Valuing European Option Under Double 3 / 2 -Volatility Jump-Diﬀusion Model With Stochastic Interest Rate and Stochastic Intensity Under Approximative Fractional Brownian Motion

. In this study, we propose a more comprehensive and realistic option pricing model based on approximative fractional Brownian motion, building upon recent advancements in this area. Speciﬁcally, we utilize the double 3 / 2 -volatility Jump-Diﬀusion model, which incorporates approximative fractional Brownian motion with 3 / 2 -volatility, stochastic interest rate, and stochastic intensity. To account for the stochastic interest rate, we employ a two-factor Vasicek model. Notably, our model accommodates negative interest rates. Consequently, we develop a multi-factor model with a stochastic interest rate structure for pricing European options and derive a closed-form pricing formula with an analytical solution by applying some algebraic calculations and Lie symmetries. In order to demonstrate the superiority of our proposed model over other classical approaches, we present numerical results that showcase the value of a European call option. This comparative analysis underscores the advantages of our model in comparison to traditional models.


Introduction
The groundbreaking study by Black & Scholes in 1973 [4] revolutionized option pricing by introducing an efficient model based on Brownian motion.Their closed-form formula for pricing European options, utilizing Brownian motion to explain the complexity of underlying asset prices, became a cornerstone in financial modeling.However, subsequent research by Duan and Wei [7] exposed the limitations of the Black-Scholes model in accurately capturing certain market phenomena, such as the asymmetric phenomenon and the volatility smile observed in real-world markets.To address these shortcomings, numerous academic researchers proposed various models that incorporate non-constant volatility alongside the Black-Scholes framework.Notable examples include the Hull and White [14] model, the Scott [22] model, the Stein and Stein [25] model, and the Wiggins [29] model.Despite these advancements, many of these stochastic volatility models remain impractical in real-world applications.
The impact of incorporating stochastic volatility lies in its ability to better reflect the complexities of financial markets.Traditional models that assume constant volatility fail to account for the observed phenomenon of the volatility smile, where options with different strikes exhibit varying implied volatilities.Stochastic volatility models allow for varying volatility over time, resulting in more accurate option pricing, especially for options with different maturities and strikes.Moreover, stochastic volatility models can capture sudden spikes or drops in volatility during periods of market uncertainty, providing a more realistic representation of market dynamics.This feature is particularly important in times of high market volatility, such as during economic crises or major events.Additionally, stochastic volatility models are better equipped to handle the phenomenon of volatility clustering, where periods of high volatility are followed by other periods of high volatility, and low-volatility periods are followed by other low-volatility periods.These characteristics of stochastic volatility models make them highly valuable in option pricing and risk management, as they offer improved hedging strategies and more accurate assessments of option prices in dynamic market conditions.
In 1993, Heston [12] presented a closed-form formula for European options, employing the Cox-Ingersoll-Ross process to model variance, which is the square of volatility.While this model represents an improvement, the limitations of single-factor models in precisely capturing the volatility smile have prompted researchers to explore multi-factor stochastic volatility models.Such models offer more realistic representations of return data, and in this study, we focus on option pricing under a twofactor stochastic volatility model, which better approximates the complexities of real-world financial markets.
Moreover, the financial market exhibits long-range persistence and self-similarity, characteristics fundamental to fractional Brownian motion.However, fractional Brownian motion is neither a semimartingale nor a Markov process, rendering traditional Ito calculus inapplicable.To overcome this limitation, Hu and Oksendal [13] introduced Wick products for analyzing fractional Brownian motion, and Xiao and Al [30] defined a fractional stochastic integral using Wick products.While the initial model lacked economic interpretation, Björk and Hult [3] addressed this concern with the introduction of mixed fractional Brownian motion [8,20,26,31].Additionally, Approximation Fractional Brownian motion [27] emerged as a viable alternative, and Thao [27] demonstrated that it constitutes a semimartingale.This has led to increased interest in fractional stochastic volatility models among experts and academics [11], with many researchers incorporating Approximation Fractional Brownian motion in constructing stochastic volatility models [6].
In recent years, hybrid models that incorporate stochastic interest rates into stochastic models have gained attention [9,10,15,17,24].Empirical studies also support the integration of stochastic interest rates into option pricing models, as they contribute to improved model results [21].The impact of incorporating stochastic interest rates lies in capturing the dynamics of interest rate fluctuations, which can significantly influence option pricing.By considering interest rate uncertainty, the model becomes more reflective of real-world market conditions and provides a more accurate representation of option prices, especially for longer-term options.Additionally, stochastic interest rates enable the model to account for the term structure of interest rates, which is crucial in pricing options with different maturities.
In this paper, we present a comprehensive option pricing model that considers volatility and interest rate fluctuations, along with the occurrence and intensity of jumps over time.We adopt the double 3/2-volatility jump-diffusion (DJD) model with approximative fractional Brownian motion, stochastic intensity, and a two-factor interest rate model in section 2. In section 3, we derive an analytical pricing formula for European call options.We provide numerical illustrations in section 4 to demonstrate the model's effectiveness.Finally, we conclude in section 5.The incorporation of the 3/2-volatility process and the use of approximative fractional Brownian motion, together with the consideration of stochastic interest rates, offer the potential to improve the model's performance in capturing the complexities of financial market dynamics, which may lead to more accurate and robust option pricing results.This research contributes to the advancement of financial modeling and provides valuable insights for market participants and investors in making informed decisions.

