Pricing Stock Loans with the CGMY Model

e empirical research shows that the log-return of stock price in nance market rejects the normal distribution and admits a subclass of the asymmetric distribution. Hence, the pricing problem of stock loan is investigated under the assumption that the log-return of stock price follows the CGMY process in this work. Under this framework, the pricing model of stock loan can be described by a free boundary condition problem of space-fractional partial dierential equation (FPDE). First of all, in order to change the original model dened in a xed domain, a penalty term is introduced, and then a rst order fully implicit dierence schemes is developed. Secondly, based on the numerical scheme, we prove the stock loan value generated by our method does not fall below the value obtained when the contract of stock loan is exercised early. Finally, the numerical experiments are implemented and the impacts of key parameters in the CGMY model on the value and optimal redemption price of stock loan are analyzed, and some reasonable explanation should be given.


Introduction
A stock loan can be treated as a contract between two parties: the bank or other nancial institution (the lender), and a client (the borrower). e borrower obtains a loan from the lender with their stocks as collateral, and the contract o ers the borrower a right to redeem the stock at any valid time. But he/ she should pay o the loan as well as the cumulative loan. As described in Ref. [1], the risk aversion investors can use stock loan to transfer the risk of holding stock to the nancial institutions, and this kind of derivatives also can establish market liquidity. erefore, the stock loan is one of the most important nancial derivatives in the nancial market and currently, both Florida Mortgage Corporation and Shelly Bay Capital specialize in providing stock loan services.
e pricing problem about the stock loan has been a popular topic in the academic since the rst research paper [1] about the stock loan published in 2007. In this literature, the risk asset was assumed to follow the Geometric Brownian motion (GBM) and the contract of stock loan is nite maturity. Following this publication, more and more researchers paid attentions to the academical topic of stock loan valuation with other conditions. For instance, Lu and Putri [2] considered the pricing problem of stock loan with nite maturity margin. Under the framework of hyper-exponential jump di usion model, Cai and Sun [3] studied the value and optimal redemption price of stock loan with in nite and nite maturity. e stock loan with nite maturity was also investigated in Ref. [4] under the case that risk-free interest rate follows the Rednleman-Bartter model without dri term.
It is well known that the classical Black-Scholes (B-S) model was established under a lot of strict assumptions, such as the log-return of stock price follows the nonmoral distribution, frictionless, and so on. However, these strict assumptions do not accord with the dynamic process of stock price in real nancial market. e empirical research shows that discontinuities or jumps are believed to be an indispensable element of nancial risk-asset price (see e.g., [5][6][7]) and the log-return of risk asset appear to be "asymmetric distribution" and "leptokurtic distribution" (see e.g., [8][9][10][11][12]). For this reason, for capturing these characters of stock price, many scholars use other complex stochastic process to drive the stock price. Prominent examples including the CEV model [13], GEV model [14], KoBoL model [15], and so on. A frequently used stochastic process is called CGMY process and it was presented by Carr et al. [16] with the aim to provide a model for the dynamic of equity log-returns. is model is rich enough to accommodate jumps of nite or in nite activity, and nite or in nite variation. In fact, this stochastic process is a particular type of pure jump Lévy process with four key parameters (e.g., , , , and ) which control its essential characteristics. e parameter may be viewed as a measure of the overall level of activity. e aggregate activity level should be calibrated through movements under the case of keeping the other three parameters constant and integrating over all that moves exceeding a small level. Parameters and control the rate of exponential decay on the right and le of the Lévy density, respectively, leading to skewed distributions when they are unequal. Parameter is particularly useful in characterizing the ne structure of the stochastic process. e CGMY process has in nite variation and nite quadratic variation if ∈ (1, 2). At present, many papers discussed pricing problem of nancial derivatives based on the CGMY process. Ballotta and Kyriakou [17] obtained the option price by using the Monte Carlo method in this case. Zhang et al. [18] presented a fast numerical method to double barrier option under framework of CGMY. Chen et al. [19] investigated European option and they obtained the explicit closed-form analytical based on the model. A better refernce value to the holder of stock loan can be provided in this case of CGMY model, a better reference value to the holder of stock loan is provided; therefore, in this paper we intend to investigate the pricing of stock loans with nite-maturity under the CGMY model. e pricing model is free-boundary problem of partial di erential equation with tempered fractional derivatives. e tempered fractional derivatives and nonlinearity associated with the free boundary add the di culty of solving the pricing model. Aiming at this problem, we consider a penalty method in which the free boundary is removed by adding a small and continuous penalty term to the governing equation. erefore, the mathematical model should be solved on a xed domain. Even if on the xed domain, the analytical solution is seldom available so that a full-implicit scheme is employed. Based on our numerical scheme, we prove that the stock loan value generated by the penalty method cannot fall below the value obtained when the stock loan is exercised early, and we also do simulation to verify the result.
is study has ve main sections. Section 1 provides the introduction. e pricing model of stock loan is set in Section 2 and the numerical method is introduced in Section 3. e simulation and discussions are presented in Section 4, and conclusion is displayed in Section 5.

