A probabilistic approach for the valuation of variance swaps under stochastic volatility with jump clustering and regime switching

The effects of stochastic volatility, jump clustering, and regime switching are con‑ sidered when pricing variance swaps. This study established a two‑stage procedure that simplifies the derivation by first isolating the regime switching from other sto‑ chastic sources. Based on this, a novel probabilistic approach was employed, leading to pricing formulas with time‑dependent and regime‑switching parameters. The formulated solutions were easy to implement and differed from most existing results of variance swap pricing, where Fourier inversion or fast Fourier transform must be per‑ formed to obtain the final results, since they are completely analytical without involv‑ ing integrations. The numerical results indicate that jump clustering and regime switching have a significant influence on variance swap prices.


Introduction
The effective management of financial risk, which is vital for market stability, is an ongoing topic in finance practice.Volatility derivatives can efficiently provide volatility exposure without having to invest in target assets directly.Such favorable properties have made these derivatives, among which variance swaps are representative, especially attractive.
To affect the risk-management process, the prices of variance swaps need to be determined accurately.A key determinant of the variance swap prices is the sampling approach, which can be continuous or discrete.Different results have been obtained when continuous sampling is adopted under various stochastic volatility models (Swishchuk 2006;Carr and Lee 2007).Despite the simplicity of this framework and the convenience of its practical implementation, it does not match existing practices in financial markets, where discrete sampling is selected.These inconsistencies have prompted research in the direction of discrete sampling.
The selection of an appropriate model that can reflect market characteristics is significant when the sampling of realized variance is discrete and the obtained solutions can greatly vary according to the chosen underlying model.For example, an analytical solution was presented by Zhu and Lian (2011) when the Cox-Ingersoll-Ross (CIR) process was used to model stochastic volatility (Heston 1993), whereas analytical and asymptotic solutions were obtained Bernard and Cui (2014) using three different stochastic volatility models.Multifactor stochastic volatility has gained attention because of its ability to produce implied volatility closer to market data (Rouah 2013;Lin et al. 2024;He and Lin 2024), leading to the consideration of variance swap valuation under this framework (Kim and Kim 2019;Issaka 2020).The stochastic interest rate is incorporated when valuing variance swaps (Cao et al. 2020;Wu et al. 2022;Badescu et al. 2019).
The abovementioned works neglect the possibility of price jumps in underlying stocks, which violates statistical/empirical observations (Bates 1996;Eraker 2004;Hu et al. 2024).This leads to an investigation of pricing variance swaps with Poisson jump-diffusion or Levy jumps (Broadie and Jain 2008;Hong and Jin 2023;Pun et al. 2015;Carr et al. 2012).However, independent increments assumed in these jump models prevented the incorporation of jump clustering, which often occurs in practice (Aït-Sahalia et al. 2015).Therefore, various research interests have guided the adoption of Hawkes jump-diffusion models for valuing financial derivatives, including vulnerable options (Ma et al. 2017), power exchange options (Pasricha and Goel 2020), and Volatility Index (VIX) options (Jing et al. 2021).Stochastic volatility and jump clustering are combined by Liu and Zhu (2019), who present an analytical valuation of variance swaps.
Another strand of research requires the consideration of the variable economic status that can greatly influence asset prices (Hamilton 1990).Consequently, regime switching has been incorporated into different dynamics in the process of pricing derivatives (He and Chen 2022;Siu and Elliott 2022) and has also been applied to determine volatility derivative prices.For instance, regime-switching stochastic volatility was considered by Elliott and Lian (2013), Lin and He (2023) when pricing variance swaps, which were extended by Cao et al. (2018) to incorporate the regime-switching stochastic interest rate further.Jump diffusion was added to a similar framework for pricing variance and volatility swaps (Yang et al. 2021).
This study attempted to solve the pricing problem of variance swaps when stochastic volatility, jump clustering, and regime switching coexist.To the best of our knowledge, this has not been studied in previous literature.A novel probability approach was employed in the valuation yielding a closed-form formula for variance swap prices.In contrast with existing literature, the formula is simple and no longer requires Fourier inversion, fast Fourier transform, or numerical integration.This unique property renders the formula extremely flexible when adopted by market participants.
A brief summary of the content is presented in the following sections.The combination of regime switching stochastic volatility and Hawkes jump diffusion is illustrated in Sect. 2. Section 3 discusses the probability approach to the analytical formulation of variance swap prices.Before concluding the paper, an implementation of this formula is presented in Sect. 4.

