The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.


Introduction
Compared to the European option, the biggest difference is that American option can be exercised any time before its maturity date. Due to the early exercise feature, the pricing of American option has long been the most challenging research topic in finance (see Karatzas [1], Rogers [2], and Haugh and Kogan [3]). The American option can be valued using an analytic approximation approach called the Barone-Adesi and Whaley method (BAW method) when the underlying asset is driven by a stochastic process with constant volatility [4]. However, the significant leptokurtic feature of the underlying asset process found from lots of empirical evidences indicates that the volatility should be randomly distributed rather than a constant, and thus, the BAW model can hardly be applied if a stochastic process drives the volatility. To solve this problem, many researchers currently resort for the stochastic volatility models. In particular, the multiscale stochastic volatility model has been proposed to deal with the fast data and slow data frequency of the underlying asset.

Multiscale Stochastic Volatility Modelling.
Most of the early studies on the American option always assume that a stochastic process with constant volatility drives the underlying asset. This assumption can simplify the problem but fails to capture the real market feature of the volatility (Giesecke et al. [5]). Numerous extensions have been made for relaxing the overstrict assumption, namely, the stochastic volatility models. The most popular stochastic volatility models are Heston's model (see [6]) and Stein and Stein's model (see [7]), which assume that the volatility is driven by a single-factor stochastic process. However, currently, empirical study shows that the volatility can be rescaled according to the frequency of the observed data. Besides, multifactor stochastic models can be used to improve the accuracy of the option pricing (see Fouque and Zhou [8] and Christoffersen et al. [9]).
The studies by Clarke and Parrott [10] and Muzy et al. [11] defined the time scale as the frequency of the observed data in different time periods. Further study of the time scale divided into fast scale and slow scale has been carried out by Chacko and Viceira [12]. The fast scale describes the short period fluctuation with high frequency, while the slow scale describes the long-term variations with low frequency. A comprehensive empirical analysis was presented by Fouque and Zhou [8]. They proposed the slow-scale volatility model to price the European option and investigated the long run time correction effect on the option pricing. A multiscale volatility model was developed to price option by Fouque et al. [13], from which the authors combined both the fastscale volatility and slow-scale volatility into the volatility processes. In Fouque et al.'s research, the analysis of the fast-scale volatility is connected with the singular perturbation theory, while the study of the slow-scale volatility is associated with the regular perturbation theory. Therefore, the asymptotic approach can be applied to approximate the volatility correction terms.
Despite the popularity of the classic European option pricing, the multiscale model has also been widely used to price other more complicated financial derivatives, such as the Asian option and the VIX future (see Fouque and Han [14] and Fouque and Saporito [15]). The multiscale volatility model is studied in Liu et al. [16], from which the authors incorporated the jump-diffusion terms in pricing European option and the variance swap and applied the finite element method to approximate the generated high-dimension partial integral differential equation (PIDE). Recently, a perpetual American option is investigated under the stochastic elasticity of variance (CEV) model, and the fast-scale correction of the variance elasticities derived by the multiscale asymptotic method [17][18][19]. Apart from the derivative pricing, the multi-scale model can also be applied in studying portfolio selection (see Fouque and Hu [20,21]).

