European options represent one of the most significant derivative instruments in the global financial markets. To accurately price these instruments, numerous scholars have developed various pricing models. However, as most of these models rely on numerical simulation methods—which demand substantial computational resources—the development of a closed-form pricing solution for such securities remains of paramount importance for practitioners in the financial industry. Since the Black and Scholes (B-S) model ([4]) was proposed, it has still been widely adopted by financial practitioners. In the B-S model, the underlying asset price follows a lognormal distribution, and there is a simple formula to calculate the price of European options. However, despite the popularity of the B-S model, some of the oversimplified assumptions it makes to achieve analytical tractability lead to potential mispricing problems. The existence of the “volatility smile” ([6]) demonstrates the unrealistic nature of the model’s assumption of constant volatility. Non-constant volatility models have been significantly developed to improve the model further. Non-constant volatility models can be divided into two main categories: local volatility models and stochastic volatility models. The former was proposed by [7]; in this model, volatility is defined as a deterministic function of the underlying price and time. However, numerous empirical studies have demonstrated that the local volatility model cannot capture the “volatility smile” ([11]). This limitation has led to the growing popularity of stochastic volatility models.

The stochastic volatility model complicates the discovery of a closed-form pricing formula for European options because it introduces volatility as a new random variable. As a result, most existing models must rely on numerical methods. [24] uses the Monte Carlo simulation to calculate the option price. [27] uses the finite difference method to calculate the option price. However, these numerical methods often need a long time to obtain the results of option pricing, making them inapplicable in actual financial markets due to the extensive time required for model calibration and option valuation. Therefore, stochastic volatility models with closed-form solutions have become a better and more relevant research direction for the needs of the financial market. [17] derived a series solution under their model in which one volatility follows another geometric Brownian motion. However, the model is still unsatisfactory. On one hand, the assumption of independence between the underlying price and volatilities violates the leverage effect, as empirical studies confirm a negative correlation between the underlying price and volatility ([1]; [18]). On the other hand, the volatility process of the model lacks a mean-reverting property, contradicting the fact that the volatility process is mean-reverting ([3]).

[16] proposed a more effective and widely utilized model among financial practitioners, which enables the derivation of a closed-form pricing formula for European call options through the Cox-Ingersoll-Ross (CIR) process. The model not only satisfies the assumption of arbitrary correlation between the underlying price and volatility but also upholds the non-negative and mean-reverting properties of the volatility process. More importantly, due to the existence of an analytical solution, the Heston model has overcome the shortcomings associated with numerical methods, which require considerable time and energy for model calibration and option valuation when applied to real financial markets. Despite its numerous advantages, the Heston model is not without flaws, as it exhibits significant nonlinear mean-reversion phenomena in asset volatility ([2]). To mitigate this issue as much as possible, [14] replace the constant term of the long-term mean volatility with a variable that follows a normal distribution. Notably, this model still retains the essential advantage of the Heston model: analytical tractability, allowing for the derivation of closed pricing formulas for European options. To model the volatility structure more flexibly and better fit empirical data to European option prices, [5] constructed the double Heston model by adding another stochastic volatility factor based on the CIR process to the Heston model. Mehrdoust et al. ([22]) and [28] support this with their research on American options. More profoundly, two mutually independent CIR mean-reversion processes are simultaneously included in the double Heston model, with computational tools very similar to those of the standard Heston model; however, the results obtained from option pricing under this model can be highly desirable.

Regime-switching models have been widely used in the finance field to simulate the prices of financial derivatives that are impacted by economic cycles ([12]; [9]). Recently, much literature has applied the regime-switching mechanism to a stochastic volatility model with regime-switching. [8] provide closed-form exact solutions for pricing discrete sampling variance swaps and volatility swaps based on the Heston stochastic volatility model featuring regime-switching. [20]) and [15]) constructed two nonlinear stochastic volatility models based on the He-Chen model ([14]) by modelling the long-term mean of volatility and introduced a regime-switching model in each, deriving closed solutions easily. [21] examined the pricing of American options under a new model developed by incorporating regime-switching at the interest rate level and mean reversion into the double Heston model.