The model
The risk-neutral measure Q was emphasized as a part of the complete probabil- ity space (�, F, Q) .A continuous-time Markov chain,1 denoted by {X t } t≥0 , is intro- duced to model the variation in economic conditions, such that its value is equal to that of the two-dimensional unit vectors.When the generator of the Markov chain is �(t) = [ ij (t)] i,j=1,2 and the Martingale increment process conditional on the filtration generated by the Markov chain is {V t } t≥0 , one can have resulting from the semimartingale representation theorem (Elliott et al. 2008;He and Lin 2023a, b).
In the considered risk-neutral world where the risk-free interest rate is denoted by r, Y t , ν t , and γ t are respectively used to denote the price of the stock, its variance, and the intensity of the Hawkes-type jump.Their dynamics in the formulated risk-neutral world are {W i,t } t≥0 , i = 1, 2 are two Wiener processes, with dW 1,t dW 2,t = ρdt representing the connection between the stock price and its variance.ν t is mean-reverting, the speed of which is p.This process is affected by the volatility, ζ .The long-term variance level to be approached is assumed to be influenced by economic conditions, and regime switching is performed to reflect such effects, thus one can compute q X t using the inner product between the Markov chain X t and the vector q = (q 1 , q 2 ) , that is, q X t =< q, X t > (He and Lin 2023c).The jump sizes with support for (−1, ∞) have identical and independent distributions, with E(J t ) = ω .H t is the Hawkes process, which is defined as: with jump time moments denoted by {t i } > 0 .The expression of jump intensity γ t con- siders the jump clustering effect, indicating that the occurrence of a jump contributes to a greater intensity at the rate of δ , and the increment then experiences exponential decay at the rate of η.
A natural transform applied to stock price dynamics is Z t = ln(Y t ) , which, when com- bined with Ito's lemma, yields the dynamics of Z t as (1) (3) where the log stock price jumps to Jt = ln(1 + J t ) .Jt is assumed to be identically and independently distributed following a normal distribution for any given t with µ and σ 2 as its mean and variance, respectively.In this setting, ω = E e Jt − 1 = e µ+ 1 2 σ 2 − 1.

Variance swap pricing
The model established in the previous section considers the effects of stochastic volatility, jump clustering, and varying economic conditions when modelling stock prices.
Although appealing, the complex structure of the model dynamics can be an obstacle to efficiently evaluating financial derivatives, and it is certainly of interest to investigate the analytical valuation of variance swaps under this framework.The related details are presented below.
Unlike other types of financial derivatives, the prices of the variance swaps to be computed are referred to as delivery prices K agreed upon by both parties listed in the contract.The target delivery prices rely on how the realized variance, denoted by RV, is defined.This study adopts the following model where the time period between the current time t = 0 and the expiry of the contract t = T is split into N sub-intervals uniformly.This yields the swap payoff (RV − K )Z , where Z denotes a constant notional value.This specific definition of realized variance was selected not only for its ability to derive an analytical solution but also due to its extensive adoption in the existing literature (Pun et al. 2015;Badescu et al. 2019;Zheng and Kwok 2014;Lin and He 2023).
Owing to the nature of the swap contract's initial zero value, we must obtain where which requires the computation of the unknown function g 1 .However, the multiple stochastic sources involved complicate analytically computing expectations, and the separation of certain stochastic variables is first performed such that the number of stochastic sources to be dealt with at one time can be reduced.
In particular, we apply the tower rule of expectation, which leads to with Obviously, the calculation of g 2 (Y 0 , ν 0 , γ 0 , s, t) is an a priori step in determining g 1 (Y 0 , ν 0 , γ 0 , X 0 , s, t) as well as the variance swap delivery prices.This first step assumes a given Markov chain, reducing the problem of finding the variance swap-pricing formula with the long-term variance q t being no longer stochastic but time-dependent.First, we present a solution to this simplified problem.

