Valuation of stock loan under uncertain environment

Soft Comput 28 22:5663 5669 https://doi.org/.7/s5-7-259-x FOCUS Valuation of stock loan under uncertain environment Zhiqiang Zhang Weiqi Liu 2,3 Jianhua Ding Published online: 5 April 27 Springer-Verlag Berlin Heidelberg 27 Abstract In this paper, within the framework of uncertainty theory, the valuation of stock loan is investigated. Different from the methods of probability theory, we solve the stock loan pricing problem by using the method of uncertain calculus. Based on the assumption that the underlying asset price follows an uncertain differential equation, we obtain the stock loan pricing formulas for uncertain stock model. Keywords Uncertainty theory Uncertain differential equation Uncertain stock model Stock loan Introduction Valuation of stock loan is a popular problem in financial fields that has been attracting the attention of both the financial market participants and academic researchers. Stock loan is a contract between a borrower and a bank in the case of the borrower obtains a loan from the bank with his or her own stock as collateral that gives the borrower the right to regain Communicated by Y. Ni. B Jianhua Ding sjdingjianhua@63.com Zhiqiang Zhang sjzhangzhiqiang@sxdtdx.edu.cn Weiqi Liu liuwq@sxu.edu.cn School of Mathematics and Computer Science, Shanxi Datong University, Datong 379, China 2 Institute of Management and Decision, Shanxi University, Taiyuan 36, China 3 Faculty of Finance and Banking, Shanxi University of Finance and Economics, Taiyuan 36, China the stock at any time prior to the loan maturity by repaying the bank the principal plus interest associated to the loan, otherwise the borrower will surrender the stock. The stock loan can afford an opportunity to hedge against a financial market downturn for a stock holder. For example, in the case of the stock price goes up, the borrower can choose to repay the loan and take back his or her stock. On the other hand, if the stock price goes down, the borrower can choose to lose the collateral rather than repaying the loan. The valuation of stock loan has been investigated by many scholars. The study of valuation of stock loan was pioneered by Xia and Zhou 27, they solved the pricing problem of stock loan under the Black Scholes model. Then, Zhang and Zhou 29 discussed the valuation problem of stock loan with regime switching. Liang et al. 2 investigated the stock loan with automatic termination clause, cap and margin. Wong and Wong 22 derived an analytical pricing formula of stock loan with stochastic volatility and optimal exercise boundary by means of asymptotic expansion. Pascucci et al. 23 gave a mathematical analysis and numerical methods for a partial differential equation model of a stock loan pricing problem. Cai and Sun 24 studied the valuation of stock loans with jump risk. Above-mentioned studies on valuation of stock loans are all within the framework of probability theory. But a lot of surveys showed that in financial practice human s belief degrees usually influence the investors judgement and decision making. For example, Kahneman and Tversky 979 found that investors often make a nonlinear transformation of probability as their basis which they based on to make decisions. In real complicated financial market, with the cognitive resources limitations, many investors usually make their belief degrees of some financial events according to the experts advise or their knowledge as their basis of decision making rather than to use the databases of extremely large 23

5664 Z. Zhang et al. size to infer the parameter estimates or probabilities. From these facts we can see that belief degrees play an important role in real financial practice. The home bias puzzle also showed that the role of belief degrees in financial practice is primary. Although many scholars try to explain the home bias puzzle see Ahearne et al. 24; Devereux and Saito 997; Lewis 999; Ueda 999, undoubtedly, investors belief degrees play an important part in real financial market. An axiomatic mathematics to deal with belief degrees called uncertainty theory was founded by Liu 27. For modeling the evolution of phenomena with uncertainty, Liu 28 gave the concept of uncertain process in 28. Liu 29 investigated a type of process