\ = &\frac{1}{1+\left(\frac{k \pi}{b-a}\right)^{2}}\left[\cos \left(k \pi \frac{x_{2}-a}{b-a}\right) e^{x_{2}}-\cos \left(k \pi \frac{x_{1}-a}{b-a}\right) e^{x_{1}}\right.\\ \\&\left.+\frac{k \pi}{b-a} \sin \left(k \pi \frac{x_{2}-a}{b-a}\right) e^{x_2}+\frac{k \pi}{b-a} \sin \left(k \pi \frac{x_{1}-a}{b-a}\right) e^{x_{1}}\right] \end{split} \end{equation}$

(3.13)

and

$\begin{equation} \begin{split} \psi_{k}\left(x_{1}, x_{2}\right)& = \int_{x_{1}}^{x_{2}} \cos \left(k \pi \frac{x-a}{b-a}\right) d x\\ \\& = \left\{\begin{array}{ll} \frac{b-a}{k \pi}\left[\sin \left(k \pi \frac{x_{2}-a}{b-a}\right)-\sin \left(k \pi \frac{x_{1}-a}{b-a}\right)\right], & k \neq 0 \\ \\x_{2}-x_{1}, & k = 0 \end{array}\right., \end{split} \end{equation}$

(3.14)

so that $V_{k}$ can be calculated as:

$\begin{split} V_{k}^{c a l l}& = \frac{2}{b-a} \int_{0}^{b} K\left(e^{y}-1\right) \cos \left(k \pi \frac{y-a}{b-a}\right) d y\\ & = \frac{2}{b-a} K\left(\chi_{k}(0, b)-\psi_{k}(0, b)\right). \end{split}$

Similarly, the European put option price can be derived as:

$\begin{split} V_{k}^{put}& = \frac{2}{b-a} \int_{0}^{b} K\left(e^{y}-1\right) \cos \left(k \pi \frac{y-a}{b-a}\right) d y\\ & = \frac{2}{b-a} K\left(-\chi_{k}(a,0)+\psi_{k}(a,0)\right). \end{split}$

4. Numerical analysis

4.1. Accurate tests

In this subsection, some numerical examples are performed to show the accuracy of the newly derived formula by comparing the results produced from the formula and those obtained from Monte Carlo simulation. Without loss of generality, European call options will be used as an example to demonstrate this, and the integration interval [a, b] is chosen as ^[13]

$[a, b] = \left[c_{1}+a_{0}-L \sqrt{c_{2}+\sqrt{c_{4}}}, c_{1}+a_{0}+L \sqrt{c_{2}+\sqrt{c_{4}}}\right]$

with $a_{0} = \ln S_{0}, L = 10$ and $c_{n}$ being the $n$ -th cumulant of $lnS_{T}$ . The number of the sample paths and time steps for Monte Carlo simulation is $100,000$ and $200$ , respectively, and $N = 2^{10}$ . The default parameter values are listed in for all our numerical examples. The computer used in the experiments equips an Intel Core $i5$ CPU with a $1.6+2.1$ GHz processor. All of our numerical examples were performed with Matlab 2020a.

Table 1. Parameter values for the numerical experiments.

Parameter	$r$	$\theta_1$	$\theta_2$	$\rho_1$	$\rho_2$	$S_{0}$	$\sigma_1$	$\sigma_2$	$v_{20}$	$v_{10}$
value	0.05	0.05	0.08	-0.5	-0.5	100	0.09	0.09	0.02	0.05

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Tables 2 and 3 indicate that the approximation formula is fairly accurate, with the absolute relative error (Abs R.E.) between European option prices obtained from our formula and those from the Monte Carlo (MC) simulation across a wide range of strike prices and different maturities being less than 0.7%. On the other hand, one can clearly observe that our approach is significantly faster than the MC simulation. These demonstrate the accuracy and efficiency of our proposed approach.

Table 2. Our price vs. Monte Carlo price when

$\kappa_1 = 12, \kappa_2 = 16$ .