The formula with time-dependent long-term variance
Before formally dealing with the problem of g 2 (Y 0 , ν 0 , γ 0 , s, t) , we first present some useful results associated with ν t and γ t that are needed in the derivation.
Proposition 1 If the stock variance v t and jump intensity γ t are governed by the stochas- tic differential equations presented in Eq. (2), with the stochastic variable q X t replaced by the time-dependent variable q t , we must have where a 1 = η − δ and a 2 = ηγ ∞ a 1 .
Appendix 1 provides the proof of Proposition 1.
With the expectations and variances of ν t and γ t presented, we are now ready to compute g 2 (Y 0 , ν 0 , γ 0 , s, t) .This requires us to calculate the conditional expectation of ln Y t Y s , which contains two random variables when s > 0 : To simplify the computa- tion, the tower rule of expectation is used, such that where Clearly, the target g 2 (Y u , ν u , γ u , u, s, t) is the expectation of g 3 (Y s , ν s , γ s , s, t) , which we can obtain if are both known.Thus, we present the corresponding results in the following proposition: Proposition 2 If the dynamics of the stock price Y t are given in Eq. (2), then we must have and where and with , and τ = t − s.
We prove Proposition 2 in Appendix 2. Now, we obtain g 3 (Y s , ν s , γ s , s, t) in Eq. ( 11) by using g 3 = (g 3,1 ) 2 + g 3,2 , providing where (13) (15) The problem that remains to be solved involves the computation of g 2 (Y 0 , ν 0 , γ 0 , s, t) using Eq. ( 10).With the solution to g 3 (Y s , ν s , γ s , s, t) given in Eq. ( 15), the target expecta- tion requires the computation of ) .Because the required solutions have already been presented in Proposition 1, the results of g 2 (Y 0 , ν 0 , γ 0 , s, t) are illustrated directly without requiring tedious calculations and rearrangements. where Once this stage is reached, one should realize that when the long-term stock variance, q ξ , ξ ∈ [0, T ] , is time dependent, Eq. ( 5) should be revised such that the corre- sponding delivery price is a result of ( 16) (17) This formula can be used with the formulation of g 2 (Y 0 , ν 0 , γ 0 , s, t) as shown in Eq. ( 17).The results associated with the time dependence of q ξ , ξ ∈ [0, T ] are not the target val- ues, and their regime-switching nature should be re-introduced to capture the varying economic states.
Thus, the solution to U 1 is presented in Proposition 3.
Proposition 3 With q X ξ , ξ ∈ [0, t] being a regime-switching parameter controlled by Markov chain X t , we have where and Proposition 3 will be validated in Appendix 3. (18 g 2 (Y 0 , ν 0 , γ 0 , t i , t i−1 ). ( Because the derivation of U 2 is similar to that of U 7 , we provide their solutions in Proposition 4.
Proposition 4 With q X ξ , ξ ∈ [0, t] being a regime-switching parameter controlled by Markov chain X t , we have where and Appendix 4 provides the proof of Proposition 4. The remaining unknown value U i , i = 3, 4, 5, 6 can be determined similarly.Their for- mulae are presented together in Proposition 5. (21 Based on Chebyshev's inequality, one can illustrate that RV − RV c converges to zero as the probability approaches one, following arguments similar to those presented in Broadie and Jain (2008), Liu and Zhu (2019): Therefore, variance swaps with continuous sampling can be expressed as: yielding

Numerical analysis
We first discuss the reliability of the formula (5) presented in the previous section by comparing its produced prices with the Monte Carlo results.We also analyze the sensitivity of strike prices to changes in the different model parameters.Here is a list of parameters that produce the plots in this section.
Figure 1 displays the strike prices produced by Eq. ( 5) compared to Monte Carlo prices.The clear pattern shown in Fig. 1a indicates a very close agreement between the corresponding results from the two methods, which is further supported by Fig. 1b, where the errors of our results relative to the benchmark are less than 0.03%.Another evidence supporting the correctness of the formula with discrete sampling is its convergence with that with continuous sampling when the sampling frequency is increased.
Model performance can be assessed once the accuracy of the formula is confirmed.We first show the difference between our model considering stochastic volatility, jump (24) clustering, and variations in economic status (HJSVRS) and that constructed by our model ignoring jump clustering (PJSVRS).This comparison is illustrated in Fig. 2 with the varying decay rates of jump clustering η .Some distance exists between the swap prices generated by the two models, and decreasing the value of η typically widens this gap.This phenomenon is reasonable, because a higher decay rate implies a less significant impact of jump clustering.The opposite phenomenon is displayed in Fig. 3, where increasing the value of δ is equivalent to increasing strike prices.A larger δ implies a higher possibility for the arrival of another jump, creating more significant jump-clustering effects and leading to the corresponding strike prices.The strike prices of PJSVRS model stay in between HJSVRS prices, which is probably due to the decrease in δ leads to the number of expected jumps in HJSVRS model falling below that in PJSVRS model.The impact of jump clustering is also examinedwhen we assign γ ∞ different values in Fig. 4.An increasing trend in variance swap prices can be observed with the increase in γ ∞ , leading to a wide difference between HJSVRS and PJSVRS model prices.The higher γ ∞ indicates a greater number of expected jumps, yielding larger risks, as well as swap delivery prices.
The impacts of varying economic statuses are crucial factors that must be studied.The HJSV model was constructed as a benchmark for this by removing the regime switching contained in HJSVRS model and plot both model prices in Fig. 5 after equating 12 and 21 to for ease of comparison.An increase in transition rates would inevitably result in an increase in strike prices, causing a greater difference between HJSVRS and HJSV model prices.The reasonableness of this observation lies in the current setting of the long-term variance, corresponding to the current state being lower than that of the other states, and the increasing transition rates actually raise the volatility level, yielding higher risks and strike prices.

Conclusion
This study solved the variance-swap pricing problem when the underlying dynamics are subject to the risks of stochastic volatility, jump clustering, and regime switching.We utilized a novel probabilistic approach and first considered a simplified case when regimeswitching parameters were replaced by time-dependent ones, whose solution acts as a given condition when working on a general case.Compared with the literature requiring Fourier inversion or fast Fourier transform, the obtained solution involved no numerical integration and was written using only fundamental functions, which greatly enhances its efficiency.Variance swap prices are also numerically shown to be significantly influenced by jumps and regime switching.

Fig. 1
Fig. 1 Verification of the correctness