that is a stationary independent increment process whose increments are normal uncertain variables. Later, this type of process was named Liu process by the academic community. The Liu integral was also introduced by Liu. The study of uncertain differential equation was initiated by Liu 28. After Liu s pioneer work, uncertain differential equation was extended by many researchers and has been widely applied in many fields, including uncertain finance, uncertain control, uncertain differential game and so on. In classical stochastic finance theory, the underlying asset price process is assumed to follow the stochastic differential equations. This assumption was challenged by many scholars.liu 23 gave a convincing paradox to show that using any stochastic differential equations to describe the stock price process is inappropriate. Liu suggested using uncertain differential equations to describe the stock price process. In 29, for the first time, uncertain differential equations were introduced into finance and an uncertain stock model was presented by Liu 29. The pricing problem of European option, American option and geometric average Asian option for Liu s uncertain stock model was solved by Liu 29, Chen 2 and Zhang and Liu 24, respectively. And Zhang et al. 26b derived the pricing formulas of power option for Liu s uncertain stock model. Many scholars also proposed other uncertain stock models, for example, Peng and Yao 2 proposed an uncertain mean-reverting stock model, Chen et al. 23 proposed a stock model with periodic dividends and derived the pricing formulas for this type of model. Yao 25a obtained the no-arbitrage determinant theorems for this type of uncertain stock model. In 25, based on the uncertain stock model with jump, the problem of option pricing was discussed by Ji and Zhou 25. Chen and Gao 23 proposed some uncertain interest term structure model. Yao 25b applied uncertain contour process to the stock model with floating interest rate. The problem of valuing interest option was discussed by Zhang et al. 26a. Using the uncertain differential equation to establish the exchange rate model, the problem of currency option pricing was studied by Liu et al. 25. Besides, uncertain differential equation also has been applied in other fields. For example, Zhu 2 introduced uncertain differential equation into optimal control, differential games with applications to capitalism and resource extraction problem by using uncertain differential equation were studied by Yang and Gao 23, and Yang and Gao 26, respectively. In this paper, different from classical stochastic finance theory, we investigate the valuation of stock loan within the framework of uncertainty theory. Based on the assumption that the stock price process follows an uncertain differential equation, the stock loan pricing formulas are derived for Liu s uncertain stock model, and the valuation of stock loans is also discussed under uncertain stock model with periodic dividends. The rest of the paper is organized as follows. In next section, we introduce some useful concepts and theorems of uncertainty theory as needed. In Sect. 3, we investigate the valuation of stock loan for Liu s uncertain stock model. In Sect. 4, we explore the valuation of stock loan for uncertain stock model with periodic dividends. Finally, we make a brief conclusion in Sect. 5. 2 Preliminary The following are some useful definitions and theorems of uncertainty theory as needed. Definition 2. Liu 27 Let Γ be a nonempty set, and let L be a σ -algebra over Γ. An uncertain measure is a function M : L, ] such that Axiom Normality Axiom M{Γ }= for the universal set Γ ; Axiom 2 Duality Axiom M{Λ}+M{Λ c }= for any event Λ; Axiom 3 Subadditivity Axiom For every countable sequence of events {Λ i } we have { } M Λ i M{Λ i }. 2. i= i= AsetΛ L is called an event. The uncertain measure M{Λ} indicates the degree of belief that Λ will occur. The triplet Γ, L, M is called an uncertainty space. In order to obtain an uncertain measure of compound event, a product uncertain measure was defined by Liu 29. Axiom 4 Product Axiom LetΓ k, L k, M k be uncertainty spaces for k =, 2,...The product uncertain measure M is an uncertain measure on the product σ -algebra L L 2 satisfying 23

