Multiscale stochastic volatility for variance swaps with constant elasticity of variance

The variance swap is one of volatility derivatives popularly used for the risk management of financial instruments traded in volatile market. An appropriate choice of a volatility model is an important part of the risk management. One of desirable considerations should be given to the fact that the volatility varies on several characteristic time scales. Stochastic volatility models can reflect this feature by introducing multiscale volatility factors. However, pure stochastic volatility models cannot capture the whole volatility surface accurately although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. So, we choose a hybrid model of constant elasticity of variance type of local volatility and fast and slow scale stochastic volatility for evaluating the fair strikes of variance swaps. We obtain a closed-form solution formula for the approximate fair strike values of continuously sampled variance swaps and compute the solution. The theoretical formula is validated through Monte Carlo simulation. The predictability of the strike price movements is discussed in terms of the sensitive effects of the stochastic volatility and the elasticity of variance parameters for a given partial information about the underlying asset and volatility.


Introduction
Volatility itself began to be considered as an asset in 1993 after the Chicago Board Options Exchange (CBOE) Volatility Index (VIX) was introduced. Volatility trading volume has been increased rapidly in real markets, and a variety of volatility derivatives have been produced since then. As volatility markets have developed, volatility and variance swaps have dominated the markets among other things. The purpose of volatility or variance swaps are predicting the future levels of volatility, trading spreads between the implied and realized volatilities, or hedging the risks provoked by volatility movements. A variance swap is a forward contract for the future realized variance of an underlying asset. This product allows an investor to speculate about the B Jeong-Hoon Kim jhkim96@yonsei.ac.kr Ji-Su Yu yujisu@yonsei.ac.kr 1 Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea spread between its realized and implied variances or hedge its volatility risk. One leg of the swap pays a fixed amount called a strike, while the other leg pays an amount reflecting the realized variance of the underlying asset returns. The difference between these two amounts would be net income for the counterparties settled in cash at the expiration of the contract. Refer to Demeterfi et al. (1999) and Carr and Madan (2001) for general understanding of this type of product.
The question of how to determine the fair value of the strikes of variance swaps has been an active research topic in mathematical finance community since its inception in 1998. For example, Little and Pant (2001) developed a finite difference method for the fair strike price of discretely sampled variance swaps in the local volatility model of Dupire (1994) and Derman and Kani (1994). Zhu and Lian (2011) extended the work of Little and Pant (2001) and obtained a closedform solution under the stochastic volatility model of Heston (1993). Wang et al. (2015) obtained a closed-form approximate solution under the constant elasticity of variance (CEV) model of Cox (1975;1996). Kim and Kim (2019) derived both semianalytic exact and analytic approximate solutions for discretely sampled variance swaps under a double lognormal stochastic volatility model.
The above-mentioned studies of pricing the fair strikes of variance swaps are based on either pure stochastic volatility or pure local volatility models. However, the popular pure stochastic volatility models such as the Heston model have a practical difficulty of capturing the whole implied volatility surface, especially for those deep in-the-money and out-ofthe money option strikes. It is because that the market implied volatility data with near-the-money strikes are usually used to calibrate the parameters of the pure stochastic volatility models. Moreover, those stochastic volatility models are far from capturing the implied volatility surface with very short and long time-to-maturities. Refer to, for instance, Clark (2011) and Tian (2013) for more details. On the other hand, the pure local volatility models developed by Dupire (1994) and Derman and Kani (1994) and the CEV model are easy to calibrate and hedging can be based only on the underlying asset. However, those models cannot reflect such a random nature of volatility itself that the volatility varies on several characteristic time scales as the only source of randomness of volatility is the underlying asset price. In this sense, it is desirable to choose a hybrid model of stochastic and local volatilities for pricing volatility derivatives traded in volatile market in order to capture the diverse characteristics of volatility.
In this paper, we choose the multiscale volatility model of Fouque et al. (2011) for pure stochastic volatility and the CEV model for pure local volatility and combine these two models to create a hybrid multiscale stochastic-local volatility model for pricing the fair strikes of continuously sampled variance swaps. In fact, Choi et al. (2013) proposed this type of hybrid model but with single fast scale stochastic volatility to evaluate the European option prices and recently Cao et al. (2021) obtained a semiclosed form approximation formula for continuously sampled variance swaps under this hybrid model. Our model chosen in this paper is a more general framework than that of Choi et al. (2013) in the following sense. There are two differences even if the CEV diffusion is commonly used as a local volatility part. First, the stochastic volatility part in Choi et al. (2013) is given only by a fast mean-reverting Ornstein-Uhlenbeck (OU) process while the stochastic volatility in this study is more generally fluctuating on a fast time-scale. It is driven by an ergodic process that has an invariant distribution. The ergodic process could be the OU process as used in Choi et al. (2013), the CIR process of Cox et al. (1985) or any other ergodic process. Second, the stochastic volatility part in our model is driven by not only the fast scale diffusion process but also a slow scale process. The addition of the slow factor to the volatility model can provide an efficient way to improve the fit to the longer timeto-maturities in option pricing as discussed in Fouque et al. (2011). Also, a recent study by Seo and Kim (2022) on the stochastic elasticity of variance suggests that a slow-scale persistent random factor is required to cope with the market behavior of underlying assets for pricing derivative products with long time-to-maturity.
The main contribution of this paper is to obtain a closedform pricing formula for the fair strikes of variance swaps under the hybrid multiscale stochastic-local volatility framework and show that there are significant effects of the slow-scale factor on the fair strike prices. To fulfill this, this paper is organized as follows. Section 2 describes a hybrid multiscale stochastic-local volatility framework and establishes a pricing problem for the fair strikes of variance swaps in terms of a partial differential equation (PDE). In Sect. 3, we solve the PDE problem by using some asymptotic analysis and obtain a closed-form formula for the fair strike prices. Section 4 validated the theoretical result via Monte Carlo simulation and discussed the effects of the multiscale stochastic volatility and the elasticity of variance on the fair strike values. Section 5 concludes this paper.