Pricing Methodologies for American Option.
It is well known that American-type option featured by early exercise can be formulated as a free boundary PDE problem, whereas the analytical solution is not available. Hence, numerous studies have been conducted to find an approximation of the American option price including the semianalytical approach, the numerical approach, and the Monte Carlo simulations (see Fouque et al. [22], Longstaff and Schwartz [23], and Stentoft [24]).
The semianalytical approach can be divided into three groups, including an analytical method of lines, the integral equation approach, and capped option approximation approach. A semianalytic valuation method for American option is developed by randomizing the maturity date, from which the maturity date is viewed as several jumps driven by standard Poisson processes [25]. The integral equation approach is built by valuing and hedging American options based on a recursive integration of the early exercise boundary [26]. Instead of approximating the early exercise boundary, the capped method is inclined to impose the lower and upper bounds on American option value [27]. Although the semianalytical methods are supposed to be more efficient, it is hardly applicable if more factors are included. Regarding the simulation approximation, various methods are applied to undertake an analysis of the field. The simplest one is the binomial tree method introduced by Cox et al. [28]. LSM method has been introduced by Longstaff and Schwartz [23], who claimed that the simulation method is favourable for multifactor models. The LSM method approximates and simulates the conditionally expected pay-off function from the cross-sectional information using the least square method. However, the method is proved to be less efficient compared to other numerical and semianalytic approaches.
According to Feynman-Kac theorem, American option pricing problems essentially described by a nonlinear PDE can be solved numerically by the finite difference method and the finite element method [29]. For the time-dependent American option, a free and moving boundary is considered due to the early exercise feature. Two approaches, namely, the fixed-point approach and the penalty approach, are commonly used to deal with the moving boundary. Merton et al. [29] applied the front-fixing transformation to incorporate the unknown boundary into the PDE and solved it numerically as a fixed boundary nonlinear PDE problem. The frontfixing method is highly efficient because it does not have to embed iteration at each time step of evolution. A detailed comparison between the front-fixing method and the penalty method has been carried out by Nielsen et al. [30]. Different from the front-fixing method, the penalty method removes the free boundary by adding a small and continuous penalty term which accurately captures the boundary properties. Nielsen et al. [31] derived a penalty method for solving the American multiasset option pricing problem, and they have proved that the semi-implicit FDM method performs better than the implicit method by avoiding the nonlinear term. However, even though the numerical method is beneficial to dealing with the PDE problem with free boundaries, the method is not perfect for a high-dimensional problem concerning the issue of computation efficiency. This paper is aimed at expanding current research on the subject of American option pricing problem with stochastic volatilities. In our research, a multiscale stochastic volatility model is incorporated to investigate the value of American option. We compare the fast-scale and slow-scale effects of the volatility, which helps to explain the investors' different behaviors when facing the short-run and long-run volatility risks. Since more factors have been taken into account in the model, it will result in a high-dimensional problem subject to moving boundaries, which is hard to solve analytically. Even though many of the numerical methods are useful in solving the higher-dimensional PDE, such as the finite element method in Liu et al. [16], it is highly time-consuming. In our research, we propose an efficient approach to deal with the problem by combining the asymptotic method with the front-fixing method. Besides, as the valuation of the Greeks 2 Journal of Function Spaces is closely related to hedging and the risk exposure of the portfolio selection, in this paper, we study the Greeks numerically to test the sensitivities of model parameters. Last but not least, we apply the real financial market data to calibrate the correction terms, which makes our research more reliable and realistic. Instead of using SPX 500 option quotes to calculate the likelihood of the underlying distribution, we follow Fouque et al.'s calibration framework as an alternative approach to analyse the linear relationship between the implied volatility and the Log-Moneyness in Fouque et al. [13]. The idea of the calibration is straightforward and more efficient with fewer calibrated parameters. The rest of the paper is organized as follows. Model Setup describes the modelling of the underlying asset price, where its volatility is driven by multiscale stochastic processes. Pricing Approximation presents the asymptotic approximation algorithm which is applied to reduce the dimension of the resulting PDE for option pricing. Numerical Analysis is the numerical approach of finite difference for the solution of the PDE problem with free boundary. Numerical illustrations are presented in Empirical Results and followed by a conclusion with future work in light of the empirical findings.

Model Setup
We assume that the price of the underlying asset S is driven by the following stochastic process in the form of stochastic differential equation, where μ is the constant drift term, W ð0Þ t is a standard Brownian motion, and f ðy, zÞ is a function of two factors y and z, which represent "fast-" and "slow-"scale volatility, respectively. The fast-and slow-scale volatilities y and z are driven by the following two stochastic processes: where 1/ξ and σ denote, respectively, the fast-scale rate and slow-scale rate of volatility; Γ 1 ðyÞ and Γ 2 ðzÞ denote the risk premium of the volatility risk under the risk-neutral assumption; and the functions αðyÞ, βðyÞ, cðzÞ, gðzÞ are smooth and at most linearly grow as y ⟶ ∞ and z ⟶ ∞.
Here, we assume that the Brownian motion ðW    The American option provides the contract holder with the right of early exercise before the expiration date T, which is the key difference from European options. Hence, there exists an optimal exercise boundary fBðτÞ, t ≤ Tg, which divides the area into the holding region and the exercise region. It is optimal to exercise the American put option with strike price K when S ≤ BðτÞ subject to the following boundary conditions: On the other hand, if S > BðτÞ, it is the holding region, and the investors choose to hold the contract until the maturity time. The resulting boundary condition in this region becomes and the terminal condition of the free boundary at expiration date is defined as