Mathematical Model
In this section, the governing equation of the stock loan with nite maturity, which is a partial di erential equation with fractional derivative, is presented under the CGMY framework. And then, nancially, the corresponding free moving boundary and terminal conditions will be given to complete the pricing model. e CGMY process assumes the log value of stock price = ln follows a stochastic di erential equation under the risk-neutral measure Q [16] where , , and are the risk free interest rate, the dividend and the current time, respectively. is a stochastic variable and its characteristic function is In addition, where the parameters , , and are constants, and ∈ (1, 2), > 0, ≥ 0, ≥ 0.

Governing Equation.
In fact, as described in Ref. [1] a stock loan problem can be regarded as an American call option with a time-varying strike price , where is the principal and is the continuously compounded loan interest rate and ≥ in general. erefore, the payo function of stock loan at maturity can be written as In addition, typically as same as the American call, at each valid time there is a particular value of the underlying, which yields a boundary between two regions: in one side(called exercising region) the investor exercise the contract of stock loan and in other side (called holding region) one should hold this contract. e particular value of the asset is called optimal redemption price and denoted by = ln . en, based on the principle of non-arbitrage pricing, in the holding region, the value of stock loan at time satis es where E Q is the conditional mean operator under the measure Q and F denotes the information ow at time . Famous authors, Cartea [20] and Del-Castillo-Negrete obtained the governing equation of ( , ) through the Fourier transformation as follows: where ∈ −∞, , and −∞ , are le and right sided Riemann-Liouville fractional derivatives.
In fact, the -order le and right fractional derivatives are closely related to CGMY process. e fractional di erentiation is nonlocal, which relates to the stock loan value in the stopping region −∞, . is is where the stock loan value is equal to its intrinsic value. e nonlocalness of this fractional operator means that over a time step Δ , the price of risk-asset should di use to value +Δ far away from , provides a way in e ciently simulating the existence of jumps of the risk-asset.

e Boundary Conditions.
In this subsection, to complete the pricing system, a set of appropriate boundary conditions will be given. First of all, as a rational investor, she/he does not redeem stock if the level of stock price is very low and here we assume the stock price is close to zero, i.e., = ln → −∞, then one can obtain a boundary condition as On the other hand, the continuity of stock loan value ( , ) at optimal redemption price should still be retained to ensure the smooth pasty of the stock loan value across the free boundary. For these reasons, we still impose the two boundary conditions as follows: And the terminal condition is the payo function To sum up, the complete pricing model for stock loan with nite maturity under CGMY process can be written as It must be noted that, a rational investor cannot redeem the stock if the value of redemption is less than the holding value; therefore, the value of the stock loan should satisfy the following inequality for all ≤ and 0 ≤ ≤ .

Model Normalization.
It must be noted that there is a time-factor in the boundary conditions (14) and (17). And we nd the time-factor should in uence numerical results, hence, we introduce a new variable system to the model (14)- (17) for avoiding the e ect of . Take then let = − , therefore we obtain And let = − , so one obtains Discrete Dynamics in Nature and Society 4 If adding the penalty term (34) to the FPDE in (26), one can obtain a nonlinear FPDE system de ned on a xed domain as follows: where exp max denotes the maximum stock price, ∈ −∞, max . And according to the results in Ref. [21], the maximum value of stock is equal to 3 or 4 times of strike price, hence we take exp max = 3 in this paper. In addition, we will omit the subscript of function ( , ) for convenience.