Multiscale stochastic volatility
Let X t be the price of a risky asset on a probability space ( , F, F t , Q) at time t, where Q is the risk-neutral probability measure. Then, the CEV model for X t is given by the stochastic differential equation (SDE) where r (risk-free rate of interest), σ (volatility coefficient) and β (elasticity of variance) are positive constants. If β = 0 in (1), then the CEV model becomes the Black-Scholes model (Black and Scholes 1973). The inequality 0 < β < 1 is required to satisfy the absorbing condition at X t = 0. The CEV model suggests that an underlying asset and its return volatility have a definite, either positive or negative, correlation depending on the choice of the elasticity of variance parameter. Empirical studies such as Harvey (2001) and Ghysels et al. (1996), however, show that the elasticity of variance is time varying. Kim et al. (2014) shows that the elasticity of variance should be treated as a random process and the CEV model has a difficulty of capturing the implied volatility smile, one of main features of volatility observed in the market. So, it is required that the volatility has a random source other than the underlying asset itself. That is, a hybrid model giving a stochastic volatility correction to the CEV model is necessary. There are two possible approaches. The first one is to replace the constant elasticity of variance with a function of a random process as developed by Kim et al. (2014Kim et al. ( , 2015. The second approach is to multiply the CEV volatility by a function of a random process as done by Hagan et al. (2002) (the SABR model) and Choi et al. (2013). We choose the hybrid model of Choi et al. here, but the model in this article is a more general framework than theirs.
Generally speaking, the microscale volatility models are expected to be not efficient enough, while the macroscale volatility models are not accurate enough. So, a reasonable compromise between accuracy and efficiency is in need of multiscale volatility. Following Fouque et al. (2011), we change the constant σ in (1) into a function, f , of not only a fast scale diffusion process, Y t , but also a slow scale diffusion process Z t : Here, Y t and Z t satisfy the SDEs respectively, where the functions α, γ , c and g are assumed to satisfy the existence and uniqueness of solutions to these SDEs and ε and δ are small positive parameters. The processes Y t and Z t are assumed to satisfy the necessary conditions stated in Fouque et al. (2016), in particular, about ergodicity and existence of a unique invariant distribution. So, the volatility of the underlying asset is given by It is driven by three processes X t , Y t and Z t . It is thought of as a CEV-like model with fast and slow variation factors of pure stochastic volatility.