The Model
We present some basic information on approximative fractional Brownian motion .At the first, we present an analysis of fractional Brownian motion (W H t ) t≥0 with the Hurst index H ∈ (0, 1).It is a Gaussian process with zero mean and the following covariance: The decomposition of a fractional Brownian motion W is as follows: where B t indicates standard Brownian motion, and Γ indicates the gamma function.It is sufficient to focus exclusively on the term: that has a long-range memory.Note that The approximation of B t is B ,H t which can be expressed as where H is a long-memory parameter, ε is non negative approximation factor.Thao [27] proved that for ε → 0, (W ε,t t ) ε converges uniformly to a non-Markov process.In addition, if ψ t is a stochastic processes expressed as where(B ψ t ) t∈[0,T ] and (B v t ) t∈[0,T ] , are independent standard Brownian motions.Let(Ω, F, (F t ) t∈[0,T ] , Q) be a complete probability space with a filtration and Q presents a riskneutral measure.The stock price S t is expressed by the following dynamic system: suppose that lnJ has an asymmetric double exponential distribution with density function pdh u (x) : where γ 1 > 1, γ 2 > 0, p, q > 0, and p + q = 1, where q and p represent the probabilities for positive and negative jumps, respectively.As a result we can obtain that We set τ = T − t, X t = lnS t , Y = lnJ, the interest rate r are determined by the sum of the two factors r 1 and r 2 (r = r 1 + r 2 ) and k = lnK, where T is the maturity date, and K is the strike price.
In the risk-neutral world, the price of a call option C(S, v 1, v 2, r, λ, t) at time t ∈ [0, T ] with strike price K and maturity date T is given by where is the price at time t of a zero-coupon bond which matures at time T (see appendix).Then, we can have the following expression: We define ) where φ S (u) denotes the characteristic function under Q S , ϕ T (u) denotes the characteristic function under Q T , and ϕ(u) denotes the discounted characteristic function under Q.Furthermore, by using Radon-Nikodym derivatives we can have the following expression: All we need to do is to derive the formula of φ(u) to have the pricing formula.
Theorem 2.1.If the asset price is governed by the dynamic system (1), the discounted characteristic function φ(u; X, v 1 , v 2 , r 1 , r 2 , λ, τ ) takes the following form: where (2.20) and where and Where M is the Kummer M function.
Proof.ϕ(u; X, v 1 , v 2 , r 1 , r 2 , λ, τ ) satisfies a PIDE by applying the Feynman-Kac theorem: 1 If we assume that φ(u; X, v 1 , v 2 , r 1 , r 2 , λ, τ ) takes the form of and substitute into Equation (2.28), we can obtain 1 for Y and Z we apply Lie's method as in [18], and by applying some algebraic calculations, we will obtain the result.

