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A probabilistic approach for the valuation of variance swaps under stochastic volatility with jump clustering and regime switching
Financial Innovation volumeÂ 10, ArticleÂ number:Â 114 (2024)
Abstract
The effects of stochastic volatility, jump clustering, and regime switching are considered when pricing variance swaps. This study established a twostage procedure that simplifies the derivation by first isolating the regime switching from other stochastic sources. Based on this, a novel probabilistic approach was employed, leading to pricing formulas with timedependent and regimeswitching 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 performed to obtain the final results, since they are completely analytical without involving 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 riskmanagement 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 CoxIngersollRoss (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 etÂ 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 etÂ al. 2020; Wu etÂ al. 2022; Badescu etÂ al. 2019).
The abovementioned works neglect the possibility of price jumps in underlying stocks, which violates statistical/empirical observations (Bates 1996; Eraker 2004; Hu etÂ al. 2024). This leads to an investigation of pricing variance swaps with Poisson jumpdiffusion or Levy jumps (Broadie and Jain 2008; Hong and Jin 2023; Pun etÂ al. 2015; Carr etÂ al. 2012). However, independent increments assumed in these jump models prevented the incorporation of jump clustering, which often occurs in practice (AÃ¯tSahalia etÂ al. 2015). Therefore, various research interests have guided the adoption of Hawkes jumpdiffusion models for valuing financial derivatives, including vulnerable options (Ma etÂ al. 2017), power exchange options (Pasricha and Goel 2020), and Volatility Index (VIX) options (Jing etÂ 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, regimeswitching stochastic volatility was considered by Elliott and Lian (2013), Lin and He (2023) when pricing variance swaps, which were extended by Cao etÂ al. (2018) to incorporate the regimeswitching stochastic interest rate further. Jump diffusion was added to a similar framework for pricing variance and volatility swaps (Yang etÂ 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 closedform 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.Â 2. SectionÂ 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.Â 4.
The model
The riskneutral measure \({\mathbb {Q}}\) was emphasized as a part of the complete probability space \((\Omega ,{\mathcal {F}},{\mathbb {Q}})\). A continuoustime Markov chain,^{Footnote 1} denoted by \(\{X_t\}_{t\ge 0}\), is introduced to model the variation in economic conditions, such that its value is equal to that of the twodimensional unit vectors. When the generator of the Markov chain is \(\Lambda (t)=[\lambda _{ij}(t)]_{i,j=1,2}\) and the Martingale increment process conditional on the filtration generated by the Markov chain is \(\{V_t\}_{t\ge 0}\), one can have
resulting from the semimartingale representation theorem (Elliott etÂ al. 2008; He and Lin 2023a, b).
In the considered riskneutral world where the riskfree interest rate is denoted by r, \(Y_t\), \(\nu _t\), and \(\gamma _t\) are respectively used to denote the price of the stock, its variance, and the intensity of the Hawkestype jump. Their dynamics in the formulated riskneutral world are
\(\{W_{i,t}\}_{t\ge 0}, i=1,2\) are two Wiener processes, with \(dW_{1,t}dW_{2,t}=\rho dt\) representing the connection between the stock price and its variance. \(\nu _t\) is meanreverting, the speed of which is p. This process is affected by the volatility, \(\zeta\). The longterm 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 \({\hat{q}}=(q_1,q_2)\), that is, \(q_{X_t}=<{\hat{q}},X_t>\) (He and Lin 2023c). The jump sizes with support for \((1,\infty )\) have identical and independent distributions, with \(E(J_t)=\omega\). \(H_t\) is the Hawkes process, which is defined as:
with jump time moments denoted by \(\{t_i\}>0\). The expression of jump intensity \(\gamma _t\) considers the jump clustering effect, indicating that the occurrence of a jump contributes to a greater intensity at the rate of \(\delta\), and the increment then experiences exponential decay at the rate of \(\eta\).
A natural transform applied to stock price dynamics is \(Z_t=\ln (Y_t)\), which, when combined with Itoâ€™s lemma, yields the dynamics of \(Z_t\) as
where the log stock price jumps to \({\tilde{J}}_t=\ln (1+J_t)\). \({\tilde{J}}_t\) is assumed to be identically and independently distributed following a normal distribution for any given t with \(\mu\) and \(\sigma ^2\) as its mean and variance, respectively. In this setting, \(\omega =E\left( e^{{\tilde{J}}_t}1\right) =e^{\mu +\frac{1}{2}\sigma ^2}1\).