Di erence Scheme.
For the FPDE of (37) is nonlinear, it results in the analytical solution of the mathematical model (37)-(39) that is laborious and even impossible to achieve even if the model is de ned on the xed domain −∞, max × [0, ]. erefore, an e ective numerical algorithm should be preferred.
Speci cally, taking Δ > 0 as spatial step such that 1 Δ = max , where 1 is a positive integer, and placing 2 + 1 uniform grids in the time direction, namely Δ = / 2 , that is and = ⋅ ⋅ ⋅ , −2, −1, 0, 1, 2, 1 + 1; = 1, 2, . . . , 2 + 1. For the rst order spatial and time, we use the following di erence scheme And the fractional derivative can be approximated by the rst-order Grnwald-Letnikov formula as [22] (36) And the equation (18) should be changed as Mathematically, the function ( , ) in the model (26)-(30) can be viewed as an American call option with strike price and free-boundary . According to the property of American call, the values of ( , ) are not less than the payo max − , 0 . And the free-boundary is monotonically increasing with respect to time to expiry of option contract. In the next section, a numerical scheme based on the penalty method will be proposed to solve this model.

Numerical Method
In this section, we rst consider a penalty approach in which the free moving boundary is removed by adding a small and continuous penalty term, so that the stock loan pricing problem can be solved on a xed domain. en, a di erent scheme is proposed and an e ective nonlinear Newton-type iteration strategy is employed. Furthermore, due to the coe cient matrixes of the nally linear system, which contains the full matrix with Toeplitz structure, the fast biconjugate gradient stabilized method (FBi-CGSTAB) is used to solve our system.

Model Transformation.
In this paper the penalty function is de ned as where is a regularization constant and 0 < ≪ 1, is a constant, where ≥ 1.

Numerical Examples and Discussions
In this part, the numerical simulation should be executed for verifying the theoretical consideration of our proposed di erence scheme. e system (48) is nonlinear, hence a practical Newton-type iteration strategy is given to realize the nonlinear di erence scheme. e special structure preserved by the coefcient matrix resulted by di erence scheme is carefully exploited. Furthermore, the fractional derivatives usually result in a dense coe cient matrix in the system. It has significant computational and storage requirements. In terms of computational cost; it is very important to use an e ective and e cient method to solve our linear system. erefore, the fast biconjugate gradient stabilized method (FBi-CGSTAB) [18] is employed. Finally, the impacts of important parameters to the optimal redemption price should be analyzed.

< ,
Discrete Dynamics in Nature and Society 6 with strike and optimal exercise boundary . en, mathematically, the values of ( , ) are not less than payo function max − , 0 and is increasing with respect to time to expiry. Figures 3 and 4 show the two facts, respectively.

Impact of Parameters.
As described in Section 2, the essential characteristics of CGMY model are controlled by four key parameters, namely, , , , and and. In other words these parameters should a ect the value and optimal redemption price of stock loan. Hence, in this subsection we employ the proposed numerical method to investigate the impacts of the four parameters on the stock loan value and capture the optimal redemption price for di erent parameters setting with some reasonable explanation. Figures 5 and 6 display the behaviours of stock loan valuation and optimal redemption price for di erent , respectively. As shown in Figure 5, the mesh surf of ( , ) is higher with the increase of . Financially, this phenomenon could be explained from the fact that except keeping the other parameters constant, the aggregate activity level of CGMY process may be calibrated through movements in and the level is increasing with respect to . In addition, the stock loan and where and I denotes the 1 − 1 × 1 − 1 identity matrix, A means the transposition of matrix A. Both A and B are Toeplitz matrix In fact, the system (60) is not solved directly for the penalty function ( ) is nonlinear with respect to , so this nonlinear system can be solved through Newton-type iteration approach, which is provided in Appendix A.3.