Problem formulation
In terms of the strike, K , and the realized variance, V re t,T , of a continuous variance swap over the time interval [t, T ], the payoff, h t,T (K ), of the swap is given by where N var is the given notional amount of the variance swap. Then, the value, V var t (K ), of the variance swap at time t is where F t denotes a given filtration. Since the fair strike, denoted by K var t , at time t is defined by the condition that the contract is fair to both parties, K var t should satisfy V var t (K var t ) = 0 and so As in Brodie and Jain (2008), we define the realized variance V re t,T by where V t denotes the instantaneous variance which is given by in our hybrid model. In practice, the real variance is discretely sampled and thus, the continuous sampling assumption may cause some approximation errors. However, the daily monitoring of term sheets of this kind of contract generally reduces the discrepancy considerably. For example, the relative difference between continuous and daily sampled variance swaps is around 0.1-0.2% under the exponential Ornstein-Uhlenbeck volatility model as shown in Kim and Kim (2019). So, we keep the idealized contract variance based on continuously sampled realized variance without loss of generality. Then, using the Ito lemma, the fair strike price K var t is expressed as Therefore, the pricing of the fair variance strike (based on a known information about the value of X t , Y t and Z t ) becomes a calculation of the conditional expectation where the Markov property of the joint process (X t , Y t , Z t ) has been applied. This can be thought of as a pricing problem for a European derivative but without discount factor. Using the Markov property of the underlying processes, we solve this problem after applying the well-known Feynman-Kac theorem to change the conditional expectation into a PDE problem given by where L ε,δ,β X ,Y ,Z is an infinitesimal generator of the joint diffusion process (X t , Y t , Z t ) given by It is very difficult to find an exact analytic solution of the PDE problem (5) for general function f , it is inevitable to seek an approximate solution for the problem. Fortunately, the assumption that the parameters ε (the inverted mean reversion rate of volatility) and β (the elasticity of variance) are small is well known to meet the market findings in literature. Refer to, for example, Fouque et al. (2011) and Kim et al. (2014).

Pricing with two scale stochastic volatility
We approximate the fair value of the strike of a variance swap when the volatility is driven by the two processes Y t and Z t in (2). This section is divided into two parts. The first part covers some preliminary lemmas required to prove the main result of this paper. The second part derives the theoretical main result, i.e., an approximate solution formula for a variance swap, by using the asymptotic analysis of Fouque et al. (2011).

Preliminary lemmas
First, we reorganize the operator L ε,δ,β as follows with respect to the small parameter δ.
The operators L β 1 and L β 2 involve the small parameter β. The Taylor expansions of x 1−β and As assumed in Sect. 2, Y t is an ergodic process with an invariant distribution. In the asymptotic analysis of this paper, notation · defined by ξ = ξ(y) (y) dy (6) is going to be hired several times, where is the invariant distribution of the process Y t and ξ can be any function as long as ξ is allowed to be integrable. The following lemma, called the centering condition, for the operator L 0 is very useful in our analysis.

Lemma 1 The Poisson equation
has a solution only if the function ξ is centered with respect to the invariant distribution of the process Y t , i.e., Proof Refer to Section 3.2 in Fouque et al. (2011).
The solution of L 0 p(t, x, y, z) = 0 is assumed to satisfy the growth condition of the following lemma for the operator L 0 .
Lemma 2 Assume that equation L 0 p(t, x, y, z) = 0 admits only solutions that do not grow as fast as Then, the solution p does not depend on variable y.
Proof Refer to Theorem 4.1 in Choi et al. (2013).
Also, we need the following two lemmas for the operator L 20 .

Lemma 3 The solution of the final value problem
can be expressed as p (t, x, z) Proof Refer to Proposition 4.2 in Cao et al. (2021).

Lemma 4 The solution of the final value problem
is given by Proof This result of a simple extension of Proposition 4.3 in Cao et al. (2021) for the function f of one variable y to the function f of two variables y and z.

Asymptotic analysis
We are interested in the solution P ε,δ,β of the PDE problem (5) in the form Inserting this expansion into Eq. (5) and collecting terms of like-powers of √ δ, we find that √ δ-term : and so on. For Eq. (7), we expand P where P ε 0,i is expanded as Then, the terms P 0,i, j are the ones corresponding to the problem only with fast varying volatility and the relevant analysis should be similar to the case of fast mean-reverting volatility studied by Cao et al. (2021). So, we omit the detailed analysis here and simply write the results as follows.
where ψ is a solution to the Poisson equation Here, Lemma 3 has been used for obtaining the term P 0,0,0 . Now, we analyze Eq. (8) with respect to the parameters β and ε. If we expand P ε,β 0 and P ε,β 1 as P ε,β 0 respectively, then we have a hierarchy as follows.