Building on previous research, this paper establishes a new model based on the double Heston model by replacing the mean reversion level of each volatility process with a stochastic long-term mean process and substituting the constant parameter of the stochastic long-term mean process with a regime-switching term governed by a two-state Markov process. If the regime-switching term degenerates into a constant parameter, we obtain a new nonlinear double without regime-switching. We refer to the new models with and without regime-switching as the Markov Regime-Switching Nonlinear Double Heston Model (MRSNDH) and the Nonlinear Double Heston Model (NDH). Both MRSNDH and NDH models retain the fundamental advantages of the He-Chen model and the double Heston model, namely, analytical tractability. In addition to the derivation details, we provide verification of the accuracy of the newly derived formulas. On this basis, the MRSNDH model is compared with the NDH model to explore the effect of introducing regime-switching on European call option prices, while the NDH model is compared with the double Heston model to investigate the impact of incorporating two stochastic long-term mean processes in the double Heston model on European call option prices.

This paper involves the estimation of numerous parameters, and selecting an appropriate estimation method to achieve rapid and accurate parameter estimation under limited computing resources poses a significant challenge. The Particle Swarm Optimisation (PSO) algorithm, proposed by electrical engineer Russell Eberhart and American social psychologist James Kennedy in 1995, is an evolutionary computing method based on swarm intelligence ([19]). In 1998, [25] introduced the inertia weight into the original PSO, using its value to represent the contribution of historical velocity to current velocity, which was later called the standard PSO. [13] proposed a particle swarm optimisation algorithm based on correcting the global optimal position to estimate the parameters of the B-S model. There is still room for improvement in the traditional PSO, so [10] introduced two hybrid optimisation techniques, the hybrid PSO algorithm and the hybrid Differential Evolution (DE) algorithm, into the parameter calibration scheme to enhance the calibration quality of the new model. Ratnaweera et al. ([23]) proposed an adaptive adjustment strategy based on the most up-to-date information from each individual. This paper will adopt the Adaptive Particle Swarm Optimisation (APSO) to estimate the parameters of MRSNDH.

The remainder of this paper is organized as follows. In Section 2, we introduce the newly proposed model, followed by the closed pricing formula for the European call option based on this model. Section 3 investigates various properties of the new formula through numerical experiments. The final section presents the conclusion. In Section 4, we utilize actual data on European call options related to the S&P 500 index to analyze the performance of different models in practical market applications and demonstrate the superiority of our MRSNDH model over others.

In this section, we propose a new model, namely the MRSNDH model, for modelling the price of the underlying asset and option pricing based on the He-Chen model ([14]) and the double Heston model ([5]). This model draws on the design idea of the He-Chen model in introducing a random long-term mean in the volatility process and makes improvements on the basis of the double Heston model. Meanwhile, it further introduces a volatility term in the volatility dynamics to enhance the model’s ability to describe market complexity. First, we introduce the He-Chen model that

Here, t ≥ 0 , S t and v t denote the underlying asset price and volatility, respectively. W t 1 and W t 2 are two standard Brownian motions with a correlation coefficient of ρ , r denotes the risk-free interest rate, and σ is the so-called volatility of volatility. κ represents the speed of mean-reversion. The long-term average of volatility is composed of a stochastic component θ t . To capture the volatility structure with greater flexibility and price European options more accurately, we propose a nonlinear double-Heston model augmented with a Markov regime-switching mechanism—hereinafter referred to as the MRSNDH model.

where, W t 1 and W t 3 have correlation ρ 1 , W t 2 and W t 4 have correlation ρ 2 . Z t 1 through Z t 4 are independent standard Brownian motions. The Markov chain X t is defined as:

Here, states 1 and 2 are employed to characterise the prevailing economic conditions. In this study, the classification of these states is determined by the level of implicit volatility observed in the actual data. The transition between states follows a Poisson process:

where, λ i j represents the transition rate of the random variable X t from state i to state j , while t i j denotes the dwell time in state j prior to transitioning to state i . ε X t , δ X t , λ X t , η X t , α X t and β X t are parameters that depend on the Markov chain, expressed as follows:

Here, ε ^ = ( ε 1 , ε 2 ) T , δ ^ = ( δ 1 , δ 2 ) T , λ ^ = ( λ 1 , λ 2 ) T , η ^ = ( η 1 , η 2 ) T , α ^ = ( α 1 , α 2 ) T , β ^ = ( β 1 , β 2 ) T , Note: u ^ T denotes the transpose of vector u ^ , and the symbol 〈 ⋅ , ⋅ 〉 indicates the inner product of two vectors. When the conversion rate is 0, i.e., λ 12 = λ 21 = 0 and ε ^ , δ ^ , λ ^ , η ^ , α ^ and β ^ are all set to zero vectors, the MRSNDH model reduces to a special case corresponding to a nonlinear two-dimensional Heston model with no state transitions, which we refer to as the NDH model.