Numerical Testing.
To ensure that both the theoretical model and method are feasible, many results in our work must be veri ed before quantitative analysis; therefore, in this part we will carry out some numerical testing. First of all, the fact that our proposed numerical algorithm satis es the discrete analogue of the positive constraint, ( , ) ≥ max − , 0 in eorem 3 will be veri ed. As shown in Figures 1 and 2, the mesh surface of ( , ) ≥ max − , 0 for the di erent time and stock price under various parameter setting is presented. And the mesh surface shows that the present di erence scheme conserves the inequality ≥ max , 0 for all , .
As described in Section 2.3, the function ( , ) in the model (36)-(39) can be recognized as an American call option   max − exp , 0 must be equal to the stock loan value and the optimal redemption price is the boundary of exercising region, the higher stock loan values should yield the bigger optimal redemption price, namely, a bigger shall yield a higher optimal redemption boundary shown in Figure 6. eoretically, the two parameters and control the rate of exponential decay on the right and le of the density of CGMY process,respectively. As for < ( > ), the le (right) tail of the distribution for this process is heavier than the right (le ) tail, which is consistent with the neutral distribution typically implied from stock loan values [16]. Also a smaller value of ( ) results in a heavier le (right) tail. Moreover, according to the results in Ref. [24], it is suggested that the stock loan should have a positive value only if there is a large decrease in the risk-asset. Hence, the stock loan value relies on the le (right) tail of the risk-neutral distribution of stock. In other words, the fatter the le (right) tail is, the more valuable the stock loan value. erefore, as shown in Figures 7 and 8, the mesh surface of ( , ) is decreasing with respect to and during with other parameters given, as depicted in the complete time dimension.
According to the analysis in explained earlier, it is not difcult to obtain that a smaller value of or would yield a higher stock loan value, hence, as a rational investor she/he should raise the redemption price at any valid time when there are smaller value of or . And in other words, the higher optimal redemption boundary is increasing with respect to and at any valid time, as shown in Figures 9 and 10.
Finally, we should focus on the impact of parameter . Carr [16] claims that parameter decides whether the up jumps and down jumps of CGMY process have a completely monotone Lévy density. Moreover, both activity and variation of this process will become larger as becomes bigger. So, the investors must be willing to obtain a higher value of contract under the case of "out-the-money", then the mesh-surface is increasing with respect to with other parameters are kept xed as shown in Figure 11. However, when stock prices are less than principal, our numerical results are consistent with the conclusion of call option calculated by the fat tails distribution under the "in-the-money" case [14]. As for the stock loan contract, the stock prices must be larger than principal in the exercising region where the value of stock loan decreases monotonically with respect to . is leads to a fact that the American call have a similar property that the higher activity level of underlying asset results in the higher value. Hence, it is reasonable to suggest that the stock loan value mesh surface ( , ) should move upwards as becomes larger. Mathematically, in the exercising region where the payo  Discrete Dynamics in Nature and Society 8 bigger the value of is, the higher the optimal redemption boundary is, as depicted in Figure 12.

Conclusions
In this paper, the stock loan valuation under the CGMY framework is investigated. In this case, the pricing mathematical model is a FPDF free-boundary problem. A nonlinear penalty term is introduced to transform the free-boundary model into one with xed domain. en, a full nonlinear implicit numerical scheme is proposed and we also proved that the numerical solution of transformed model satis ed the inequality ( , ) ≥ max exp( ) − , 0 . In addition, to improve e ciency of computation, the FBi-CGSTAB method was employed.
In numerical simulation, the impacts of key parameters , , , and on the stock loan value and optimal redemption price are analyzed. Parameter decides activity level of underlying asset so that both stock loan value and optimal redemption price are increasing with respect to . For the two parameters and , they control the rate of exponential decay on the right and le of the Lévy density, respectively; in other words, the bigger value of ( ) results in the fatter le (right) (A.11) tail. For this reason, both mesh surface of ( , ) and optimal redemption boundary decrease with respect to and . For parameter , in the case of "out-of-the-moneyness", the bigger can yield the higher mesh surface and optimal redemption price. However, this phenomenon would disappear if stock price became less than the strike price.

A.2. Proof Process of Theorem 3
Proof. Without loss of generality, we prove this theorem in two steps. e scheme (48) can be rewritten as In order to prove ≥ , for all , , we introduce and it is straightforward to Hence, by substituting into (A.3), it yields Noticing the formula in the square brackets in the above equation is more than 0, and 2 +1 = max exp − , 0 ≥ 0 for all , namely 2 +1 ≥ 0. Hence, by mathematical induction, one obtains for all , , which completes the proof.
(A.28) ≥ 0,   where = 1, 2, . . ., with the initial value 0 = U +1 for each time level as the given initial guess and = − −1 . J F is Jacobian matrix of column vector and ∈ (0, 1) is a damping parameter. We choose U = , if ᐉ ᐉ ᐉ ᐉ ᐉ − −1 ᐉ ᐉ ᐉ ᐉ ᐉ∞ ≤ for some as the stopping criterion, where is a su ciently small positive control tolerance number. In this paper, we take = 0.2, = 10 −4 , = 10 −5 and 3 = 0. Now the challenging point that should be emphasized is that both matrices A and A are dense matrix, and the storage requirement and computational e orts are very high, which presents di culty in capturing the optimal exercise boundary under the CGMY framework. So, the FBi-CGSTAB method should be employed to overcome the challenging point. And nally, the total storage requirement and computational costs have been signicantly reduced from Data Availability ere isn't any data in our manuscript, but for the Matlab code, we can provide to anyone who want.