Applying Lemma 1, this equation becomes
Using Lemma 4 with the finial condition P 2,0,0 (T , x, z) = 0, its solution is given by We put the above results together to obtain the following main result for the fair strike price.
Proposition 1 If the underlying asset price X t follows the risk-neutral model (1) with the multiscale stochastic volatility dynamics given by (2) and the required assumptions stated in Sects. 2 and 3 are satisfied, then the fair strike price at time t is approximated by K multi t which is K multi

Numerical experiments
In this section, we use numerical experiments to discuss the validity of our analytic result, the impact of the slow scale volatility factor on the fair strike values of variance swaps, and the leverage effect directly caused by the elasticity of variance parameter. Specially, we use Monte Carlo (MC) simulations which are computational algorithms that depend on repeated random sampling to obtain numerical results. In this paper, we use Euler-Maruyama scheme to generate sample paths of the given stochastic processes. This scheme approximates discretized sample paths of X t up to time T with N sub-intervals, which is given by 1 and j = 1, 2, . . . , N . Also, we use Gauss-Legendre quadrature to calculate the variance swap prices given by our formula. There are integral parts in our formula that are difficult to calculate directly. So, we use the Gauss-Legendre quadrature to approximate the integral. This section is divided into two parts. The first part covers three specific models of fast mean-reverting volatility. The second part deals with a specific model with the volatility possessing both fast and slow mean-reverting factors.

Fast mean-reverting volatility
One can specify the pure volatility part f (Y t ) of the hybrid volatility f (Y t )X −β t in several ways. We choose three different specific volatility models which have the same mean-reverting property but three different invariant distributions and discuss about the impacts of the mean-reversion rate and the model choice on the fair strike prices of variance swaps. The parameters selected below provide the same initial value of the volatility among the three models.
The first volatility model is given by where θ = 0.07597, σ = 0.48665, r = 0.05 and W x t and W y t are Brownian motions with correlation coefficient ρ xy = −0.67099. We call this the CIR-CEV volatility model. The parameters come from Mrazek et al. (2017). The second volatility model is given by where θ = 0, m = 0.275, ν = 0.3239 and r = 0.05 are chosen and W 1 x and W y t are Brownian motions with correlation coefficient ρ = −0.45. The model is called the ExpOU-CEV volatility model. The parameters are cited from Kim and Kim (2019).
The third model that we consider is given by where θ = 0, ν = 1, r = 0.05 and W x t and W y t are Brownian motions with correlation coefficient ρ xy = −0.4. We call this model the OU-CEV volatility model in brief. The parameters are cited from Cao et al. (2021).
For comparison, we use S & P 500 market data in 2020 and 2021 and choose five different values of X 0 between 3200 and 4000 and two different values, 0.005 and 0.05, of the parameter ε. Since β affects only the underlying process X t , we fix β = 0.01. As shown in Table 1, the increase in the mean-reversion rate from ε = 0.05 to ε = 0.005 leads to the increase in the fair strike value in the CIR-CEV and ExpOU-CEV volatility models. It is due to the difference between the volatility functions in the models. Technically, the CIR process Y t follows a non-central χ 2 distribution in the CIR-CEV model and as ε decreases, the non-centrality parameter increases. Then, the increasing expectation of volatility follows. On the other hand, the OU process Y t in the ExpOU-CEV model is normally distributed and as ε becomes smaller, E(e Y t ) gets larger and thus, the fair strike value increases. However, the OU-CEV model presents a different situation. In Table 1, (+) indicates that the initial value Y 0 is greater than the long-run mean level of Y t while (−) means that Y 0 is less than that. Specifically, the long-run mean is 0 and the initial values of Y t are 100 and −100, respectively. The fair strike prices would be impacted by the long-run mean and the initial value of Y t . In fact, if the value of ε is very small like ε = 0.005 and Y 0 is smaller than the long-run level, then Y t is fast mean reverting and converges to the long-run mean level and thus, the volatility becomes higher in short time. If the value of ε is very small, but Y 0 is larger than the long-run level, then the volatility becomes smaller.
Under the OU-CEV volatility model (29), we check the validity of our analytic formula via Monte Carlo (MC) simulation. For the MC simulation, 100,000 sample paths are used for the variance swap. We calculate the square root of the fair strike price as a percentage (SRVS(%)) for four different values of ε. We compute the values based on the S & P 500 market data in 2020, and the initial underlying value X 0 lies between 2500 and 3500. In Fig. 1, the solid line represents the theoretical strike prices and the circles indicate the MC simulation outcomes. The theoretical results match well those from the MC simulation regardless of the choice of the mean-reversion rate ε. Similarly, one can show that the analytic solutions match well those from the MC simulation in the CIR-CEV and ExpOU-CEV volatility models also although the graphical representations are omitted here.
The OU-CEV volatility model (29) can be reduced to a previously known single-factor (local or stochastic) volatility model. More specifically, the model (29) becomes a CEV model when ε goes to zero and it becomes a (fast meanreverting) OU volatility model when β goes to zero. We compute the fair strike prices under the CEV, OU-CEV and OU volatility models for six different choices of the initial value X 0 in order to check if there is any difference among those models in terms of fair strike price level. Here, β is chosen to be 0.01 in the CEV model and β and ε are 0.01 in the OU-CEV model. As shown in Table 2, the fair strike prices with the OU volatility model are higher than the corresponding prices under the CEV model and the OU-CEV prices are between those two prices. This suggests that the pure stochastic volatility plays a role in enhancing the value, while the local volatility has the effect of lowering the value when both stochastic and local volatility factors are involved in a such hybrid volatility model as in the OU-CEV volatility case. This flexibility is a strong merit of hybrid stochasticlocal volatility models in general.