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 subintervals uniformly. This yields the swap payoff \((RVK)Z\), 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 etÂ al. 2015; Badescu etÂ al. 2019; Zheng and Kwok 2014; Lin and He 2023).
Owing to the nature of the swap contractâ€™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 \(g_2(Y_0,\nu _0,\gamma _0,s,t)\) is an a priori step in determining \(g_1(Y_0,\nu _0,\gamma _0,X_0,s,t)\) as well as the variance swap delivery prices. This first step assumes a given Markov chain, reducing the problem of finding the variance swappricing formula with the longterm variance \(q_t\) being no longer stochastic but timedependent. First, we present a solution to this simplified problem.
The formula with timedependent longterm variance
Before formally dealing with the problem of \(g_2(Y_0,\nu _0,\gamma _0,s,t)\), we first present some useful results associated with \(\nu _t\) and \(\gamma _t\) that are needed in the derivation.
Proposition 1
If the stock variance \(v_t\) and jump intensity \(\gamma _t\) are governed by the stochastic differential equations presented in Eq.Â (2), with the stochastic variable \(q_{X_t}\) replaced by the timedependent variable \(q_t\), we must have
where \(a_1=\eta \delta\) and \(a_2=\frac{\eta \gamma _{\infty }}{a_1}\).
Appendix 1 provides the proof of PropositionÂ 1.
With the expectations and variances of \(\nu _t\) and \(\gamma _t\) presented, we are now ready to compute \(g_2(Y_0,\nu _0,\gamma _0,s,t)\). This requires us to calculate the conditional expectation of \(\ln \left( \frac{Y_t}{Y_s}\right)\), which contains two random variables when \(s>0\): To simplify the computation, the tower rule of expectation is used, such that
where
Clearly, the target \(g_2(Y_u,\nu _u,\gamma _u,u,s,t)\) is the expectation of \(g_3(Y_s,\nu _s,\gamma _s,s,t)\), 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.Â (2), then we must have
and
where
and
with \(a_3=\omega \mu\), \(a_4=\sigma ^2+\left( \mu \frac{\delta a_3}{a_1}\right) ^2\), and \(\tau =ts\).
We prove PropositionÂ 2 in Appendix 2.
Now, we obtain \(g_3(Y_s,\nu _s,\gamma _s,s,t)\) in Eq.Â (11) by using \(g_3=(g_{3,1})^2+g_{3,2}\), providing
where
The problem that remains to be solved involves the computation of \(g_2(Y_0,\nu _0,\gamma _0,s,t)\) using Eq.Â (10). With the solution to \(g_3(Y_s,\nu _s,\gamma _s,s,t)\) given in Eq.Â (15), the target expectation requires the computation of \(E(\gamma _s{\mathcal {F}}_{0}^{H})\), \(E\left[ (\gamma _sa_2)^2{\mathcal {F}}_{0}^{H}\right]\) and \(E(\nu _s{\mathcal {F}}_{0}^{W_2})\). Because the required solutions have already been presented in PropositionÂ 1, the results of \(g_2(Y_0,\nu _0,\gamma _0,s,t)\) are illustrated directly without requiring tedious calculations and rearrangements.
where
with \(A_5=h_2(\gamma _0,0,s)a_2=(\gamma _0a_2)e^{a_1s}\).
Once this stage is reached, one should realize that when the longterm stock variance, \(q_\xi ,~\xi \in [0,T]\), is time dependent, Eq.Â (5) should be revised such that the corresponding delivery price is a result of
This formula can be used with the formulation of \(g_2(Y_0,\nu _0,\gamma _0,s,t)\) as shown in Eq.Â (17). The results associated with the time dependence of \(q_\xi ,~\xi \in [0,T]\) are not the target values, and their regimeswitching nature should be reintroduced to capture the varying economic states.
The formula with regime switching longterm variance
By removing the assumption that the Markov chain is foreseeable, the variance swap delivery prices should be computed using Eq.Â (5), implying that the calculation of \(g_1(Y_0,\nu _0,\gamma _0,X_0,s,t)\) using Eq.Â (7) is required. The Markov chain is the only stochastic source when deriving \(g_1(Y_0,\nu _0,\gamma _0,X_0,s,t)\) using the expression \(g_2(Y_0,\nu _0,\gamma _0,s,t)\). Therefore, the terms that one needs to find are \(U_i=E(M_i{\mathcal {F}}_{0}^{X}), i=1,2,...,7\), and we now solve them individually. Before presenting these results, we first provide some frequently used notations. We define three general functions asfollows:
and also define \(d_1=(q_1^2q_1q_2)\frac{\lambda _{21}}{\lambda }+q_1q_2,~d_2=\frac{\lambda _{12}}{\lambda }(q_1^2q_1q_2),~d_3=(q_2^2q_1q_2)\frac{\lambda _{12}}{\lambda }+q_1q_2,~d_4=\frac{\lambda _{21}}{\lambda }(q_2^2q_1q_2), d_5=\frac{d_1d_3}{\lambda }, d_6=\frac{d_2d_4}{\lambda }\). Thus, the solution to \(U_1\) is presented in PropositionÂ 3.