T	K	FC method	MC simulation	Abs R.E.
	$90$	$12.1669$	$12.2143$	0.39%
	95	$8.6172$	$8.6609$	0.50%
1/6	$100$	$5.7773$	$5.8104$	0.57%
	105	$3.6629$	$3.6830$	0.55%
	$110$	2.1979	$2.2092$	0.51%
	Time(sec.)	0.935	3.311
	$90$	$13.4270$	$13.4877$	0.45%
	95	10.1043	10.1586	0.53%
1/4	$100$	7.3614	7.4044	0.58%
	105	5.1948	5.2250	0.58%
	$110$	3.5547	3.5744	0.55%
	Time(sec.)	1.092	2.968
	$90$	16.6070	16.6933	0.52%
	95	13.6062	13.6841	0.57%
1/2	$100$	11.0102	11.0760	0.59%
	105	8.8058	8.8598	0.61%
	$110$	6.9664	7.0081	0.59%
	Time(sec.)	1.076	2.788
	$90$	21.4996	21.6192	0.55%
	95	18.7751	18.8837	0.58%
1	$100$	16.3295	16.4282	0.60%
	105	14.1503	14.2381	0.62%
	$110$	12.2214	12.2968	0.61%
	Time(sec.)	1.093	2.772

| Show Table

DownLoad: CSV

Table 3. Our price vs. Monte Carlo price when

$\kappa_1 = 2, \kappa_2 = 3$ .

T	K	FC method	MC simulation	Abs R.E.
	$90$	11.7335	11.7755	0.36%
	95	8.0186	8.0582	0.49%
1/6	$100$	5.0906	5.1190	0.55%
	105	2.9922	3.0061	0.46%
	$110$	1.6279	1.6341	0.38%
	Time(sec.)	1.638	4.443
	$90$	12.8147	12.8671	0.41%
	95	9.3383	9.3862	0.51%
1/4	$100$	6.5101	6.5461	0.55%
	105	4.3404	4.3605	0.46%
	$110$	2.7700	2.7818	0.42%
	Time(sec.)	1.581	4.648
	$90$	15.8152	15.8931	0.49%
	95	12.7023	12.7719	0.54%
1/2	$100$	10.0342	10.0906	0.56%
	105	7.8015	7.8421	0.52%
	$110$	5.9750	6.0040	0.48%
	Time(sec.)	1.613	4.613
	$90$	20.7803	20.8963	0.56%
	95	17.9864	18.0928	0.59%
1	$100$	15.4882	15.5807	0.59%
	105	13.2740	13.3527	0.59%
	$110$	11.3273	11.3926	0.57%
	Time(sec.)	1.970	4.637

| Show Table

DownLoad: CSV

4.2. Sensitivity analysis

In this subsection, we will assess the effect of the following parameters on the prices of European call options: (ⅰ) the underlying asset price $S_{0}$ and time to maturity $T$ with $t = 0$ ), (ⅱ) correlation coefficients $\rho_1$ and $\rho_2$ , (ⅲ) the long-term mean level $\theta_1$ and $\theta_2$ .

shows the variation of European call option prices with respect to the underlying asset prices $S_{0}$ and time to maturity $T-t$ . Clearly, both a higher $S_{0}$ and a higher option remaining time result in a higher European call price, which is expected since the final return of the European call option increases with the increase of $S_{0}$ , and the time value of the option increases with time.

Figure 1. European option prices for different

$S_{0}$ and

$T$ .

DownLoad: Full-Size Img PowerPoint

displays the effects of $\rho_1$ and $\rho_2$ on call option prices, and one can clearly observe that option prices increase with either $\rho_1$ or $\rho_2$ , which is as expected since a higher correlation between the underlying and volatility implies that a positive increase in the volatility will result in a greater climb in the underlying price, leading to a higher option premium. On the other hand, depicted in is the sensitivity of call option prices with respect to the long-term mean of the volatility $\theta_1$ and $\theta_2$ . It is not difficult to find that an increase in the long-term mean typically contributes to higher option, and this can be understood from the fact that increasing the level of the long-term mean is equivalent to raising the level of volatility in the long run, which implies higher risk and in turn leads call options to be more expensive.

Figure 2. European option prices for different

$\rho_1$ and

$\rho_2$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. European option prices for different

$\theta_1$ and

$\theta_2$ .

DownLoad: Full-Size Img PowerPoint

To further investigate the effect of the initial price of the underlying asset and the time to expiry on the option price, we make $S_0$ vary between 90 and 115 with other parameters kept unchanged to produce Figure 4(, ) is plotted by assuming that the maturity time $T$ changes between 0.1 and 1. One can observe that our price is larger than the single-factor model price under the current parameter settings. Of course, the comparison made between our model and the single-factor model in this section is based on the fact that the corresponding parameters in both models are kept the same, which is not the case in practice where models need to be calibrated so that model parameters can be determined from market data. Thus, we are still not sure about the performance of our model in real markets, and this will be discussed in the next subsection.

Figure 4. Our price vs single-factor model price with respect to different

$S_0$ and

$T$ .