Theorem 1:Let U ( S , v 1 , v 2 , θ 1 , θ 2 , X t , t ) be the European call option price satisfying the MRSNDH model.Then

where

Proof. Let K represent the strike price and y denote the logarithm of the asset price ( ln ( S ) ). The process of deriving European call options U ( S , v 1 , v 2 , θ 1 , θ 2 , X t , t ) is as follows:

Here, E represents the formula for calculating the expected value. 1 s t > k . p ( y ) represents the density function of y t . If this probability density function can be directly calculated, determining the option price becomes straightforward; however, in practice, deriving the density function of the underlying asset’s price is challenging. Consequently, based on the Gil-Peláez theorem, further derivations can be carried out.

where, Re [ a + b i ] denotes real part a of complex number a + b i . f ( ϕ ; τ , y , v 1 , v 2 , θ 1 , θ 2 , X t ) denotes the characteristic function of y T . It is important to note that we define a new measure, Q S :

where, B t = e ∫ 0 t r d u = e r t . Therefore, we can derive

Under the risk-neutral ( Q ) measure, asset prices evolve deterministically at the

risk-free rate; consequently, e − r ( T − t ) E Q [ S T 1 S T > K ] simplifies to

According to Formula (6), the new density function p S ( y ) is defined as p ( y ) via the Radon-Nikodym derivative with respect to

Therefore, the characteristic function of p S ( y ) is given by:

where E Q [ e y T ] is a constant; moreover, given that the characteristic function of y T is f ( ϕ ; τ , y , v 1 , v 2 , θ 1 , θ 2 , X t ) = E Q [ e i ϕ y T ] , it follows that

and

Therefore, the characteristic function of the density function p S ( y ) is given by

For f ¯ ( ϕ ; τ , y , v 1 , v 2 , θ 1 , θ 2 , X t ) , the Gil-Peláez theorem yields:

Our next objective is to define the characteristic function f ( ϕ ; τ , y , v 1 , v 2 , θ 1 , θ 2 , X t ) , which will allow us to determine the final price of the European call option. To determine the analytical solution of the characteristic function, we begin with its definition.

based on the expected value formula, this characteristic function can also be represented as

this expression suggests that we should initially compute the inner expectation, assuming the Markov chain is known prior to the maturity date. The inner expectation can be defined as

Based on the data from the Markov chain, parameters κ 1 , κ 2 , σ 1 , σ 2 , ρ 1 and ρ 2 are deterministic functions dependent solely on time. Consequently, these parameters can be denoted as κ 1 ( t ) , κ 2 ( t ) , σ 1 ( t ) , σ 2 ( t ) , ρ 1 ( t ) and ρ 2 ( t ) , respectively. According to the Feynman-Kac theorem, f must satisfy the following Partial Differential Equation (PDE):

Given the previously established conditions, we obtain

Based on the results of previous researchers ([14]; [5]; [23]), the internal function can be structured as follows.

when substituted into the PDF equation above, we obtain the following five Ordinary Differential Equations (ODEs).

The initial conditions are C ( 0 ; ϕ ) = D 1 ( 0 ; ϕ ) = D 2 ( 0 ; ϕ ) = E 1 ( 0 ; ϕ ) = E 2 ( 0 ; ϕ ) = 0 . Similar to the results of previous researchers ([16]; [5]; [20]), it is not difficult to solve the analytical solutions of the Riccati equations D 1 ( τ ; ϕ ) and D 2 ( τ ; ϕ ) . Subsequently, C ( τ ; ϕ ) , E 1 ( τ ; ϕ ) , and E 2 ( τ ; ϕ ) can be integrated from the remaining ODEs. Specifically, C ( τ ; ϕ ) can be solved as

where C ¯ ( τ ; ϕ ) is defined as

Now that we have derived the internal expected value, the next step is to compute the external expectation in order to obtain the characteristic function.