Numerical Discussion
In this section, we analyze European option prices under the Double 3/2-Volatility Jump-Diffusion (DJD) model with a two-factor stochastic interest rate model.The model parameters used in our analysis are listed in Table (1).As the interest rate r increases, the option price also rises, particularly when β and ρ r are significant factors (5).Further research can explore its performance under diverse market conditions, solidifying its position as a significant contribution to financial modeling.2 + ρ r η 1 η 2 ∂ 2 P ∂r 1 ∂r 2 − (r 2 + r 2 )P = 0 P (r 1 , r 2 , T, T ) = 1 (4.1) If we assume that P (r 1 , r 2 , t, T ) takes the form of P (r 1 , r 2 , t, T ) = e [A(τ )−B 1 (τ )r 1 −B 1 (τ )r 2] (4. 2) and substitute it into PDE (4.1), we can obtain : with the terminal condition B 1 (0) = B 2 (0) = A(0) = 0 Then we have : Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.

. 10 )
We convert measure Q to the measure Q S and the T forward measure Q T .By applying Radon-Nikodym derivatives,

Figure 1 .
Figure 1.The impact of k λ and θ on call option prices for T = 1.

Figure 2 .Figure 3 .
Figure 2. The impact of the existence of the jump intensity process on call option prices for T = 1.

Figure 4 .
Figure 4.The European call option price with respect to different values of r 1

Figure ( 2 )Figure 5 .
Figure(2) presents the effect of the presence of the jump intensity process on call option prices.It clearly demonstrates that the call option price with stochastic jump intensity is higher than that with Conversely, we proceed to analyze the impact of incorporating a two-factor stochastic interest rate model with approximate fractional Brownian motion and stochastic intensity using the theoretical conclusions derived from the pricing formula.Evidently, our proposed price model surpasses that of the traditional Heston model.Specifically, Figure(3) displays the option prices with varying time to expiry.Notably, our model's price and the Heston model's price are nearly equivalent when the time to expiry increases, but the discrepancy between the two widens with extended time to expiry.This phenomenon is attributed to the increased time duration for interest rate changes, which allows our model to reflect the expanded divergence more effectively.4.ConclusionIn conclusion, our research introduces a comprehensive pricing model for European options, combining a two-factor Vasicek model with interest rate dynamics, a stochastic process for jump intensity, and the double 3/2-volatility jump-diffusion model.By leveraging the power of approximate fractional Brownian motion, our approach captures long-range dependence and self-similarity in asset volatility.Numerical results demonstrate the superiority of our model over the double Heston and Heston models in pricing European call options.The closed-form solution enhances efficiency and accuracy, making our approach valuable for option pricing and risk management in real-world financial markets.
N t represents Poisson process with intensity λ t and J represents the jump size, and we t = ρ r dt.Any other Brownian motions are pairwise independent.v 1 and v 2 are variances, and λ t is the jump intensity.k 1 , k 2 and k λ are mean reversion rates, θ 1 , θ 2 and θ are mean reversion levels, σ 1 , σ 2 and σ are the volatilities of the variances.and the short rate follows the two-factor Vasicek model where the short rate is given as a sum of two factors r 1 and r 2 , where α 1 , α 2 are their mean-reversion , β 1 , β 2 are their mean-reversion speed, η 1 , η 2 are their volatilities,

Table 1 .
Values of parameters. ) illustrates the significant impact of changes in the mean-reversion level θ λ on call option prices, while changes in the mean-reversion rate k λ have a relatively minor effect on call option prices.