Pricing Approximation
The asymptotic approximation method for American option pricing can refer to the perturbation theorem, which aims at correcting solutions of a PDE with respect to the small variation of the coefficient. We generate the perturbation theorem by introducing fast-scale correction and low-scale correction at the same time into the process of approximation. The perturbation theorem can be grouped into the singular perturbation and the regular perturbation. The PDE degenerates at the singularity point for the singular perturbation, whereas the regular perturbation will not change the nature of the PDE. In order to approximate the altering of the solution under perturbation theorem, the asymptotic approximation is adapted in the series expansion. In our research, the correction of fast-scale stochastic volatility is regarded as a singular perturbation, while the correction of slow-scale stochastic volatility is considered to be a regular perturbation similar to Fouque et al. [13]. Therefore, American option price P ξ,σ can be decomposed into the leading term and two correction terms, i.e., where P 0,0 denotes the option price without the volatility correction, P ξ 1,0 = ffiffi ffi ξ p P 1,0 the fast-scale volatility correction, and P σ 0,1 = ffiffiffi σ p P 1,0 the slow-scale volatility correction. The error term Error is oðξ Þ with reference to Fouque et al. [32].
Under the proposed fast-scale and slow-scale stochastic volatility model, the price of an American put option PðS, y, z, tÞ satisfies the following PDE problem with free boundary conditions: where t ∈ ½0, T. The operator L ξ,σ is given by To construct a singular perturbation expansion, the asymptotic price P ξ,σ is expanded in the form of To construct a regular perturbation expansion, the asymptotic price P σ is expanded in the form of Similarly, the asymptotic price of free boundary Bðy, z, tÞ is of the form We substitute (10) and (11) into L ξ,σ P ξ,σ = 0 from (7) and collect the similar terms up to order1/2; the results are shown as below: From (9), since the operators L 0 and L 1 only contain the derivatives with respect to y, it can be concluded that P 0,0 , P 1,0 , P 0,1 , and P 1,1 are independent of y according to (14). As a result of the fact that L 1 P 1,0 = 0, equation (15) is a Poisson equation of the form with hGi ≔ Ð gðyÞΠðdyÞ = 0 according to the centring resolvability of Poisson equation, where Π is an invariant distribution to y. Thus, we obtain Correspondingly, the free boundary conditions are necessary to be expanded in terms of ffiffi ffi ξ p and ffiffiffi σ p . By combining (12) and (13) into (7) and applying Taylor expansion, we obtain Journal of Function Spaces The boundary conditions can be derived from (20), (21), and (22). The terminal condition can be rewritten as follows: Here, we propose a series of theorems correspondingly based on the assumptions above to assist for the pricing of American option. Theorem 1. Assume p = P 0,0 to be the leading term without the volatility correction; p can be determined by solving PDE (24): where δ 2 = hf 2 ðy, zÞi ≔ Ð f 2 ðy, zÞ Q ðdyÞ = 0 denotes the mean historical volatility of stock and is subject to the terminal and boundary conditions: where B 0 = B 0,0 denotes the moving boundary of the leading term.
Proof. The boundary conditions are easily derived from (20).☐ From Theorem 1, the leading term p can be obtained by solving a one-dimensional PDE with free boundary conditions. The analytical solution is impossible to obtain; thus, we resort to the numerical solution presented in the next section. Apart from the leading term, the fast-scale volatility and slow-scale volatility terms can be determined by Theorem 2 and Theorem 3.

Theorem 2.
Let P ξ = P 1,0 denote the fast-scale volatility correction term which can be solved by h·i denotes the integral with respect to y, subject to the terminal and boundary conditions Proof. From (9) and (17), we obtain Subtracting the first equation of (15) by (17) and setting we obtain where ϕ = ϕðt, S, y, zÞ.