According to [8], this expectation can be further derived as

where

and M = A T τ + B . Here, A = ( − λ 12 λ 12 λ 21 − λ 21 ) represents the transition rate matrix of the Markov chain X t and B = ( p 1 0 0 p 2 ) with p 1 = ∫ 0 τ Λ 1 ( s ; ϕ ) d s , p 2 = ∫ 0 τ Λ 2 ( s ; ϕ ) d s . The functions Λ 1 ( s ; ϕ ) and Λ 2 ( s ; ϕ ) are represented as follows.

After deriving the closed-form solution for the new model, we will investigate its properties via comprehensive numerical experiments. Prior to this, it is crucial to conduct a numerical comparison between our analytical results and those obtained from Monte Carlo simulations. This step ensures the absence of algebraic errors and validates the robustness of our conclusions. Additionally, the following section will delve into the implications of integrating two long-term mean-reverting processes into the double Heston model, along with the introduction of a transformation mechanism.

This section examines various properties of our newly derived formulas through numerical experiments. Initially, we compare European option prices calculated using Monte Carlo simulations with those from our formula to verify the pricing accuracy of the MRSNDH model and the NDH model. Next, by comparing option prices between the NDH model and the double Heston model, we evaluate the effect of including a stochastic long-term mean process. Finally, by contrasting the MRSNDH model with the NDH model, we explore how the state transition model influences option prices.

In this section, the initial parameters are set as follows: the current state is set to 1, the risk-free rate ( r ) is 0.05, the mean reversion speeds ( k 1 and k 2 ) are 6 and 8, respectively, the constant parts of the long-term mean ( v ¯ 1 and v ¯ 2 ) are specified, the initial values of the volatility process ( v 1 and v 2 ) are 0.2 and 0.1, respectively, the initial values of the stochastic long-term mean ( θ 1 and θ 2 ) are 0.12 and 0.08, respectively, the volatility terms ( σ 1 and σ 2 ) for asset volatility are 0.1 and 0.2, respectively, the correlation coefficients ( ρ 1 and ρ 2 ) are 0.1 and 0.2, respectively, the two transition rates are set to ( λ 12 = 10 and λ 21 = 20 ), the option strike price ( K ) is 10, and the parameter values for ( X t ) related to the Markov chain will be explained in the figure’s title. Consistent with the standard Monte Carlo simulation framework, we generate 500,000 independent sample paths; each path yields a single option price estimate, and the final Monte Carlo estimator is obtained as the arithmetic mean of these 500,000 simulated prices.

The MRSNDH model’s pricing formula, under initial parameters, yields prices closely matching those from Monte Carlo simulations, as illustrated in Figure 1. Specifically, Figure 1(a) shows that both prices rise with increasing underlying asset value, aligning with expected financial behaviour. Figure 1(b) indicates that the relative error between the two methods is below 0.6%. Similarly, Figure 2(a) compares option prices derived from our formula with those from Monte Carlo simulations under the NDH model.

Note: Time-related parameters have been set to: τ = 1 , ε 1 = 0.01 , ε 2 = 0.02 ; δ 1 = 0.015 , and δ 2 = 0.01 ; λ 1 = 0.03 , λ 2 = 0.01 ; η 1 = 0.01 , η 2 = 0.015 ; α 1 = 0.01 , α 2 = 0.02 ; β 1 = 0.01 , β 2 = 0.015 .

Figure 1. Verification of MRSNDH model accuracy against the benchmark: (a) Absolute price comparison, (b) Relative error.

When the MRSNDH model is simplified to the NDH model, variables ε , δ , λ , η , α and β assume the values associated with state 1 in Figure 1. Figure 2(a) shows that the prices are very similar, and the price of the European call option increases monotonically with the underlying asset’s value. Figure 2(b) indicates that the relative error between the two does not exceed 0.3%. Figure 1 and Figure 2 demonstrate that the formulas derived in this chapter are accurate, regardless of whether a state transition mechanism is included in the model.

Note: The parameters involved in the NDH model are: τ = 1 , ε = 0.01 , δ = 0.015 , λ = 0.03 , η = 0.01 , α = 0.01 , β = 0.01 .

Figure 2. Verification of the NDH model accuracy against the benchmark: (a) Absolute price comparison, (b) Relative error.