Fast and slow mean-reverting volatility
We consider a specific multiscale volatility model given by  which is an extended version of the ExpOU-CEV volatility model (28) in terms of the slow scale volatility factor Z t . For computation, the parameters are cited from Fouque et al. (2004) and they are r = 0.1, θ = −0.8, ν 1 = 0.5, δ = 0.05, m = −0.8 and ν 2 = 0.8. The Brownian motions W x t , W y t and W z t are correlated by ρ xy = −0.2, ρ yz = 0 and ρ zx = −0.2.
In Table 3, the numbers are the fair strike prices given by the analytic formula in Proposition 1 and the numbers in parentheses are the MC simulation results based on S & P 500 market data in 2020. The theoretical and MC simula-tion results are well matched regardless of the choice of ε. We notice from the comparison between Tables 1 and 3 that the addition of the slow scale volatility factor leads to a significant difference between the fair strike values in the fast scale volatility model and the ones in the multiscale volatility model. Table 4 displays the fair strike prices given by the analytic formula and the MC simulation results (the numbers in parentheses) for the fixed ε = 0.01 and the four different values 0.03, 0.01, 0.005 and 0.001 of β. The leverage effect in finance is that rising asset prices are accompanied by declining volatility, and vice versa. It is one of stylized effects of volatility observed in market. The table shows well that β controls directly the degree of the leverage effect.

Conclusion
In this paper, we employ asymptotic analysis to derive a closed form solution formula for the approximate fair strike values of continuously sampled variance swaps under a hybrid framework of constant elasticity of variance and multiscale stochastic volatility in a risk-neutral market environment. Given the invariant distribution of the process driving fast scale fluctuations of the volatility, the fair strike prices of variance swaps are obtained in an affordable manner. Our analytic result can provide the prediction of the strike price movements from the information about the elasticity of variance and the multiscale stochastic volatility. Our analysis shows that the fair variance strikes produced by the theoretical formula match well with those by the Monte Carlo simulation. Based on some specific examples of the hybrid volatility to compute the fair strike prices, we investigate the impacts of the slow scale stochastic volatility factor and the local volatility factor (the elasticity of variance parameter) on the fair strike prices of variance swaps. First, we find that the slow scale stochastic volatility factor gives a significant effect on the fair strike value. Second, the local volatility component lowers the fair strike value while the pure stochastic volatility component enhances the value. Third, the elasticity of variance parameter is shown to control directly the degree of the leverage effect. Fourth, the fair strike value increases (decreases) as the mean-reversion rate increases when the long-run mean level of volatility is higher (lower) than the initial volatility. The flexibility of the hybrid stochastic-local volatility model may allow you to take full advantage of a large sample size of volatility data and assess appropriately the variance swaps used for the risk management in volatile market.
Author Contributions All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by J-SY and J-HK. The first draft of the manuscript was written by J-SY and then finalized by J-HK. All authors read and approved the submitted manuscript.
Funding The research of J.-H. Kim was supported by the National Research Foundation of Korea NRF2021R1A2C1004080.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.