Valuation of stock loan under uncertain environment 5665 { } M Λ k = M k {Λ k } 2.2 k= k= where Λ k are arbitrarily chosen events from L k for k =, 2,..., respectively. Definition 2.2 Liu 27 An uncertain variable is a measurable function from an uncertainty space Γ, L, M to the set of real numbers, i.e., {ξ B} is an event for any Borel set B. Definition 2.3 Liu 27 The uncertainty distribution of an uncertain variable ξ is defined by x = M{ξ x} 2.3 for any real number x. Definition 2.4 Liu 27 An uncertain variable ξ is called normal if it has a normal uncertainty distribution x = e x + exp 2.4 3σ denoted by N e,σ where e and σ are real numbers with σ>. Definition 2.5 Liu 2 An uncertainty distribution x is said to be regular if it is a continuous and strictly increasing function with respect to x at which < x <, and lim x =, lim x x =. 2.5 x + Definition 2.6 Liu 2Let ξ be an uncertain variable with regular uncertainty distribution x. Then the inverse function is called the inverse uncertainty distribution of ξ. Definition 2.7 Liu 27 Letξ be an uncertain variable. Then the expected value of ξ is defined by + Eξ] = M{ξ r}dr M{ξ r}dr 2.6 provided that at least one of the two integrals is finite. Theorem 2. Liu 27 Let ξ be an uncertain variable with uncertainty distribution. If the expected value exists, then + Eξ] = xdx xdx. 2.7 Theorem 2.2 Liu 2 Let ξ be an uncertain variable with regular uncertainty distribution. Then Eξ] = d. 2.8 Theorem 2.3 Liu 2 Let ξ,ξ 2,...,ξ n be independent uncertain variables with regular uncertainty distributions, 2,..., n, respectively. If the function f x, x 2,..., x n is strictly increasing with respect to x, x 2,...,x m and strictly decreasing with respect to x m+, x m+2,...,x n, then the uncertain variable ξ = f ξ,ξ 2,...,ξ n 2.9 has an inverse uncertainty distribution Ψ = f,..., m,,..., m+ n. 2. Liu and Ha 2 proved that the uncertain variable ξ = f ξ,ξ 2,...,ξ n has an expected value Eξ] = m+ f,..., m,,..., d. 2. An uncertain process is a sequence of uncertain variables indexed by a totally ordered set T. A formal definition is given below. Definition 2.8 Liu 28 LetΓ,L, M be an uncertainty space and let T be a totally ordered set e.g., time. An uncertain process is a function X t γ from T Γ, L, M to the set of real numbers such that {X t B} is an event for any Borel set B at each time t. Definition 2.9 Liu 29 An uncertain process C t is said to be a Liu process if i C = and almost all sample paths are Lipschitz continuous, ii C t has stationary and independent increments, iii every increment C s+t C s is a normal uncertain variable with expected value and variance t 2. In order to deal with the integration and differentiation of uncertain processes, Liu 29 proposed an uncertain integral with respect to Liu process. Definition 2. Liu 29 LetX t be an uncertain process and C t be a Liu process. For any partition of closed interval a, b] with a = t < t 2 < < t k+ = b, themeshis defined as = max i k t i+ t i. n 23

5666 Z. Zhang et al. Then the Liu integral of X t is defined as b a X t dc t = lim i= k X ti C ti+ C ti 2.2 provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be Liu integrable. Definition 2. Chen and Ralescu 23 LetC t be a Liu process and let Z t be an uncertain process. If there exist uncertain processes μ t and σ t such that t t Z t = Z + μ s ds + σ s dc s 2.3 for any t, then Z t is called a Liu process with drift μ t and diffusion σ t. Furthermore, Z t has an uncertain differential dz t = μ t dt + σ t dc t. 2.4 Liu 29 verified the fundamental theorem of uncertain calculus, i.e., for a Liu process C t and a continuous differentiable function ht, c, the uncertain process Z t = ht, C t is differentiable and has a Liu differential dz t = h t t, C tdt + h c t, C tdc t. 2.5 Definition 2.2 Yao and Chen 23 Let be a number with <<. An uncertain differential equation dx t = f t, X t dt + gt, X t dc t 2.6 is said to have an -path Xt if it solves the corresponding ordinary differential equation dx t = f t, X t dt + gt, X t dt 2.7 where is the inverse standard normal uncertainty distribution, i.e., = 3. 2.8 Theorem 2.4 Yao and Chen 23 Let X t and Xt be the solution and -path of the uncertain differential equation dx t = f t, X t dt + gt, X t dc t, 2.9 respectively. Then M { X t Xt, t} =, 2.2 M { X t > Xt, t} =. 