Note: The relevant parameters at this time are: τ = 1 , ε = ε 1 = 0.01 , ε 2 = 0.02 ; δ = δ 1 = 0.015 , δ 2 = 0.01 ; λ = λ 1 = 0.03 , λ 2 = 0.01 ; η = η 1 = 0.01 , η 2 = 0.015 ; α = α 1 = 0.01 , α 2 = 0.02 ; β = β 1 = 0.01 , β 2 = 0.015 ; λ 12 = 10 z , λ 21 = 20 z .

Figure 3. MRSNDH model versus NDH model with respect to a scale parameter z.

After the accuracy of the pricing formula has been verified, the impact of introducing the state transition model can be demonstrated. A scaling parameter z is introduced to adjust the transition rates λ 12 = 10 and λ 21 = 20 , as depicted in Figure 3. When z equals 0, due to the lack of state transitions, the model with state transition mechanisms simplifies to one without state transitions, leading to identical option prices between the MRSNDH model and the NDH model. It is important to note that under current parameters, as the scaling parameter z increases, the European call option price of the MRSNDH model decreases, with the rate of decrease slowing down.

In Figure 4 that follows, the European call option prices of the NDH model and the double Heston model are compared. Initially, their prices are similar; however, as the time to maturity increases, the difference between them becomes more pronounced, with the option prices from the NDH model consistently higher than those from the double Heston model.

Figure 4. Comparison of NDH model and the double Heston model option prices at different expiration times. Where, ε = 0.0 1 ; δ = 0.0 15 , δ 2 = 0.0 1 ; λ = 0.03 ; η = 0.0 1 ; α = 0.0 1 ; β = 0.0 1 .

Figure 5 illustrates the impact of introducing a state transition mechanism into the NDH model. Option prices in the NDH model are consistently higher than those in the MRSNDH model. To investigate potential parameter reversal effects between the MRSNDH and NDH models, we compared versions with reversed parameters to their original counterparts. Our analysis shows that parameter reversal does not cause option prices in the MRSNDH model to exceed those in the NDH model; instead, it reduces them further. Note that as options approach expiration, prices converge across all three models due to decreasing likelihood of state transitions and significant price fluctuations. Conversely, differences between the models become more pronounced with longer time to expiration.

Figure 5. The comparison of option prices with different expiration dates between the NDH model after parameter update and the MRSNDH model. The reset parameters are as follows: σ 1 = 0.5 , σ 2 = 0.3 ; λ = λ 1 = 0.07 , λ 2 = 0.01 ; η = η 1 = 0.025 , η 2 = 0.005 ; α = α 1 = 0.035 , α 2 = 0.01 ; β = β 1 = 0.005 , β 2 = 0.03 ; ε = ε 1 = 0.001 , ε 2 = 0.002 ; δ = δ 1 = 0.002 , δ 2 = 0.001 . When the parameters are reversed, only the parameters with Markov state transitions will be swapped between 1 and 2.

In this section, we conduct an empirical study to assess the performance of the proposed model relative to both the double Heston model ([5]) and the He-Chen model ([14]). The purpose is to demonstrate the importance of integrating a series of regime-switching factors and stochastic long-term means, as well as additional stochastic volatility components, within the double Heston framework and to assess the performance of this new model when applied to real financial data. We first describe the dataset and the key screening conditions used in model calibration, and then detail the parameter estimation method. Finally, we present the empirical results, which clearly illustrate the relative performance of the three models in fitting real financial data.

In this paper, we propose the MRSNDH model by incorporating an additional volatility term into the volatility process and modeling the long-term mean of the double Heston framework through another stochastic process. This structure enhances the model’s ability to fit real-world financial data. After deriving a closed-form pricing formula for European options under the MRSNDH model, we numerically validate its accuracy by comparing the results with those obtained from Monte Carlo simulations. Furthermore, we conduct a comparative numerical analysis of option prices generated by the Heston model and our proposed model to highlight their differences. Finally, an empirical study based on S&P 500 index options demonstrates that the MRSNDH model consistently outperforms both the double Heston model and the He-Chen model, particularly for in-the-money options. These findings suggest that the MRSNDH model can serve as a more effective alternative to the double Heston model in practical financial applications. Future research may explore extending the model to other types of derivative instruments and investigating alternative state-switching mechanisms.