Numerical Analysis
For the convenience of numerical computation, both fastscale stochastic volatility and slow-scale stochastic volatility processes are assumed to be mean-reverting Ornstein-Uhlenbeck (OU) processes. Thus, the stochastic volatility models under the risk-neutral adjustment are where r is the risk-free interest rate; m * and k * are mean reversion level; y and z are the fast-scale stochastic volatility and slow-scale stochastic volatility, respectively; v 1 and v 2 are the volatility of volatility, i.e., vol of vol; and W * ðiÞ t , i = 0, 1, 2, are the standard Brownian motions under the risk neutral measure.
As described in the previous section, the option price could be approximated by the summation of the leading term, the fast-scale correction, and the slow-scale correction terms. In this section, the leading term is approximated numerically by applying the front fixing method (FDM) in the following work.
4.1. The Sensitivity Study of the Leading Term. Generally, to obtain the leading term of the original American put option p = P 0,0 ðS, tÞ, we first have to solve the following problem numerically: By applying the front fixing method (FDM) and letting x = S/BðtÞ, equation (40) becomes subject to the terminal and the boundary conditions: Discretizing (41) gives which can be simplified to where for simplification, let l = dt/2dx and k = dt/dx 2 : The terminal and the boundary conditions (42) have been reduced to Now, let Journal of Function Spaces then the moving boundary can be approximated by applying Newton's method: where r denotes the rth iteration at time n, and the iteration stops once the solution converges. Option Greeks are of great importance because Greeks measure the evolution of the option price along with the change of the model parameters, such as volatility and stock price. Also, Greeks are useful tools for hedging purpose and estimation of the risk exposure in portfolio selection. Here, we study the Greeks including Delta, Gamma, and Vega of the leading term, and the derived results will be used later in the approximation of the first-order correction terms.

The Study of Delta.
Let D = ∂p/∂S denote the Delta, which measures the rate of change of the option value for the stock price. It then can be determined by solving the following PDE: subject to the terminal and the boundary conditions: Using a similar approach as described in (41) and letting x = S/BðtÞ, equation (49) subject to the terminal and the boundary conditions (50) becomes The terminal and the boundary conditions have been rewritten as follows: Discretizing (51) gives which can be simplified to where a i , b i , and c i are the same as described in (45) and the force term is Formula (54) is an implicit scheme given the terminal condition. It has been approved that the implicit scheme is unconditionally stable, and for a parabolic type of PDE, the convergence of the FDM requires that dt < <dx is satisfied.

The Study of Gamma.
Let G = ∂ 2 p/∂x 2 = ∂D/∂x denote the Gamma, which measures the rate of change of Delta, and it can be approximated numerically by solving the following PDE: subject to the terminal and the boundary conditions: By using the similar approach as in (41), we let x = S/B ðtÞ, and then, equation (56) subject to the terminal and the boundary conditions (57) becomes 7 Journal of Function Spaces subject to the terminal and the boundary conditions: Discretizing (58) gives which can be simplified to where a i ,b i , and c i are the same as defined in (45) and the force term is 4.4. The Study of Vega. The Vega measures the sensitivity of the option value for the volatility, which can be obtained by solving the following PDE: subject to the terminal and the boundary conditions: where V = ∂p/∂ δ denotes the Vega. Applying the approach as used in (41), letting x = S/BðtÞ, equation (63) subject to the terminal and the boundary conditions (64) becomes subject to the terminal and the boundary conditions: Discretizing (65) gives which can be simplified to where a i ,b i , and c i are the same as defined in (45) and the force term is

Empirical Results
In order to study our model effectively and efficiently, we firstly calibrate the proposed model and fit it to the financial market data. The source of the calibrated data in this section is from the Chicago Board Options Exchange (see website [33]), and the method we applied here is the least square method.

Numerical
Analysis of the Leading Term. Figure 1(a) is the profile of the American option value without volatility correction with different interest rates. Figure 1(b) is the profile of the moving boundary with different interest rates. The region to the left of the free boundary is the holding region, at which the investor will continue to hold the American put option instead of executing the option. The region to the right of the free boundary is the exercising region. The investor will exercise the American put option as soon as the stock price reaches the optimal exercising boundary.
Interest rates play a key role in determining the optimal exercising boundary. The opportunity cost of holding the stock increases with the growth of the interest rate. When r increases, holding stock becomes less attractive, and the investors intend to hold the put option longer. As a result, the put option price intends to decline. The leading term p = P 0,0 , which can be solved numerically from (40), is shown in Figures 2(a) and 2(b) with different interest rates.
To ensure the convergence of FDM scheme, we let the time step size dt = 10 −4 and the space step size dx = 0:128. The trajectories of moving boundaries are compared in Figures 3(a) and 3(b). As shown in Figure 3(a), the value of American put option is increasing along with the volatility, which means that the put option has more chance to be profitable as the volatility grows.