DownLoad: Full-Size Img PowerPoint

4.3. Model calibration

In this subsection, we use the SSE 50ETF option trading data to calibrate the proposed model. We define the relative mean error sum of squares (RMSE) loss function as follows:

$RMSE = \frac{1}{NT \times NK} \sum\limits_{t = 1}^{NT} \sum\limits_{k = 1}^{NK}\left(P_{tk}^{\Theta}-P_{tk}\right)^{2} / P_{tk}$

where $P_{tk}^{\Theta}$ and $P_{tk}$ means the tth option prices obtained from the model and market with maturity time $T(t)$ and strike price $K(k)$ , respectively. ${NT}$ is the number of strike prices, and ${NK}$ is the number of maturity times. Hence, the parameters can be estimated by solving the following nonlinear optimization problem:

$\Theta^{*} = \arg \min R M S E,$

where $\Theta^{*}$ is the optimal parameter vector. The risk-free interest rate is set to be $0.15$ . We choose the SSE 50ETF options as of 2 January 2020 for the calibration of the two-factor non-affine stochastic volatility model and the single-factor one, the estimated parameters for which are listed in Table 4.

Table 4. Parameter values of the two-factor non-affine stochastic volatility model and the single-factor model calibrated to SSE 50ETF options as of 2 January 2020.

Parameter value	Single-factor model	Two-factor model
$\kappa_1$	0.0846	0.0219
$\theta_1$	0.261	0.9842
$\sigma_1$	0.1432	0.0127
$\rho_1$	-0.9974	-0.9840
$v_{10}$	0.0246	0.0012
$\kappa_2$		0.0106
$\theta_2$		0.0286
$\sigma_2$		0.2455
$\rho_2$		-0.2183
$v_{20}$		0.0236

| Show Table

DownLoad: CSV

To test the performance of the proposed model, we give the following two measures, i.e., the relative mean absolute error (RAE) and the relative mean squared error (RSE), which are respectively defined as

$\begin{aligned} RAE & = \frac{1}{N} \sum\limits_{k = 1}^{N} \frac{\left|P_{k}^{\Theta}-P_{k}\right|}{P_{k}}, \\ RSE & = \frac{1}{N} \sum\limits_{k = 1}^{N} \frac{\left(P_{k}^{\Theta}-P_{k}\right)^{2}}{P_{k}}, \end{aligned}$

where $P_{k}$ and $P_{k}^{\Theta}$ denote the kth market price and model price, respectively, and $N$ means the number of options used in calibration. We compute the above three errors of the two-factor non-affine stochastic volatility model and those of the single factor one using the obtained calibrated parameters listed in Table 4, the results of which are provided in Table 5. We can easily observe that our two-factor non-affine stochastic volatility model provides better performance when pricing European options, compared with the single-factor model.

Table 5. Comparison of the RAE and RSE between the two-factor non-affine stochastic volatility model and the single factor one for pricing SSE 50ETF options as of 2 January 2020.

	RAE		RSE
	In-sample	Out-of-sample	In-sample	Out-of-sample
One-factor model	0.0328	0.0405	0.1067	0.1116
Two-factor model	0.0326	0.0400	0.1066	0.1113

| Show Table

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5. Conclusions

Motivated by the nonlinear characteristics of the volatility as well as the advantages of multi-factor stochastic volatility models, this paper proposes a two-factor non-affine stochastic volatility model for option pricing. Based on the Taylor expansion and COS method, we derive an analytical approximation formula for European option prices, after the characteristic function of the underlying log price is successfully derived. Through numerical experiments, we verify our formula by comparing it against Monte Carlo simulation, and the influence of main model parameters on option prices under the newly proposed model is also shown. We also show through some empirical analysis that our two-factor non-affine stochastic volatility model performs better than the single-factor model does for the pricing of European options.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 12101554), the Fundamental Research Funds for Zhejiang Provincial Universities (No. GB202103001), the Advanced Research Funds of Zhejiang University of Technology (No. SKY-ZX-20220212) and Doctoral Scientific Fund Project of Anshun University (No. asxybsjj202202).

Conflict of interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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AIMS Mathematics

Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model

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Abstract

1. Introduction

2. Model specification

3. The pricing of European options

3.1. An approximate characteristic function

3.2. An analytical pricing formula

4. Numerical analysis

4.1. Accurate tests

4.2. Sensitivity analysis

4.3. Model calibration

5. Conclusions

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model

Related Papers:

Abstract

1. Introduction

2. Model specification

3. The pricing of European options

3.1. An approximate characteristic function

3.2. An analytical pricing formula

4. Numerical analysis

4.1. Accurate tests

4.2. Sensitivity analysis

4.3. Model calibration

5. Conclusions

Acknowledgments

Conflict of interest

References

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