Proposition 3
With \(q_{X_\xi }, \xi \in [0,t]\) being a regimeswitching parameter controlled by Markov chain \(X_t\), we have
where
and
PropositionÂ 3 will be validated in Appendix 3.
Because the derivation of \(U_2\) is similar to that of \(U_7\), we provide their solutions in PropositionÂ 4.
Proposition 4
With \(q_{X_\xi }, \xi \in [0,t]\) being a regimeswitching parameter controlled by Markov chain \(X_t\), we have
where
and
Appendix 4 provides the proof of PropositionÂ 4.
The remaining unknown value \(U_i, i=3,4,5,6\) can be determined similarly. Their formulae are presented together in Proposition 5.
Proposition 5
With \(q_{X_\xi }, \xi \in [0,t]\) being a regimeswitching parameter controlled by Markov chain \(X_t\), we have
where
and
Appendix validation propositionÂ 5.
Once we have determined \(U_i, i=1,2,...,7\), the results of target \(g_1(Y_0,\nu _0,\gamma _0,X_0,s,t)\) are summarized in the following theorem.
Theorem 6
With the formulae of \(U_i, i=1,2,...,7\) being provided in Propositions 3.3\(\)3.5, the function \(g_1(Y_0,\nu _0,\gamma _0,X_0,s,t)\), when \(Y_t\) follows Eq.Â (2), has a solution of
Proof
The proof of this theorem is straightforward after considering the expectations of \(g_2(Y_0,\nu _0,\gamma _0,s,t)\). \(\square\)
The strike prices of the variance swaps can now be analytically computed using Eq.Â (5) when the effects of jump clustering, stochastic volatility, and varying economic conditions are incorporated. Note that this formula does not involve Fourier inversion or any other integration, which can significantly improve efficiency in practice.
It would also be interesting to investigate how variance swaps with continuous sampling behave under the considered model, which can provide an additional check on the validity of the proposed formula. By defining
Based on Chebyshevâ€™s inequality, one can illustrate that \(RVRV^c\) 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
from the results of propositionsÂ 1 andÂ 5.
Formulae with discrete and continuous sampling are numerically implemented in the next section to demonstrate the impact of the three stochastic factors.
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.
FigureÂ 1 displays the strike prices produced by Eq.Â (5) compared to Monte Carlo prices. The clear pattern shown in Fig.Â 1a indicates a very close agreement between the corresponding results from the two methods, which is further supported by Fig.Â 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 clustering, and variations in economic status (HJSVRS) and that constructed by our model ignoring jump clustering (PJSVRS). This comparison is illustrated in Fig.Â 2 with the varying decay rates of jump clustering \(\eta\). Some distance exists between the swap prices generated by the two models, and decreasing the value of \(\eta\) 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.Â 3, where increasing the value of \(\delta\) is equivalent to increasing strike prices. A larger \(\delta\) implies a higher possibility for the arrival of another jump, creating more significant jumpclustering 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 \(\delta\) 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 \(\gamma _\infty\) different values in Fig.Â 4. An increasing trend in variance swap prices can be observed with the increase in \(\gamma _\infty\), leading to a wide difference between HJSVRS and PJSVRS model prices. The higher \(\gamma _\infty\) 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.Â 5 after equating \(\lambda _{12}\) and \(\lambda _{21}\) to \(\lambda\) 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 longterm 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 varianceswap 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 timedependent 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.
Availability of data and materials
No datasets were generated during and/or analysed during the current study.
Notes
It is assumed to be of two states for the ease of discussion, but the results with \(N>2\) states can easily be obtained in a similar manner.
Abbreviations
 CIR:

Coxâ€“Ingersollâ€“Ross
 VIX:

Volatility index
 HJSVRS:

Model with stochastic volatility, jump clustering, and regime switching
 PJSVRS:

Model with stochastic volatility, Poisson jumps, and regime switching
 HJSV:

Model with stochastic volatility and jump clustering
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Acknowledgements
The authors gratefully acknowledge the anonymous refereesâ€™ constructive comments and suggestions, which greatly helped improve the correctness and readability of the manuscript.