Sensitivity Analysis.
The approximation of Greeks of the leading term is studied in this subsection. Also, the option valuation is intimately related to hedging and the risk exposure of the portfolio selection, and it is interesting to see the evolution of the most critical sensitivities. Here, Delta, Gamma, and Vega will be studied numerically by using the front-fixing method to be applied later in the formation of the first-order correction terms.  (63), which represents the sensitivity of volatility. The Greeks will be applied subsequently to solve the first-order correction terms.

Numerical Analysis of Correction Terms.
In this subsection, more details on the numerical analysis of the correction terms are provided. According to Theorem 2, the fast-scale correction term can be obtained by solving the following. PDE: subject to the initial and the boundary conditions: where V 0 = ρ 1 σhβðyÞihΦ y i. According to Theorem 3, the slow-scale correction term can be solved from the following PDE: subject to the initial and the boundary conditions The discretizing scheme is the same as (49) with different force terms. Similarly, the front-fixing method is applied to solve the fast-scale volatility and the slow-scale volatility terms P ξ and P δ . From Figures 7(a) and 7(b), they conclude that the inclusion of the fast-scale volatility will result in the increasing of the option price. Figures 8(a) and 8(b) show that the inclusion of the slow-scale volatility also corrects the leading term of American option price positively.
Moreover, it can be summarized from Figures 7(a) and 7(b) that volatility correction terms grow rapidly when the underlying asset price is close to the strike price, and start to decrease quickly when it is close to maturity. Table 4 provides the comparison of the American option price with and without scale correction at the point where the underlying asset price is close to the strike price (at the money). Assuming S = 13:0216 when the strike price K equals to 14 and ϵ = σ = 0:1, we can conclude that the short-term fast-scale volatility has more significant effects on American option pricing than the long-term slow-scale volatility. The possible explanation behind this phenomenon is that the investors are normally sensitive to the price change in the short run than in the long run. In the long run, investors care more about the quality of the investment rather than the shortterm economic indicators.       Journal of Function Spaces

Conclusions
This paper has investigated the correction of multiscale stochastic volatility to American put option pricing. The application of the fast-scale volatility and slow-scale volatility is more practical to capture the behavior of the volatility in a different time periods. The fast-scale volatility is modelled to describe the highly oscillated fluctuation of the return process in the short run. In contrast, the slow-scale volatility makes a description of the slow-varied oscillation fluctuation of long-run return process. The empirical study conducted by Christoffersen et al. [9] offers a better explanation of the leptokurtosis feature of implied volatility surface. In fact, the effects of the multiscale volatility and the early exercising feature of American option can be transferred to a high-dimensional nonlinear partial differential equation subject to a moving boundary. However, this model is impossible to solve analytically. In this paper, we reduce the high-dimensional PDE to three one-dimensional PDEs with moving boundary by applying the asymptotic approximation method. The asymptotic approximation can be applied because the scales of the fast-scale volatility and slow-scale volatility can be viewed as a singular perturbation and a regular perturbation, respectively. The resulted moving boundary problem has been solved numerically by using the front-fixing method. Moreover, the coefficients in the numerical study are calibrated by the implied volatility derived from S&P 500 options. The numerical results of finite difference show that multiscale volatilities have a significant influence on the American option pricing. The incorporation of the fastscale volatility and the slow-scale volatility will increase the value of American option pricing, and the effects are more evident in the long run than in the short run. In addition, Greeks are also studied in this paper to show the sensitivity of the model parameters. The first-order correction terms are proved to be related to the sharpness of the leading term. Even though we have successfully applied the multiscale model in studying the pricing problem of the American option, the application of the multiscale volatility model on other path-dependent financial derivatives is also worth to be evaluated in the future research.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.