Funding
This work is supported by the National Natural Science Foundation of China (Nos. 12101554, 12301614), the Fundamental Research Funds for Zhejiang Provincial Universities (No. GB202103001), Zhejiang Provincial Natural Science Foundation of China (No. LQ22A010010), and Ministry of Educational Social Science Foundation of China (No. 21YJC880050).
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Appendices
Appendix 1
Here, we prove PropositionÂ 1. To determine the first and second moments associated with \(\nu _t\), we first apply Itoâ€™s lemma to derive
The integration of from s to t leads to the expression \(\nu _t\) asfollows:
The expectation of \(\nu _t\) is straightforward, because \(\int ^t_s\sqrt{\nu _\xi }e^{p(t\xi )}dW_{2,\xi }\) is a martingale whose expectation is zero. The variance of \(\nu _t\) can be computed as
With
one can then obtain
We also need to derive the first and second moments associated with \(\gamma _t\). We first adjust the jump process to form a martingale through \(N_t=H_t\int _0^t\gamma _\xi d\xi\) so that the expectation of \(dN_t\) is zero. As a result, we obtain
One can further compute the dynamics of \(d\left( e^{a_1(t\xi )}\gamma _\xi \right)\) with Ioâ€™s Lemma, which provides a representation of \(\gamma _t\) as
This directly gives the result of \(E(\gamma _t{\mathcal {F}}_{s}^{H})\) due to \(\int _s^te^{a_1(t\xi )}dN_\xi\) being a martingale with its mean as zero. Moreover, we can also obtain
which leads to the desired explicit result after working out the integration.
Appendix 2
Here is the proof of Proposition 2.
To compute the expectation and variance of \(\ln \left( \frac{Y_t}{Y_s}\right)\), a necessary step is to figure out its expression. In particular, Integrating both sides of Eq.Â (4) from s to t should yield
A simple treatment gives
which directly leads to
Here,
Taking the expectation on both sides of Eq.Â (30) and using \(E(L_t)=\mu\), we obtain the desired result in Eq.Â (13).
However, considering the variance on both sides of Eq.Â (30) results in:
One can directly have
which is the result of \(J_1\) being constant given \({\mathcal {F}}_{s}^{H}\). Based on the results of Proposition 1, we can further compute
The third unknown term is calculated as follows:
With
one can arrive at
From the results in PropositionÂ 1, the fourth unknown term can be derived as
We can further compute the fifth unknown term using the results in PropositionÂ 1, leading to
where
The solution to \(g_{3,2}\) in Eq.Â (31) can then be found by substituting the results in Eqs.Â (32)â€“(37), as well as some simplifications.
Appendix 3
Here, we prove PropositionÂ 3.
We can formulate
where the final step is the result of the tower rule of expectations. This demands the solution to \(I_{11}={\mathbb {E}}\left\{ q_{u} e^{k(su)}\cdot \int ^{t}_{s}{\mathbb {E}}\left[ q_\xi \left( 1e^{p(t\xi )}\right) {\mathcal {F}}_{u}^{X}\right] d\xi {\mathcal {F}}_{0}^{X}\right\}\). Let
where \(\lambda =\lambda _{12}+\lambda _{21}\). We further write
and denote
We can then directly obtain the inner expectation of \(U_{10}\) as
with which we can obtain
where
Simple manipulation yields
where
Substituting Eq.Â (39) into (38) yields:
One can then reach The desired result can be obtained after determining the integrations involved.
Appendix 4
Here, we prove PropositionÂ 4. The computation of \(U_2\) can be expressed as
where
This requires calculation of \(U_{21}\) and \(U_{22}\). Specifically,
where
Following a similar argument, one can also find that
where
Consequently, we arrive at the following hypotheses:
Determining the integration contained in the above formula leads to a solution for \(U_2\).
Similarly, we can reformulate \(U_7\) as
where
Thus, we have
and this further yields
to obtain the desired formulation.
Appendix 5
Here, we prove PropositionÂ 5.
The calculation of \(U_3\) is straightforward through
The corresponding solutions were then obtained.
Similarly, we compute \(U_4\) using
Thus, the solution is as follows:
\(U_5\) can be calculated using
yielding the final solution.
The calculation of \(U_6\) can be performed using:
leading to the final expression.
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He, XJ., Lin, S. A probabilistic approach for the valuation of variance swaps under stochastic volatility with jump clustering and regime switching. Financ Innov 10, 114 (2024). https://doi.org/10.1186/s40854024006404
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DOI: https://doi.org/10.1186/s40854024006404