2.2 Theorem 2.5 Yao and Chen 23 Let X t and Xt be the solution and -path of the uncertain differential equation dx t = f t, X t dt + gt, X t dc t, 2.22 respectively. Then the solution X t has an inverse uncertainty distribution t = X t. 2.23 Theorem 2.6 Yao 23 Let X t and X t be the solution and -path of the uncertain differential equation dx t = f t, X t dt + gt, X t dc t, 2.24 respectively. Then for any time s > and strictly increasing function Jx, the remum JX t 2.25 t s has an inverse uncertainty distribution s = JXt ; 2.26 t s and the infimum inf JX t 2.27 t s has an inverse uncertainty distribution s = inf JX t. 2.28 t s 3 Valuation of stock loan for Liu s uncertain stock model Different from classical stochastic finance theory, Liu 29 suggested to describe the stock price process by using an uncertain differential equation and proposed an uncertain stock model as follows { dxt = rx t dt 3. ds t = μs t dt + σ S t dc t where X t is the bond price, S t is the stock price, r is the risklessinterest rate, μ is the log-drift, σ is the log-diffusion, and C t is a Liu process. It follows from the Eq. 3. that the stock price is S t = S expμt + σ C t 3.2 23

Valuation of stock loan under uncertain environment 5667 whose inverse uncertainty distribution is t = S exp. 3.3 The stock loan problem can be described as follows. A borrower obtains amount K from a bank with one share of his or her stock as collateral. After paying a service fee c < c < K to the bank, the borrower receives the amount K c. The borrower has the right to regain the stock at any time prior to the loan maturity T by repaying the bank the principal plus interest associated to the loan that is K expθt, where θ > r is the loan interest rate. This means that the borrower pays S K c to buy an American option with a time-varying strike price K expθt and maturity T at time. The present value of the payoff of the borrower is exp rts t K expθt] +. 3.4 Thus, the value of the stock loan should be the expected present value of the payoff. Definition 3. Assume a stock loan has loan amount K, loan interest rate θ and loan maturity T.Let f denote the value of the stock loan. Then the value of the stock loan is ] f = E exp rt S t K expθt ] +. 3.5 For rationally determining the parameters K, c and θ, the evaluation of the value of the stock loan is needed. In this paper, our main goal is to evaluate the stock loan value defined in 3.5 and 4.4. Theorem 3. Assume a stock loan for the stock model 3. has loan amount K, loan interest rate θ and loan maturity T. Then the value of the stock loan is f = exp rt S exp K expθt] + d. 3.6 Proof The uncertain differential equation ds t = μs t dt + σ S t dc t has an -path S t = S exp μt + σ t 3.7 where is the inverse standard normal uncertainty distribution. Since Jx = exp rtx K expθt] + is an increasing function, it follows from Theorem2.6 that JS t = exp rts t K expθt] + has an inverse uncertainty distribution + exp rt S exp μt+ σ t 3 K expθt]. 3.8 Therefore, the value of the stock loan is f = exp rt S exp K expθt] + d. 3.9 Example 3. Suppose that the stock price follows the uncertain stock model 3. with parameters μ =.7 and σ =.35. Assume the riskless interest rate r =.6, the initial stock price S = 4, the loan amount K = 28, loan interest rate θ =.8, the maturity time T =. By the formula of Theorem 3., we can calculate out that the value of stock loan is f = 3.96. 4 Valuation of stock loan with periodic dividends In above section, we do not consider the case of the stock with dividends. In most cases, the dividends are paid by enterprises that will affect the price of their stock. Chen et al. 23 proposed a stock model with periodic dividends to describe the case of the equity pays a dividend of a fraction δ of the stock price at deterministic T, T 2,...,the stock model can be written as follows { Xt = X exprt S t = S δ nt] 4. expμt + σ C t where X t is the bond price, S t is the stock price, r is the riskless interest rate, μ is the expected return rate, σ is the volatility, and C t is a Liu process, nt] =max{i : T i t} is the number of dividend payments made by time t. Thus, the value of dividends at time t can be described by I t = S expμt + σ C t S δ nt] expμt + σ C t = S δ nt] 4.2 expμt + σ C t. Assume the dividends associated to the stock are gained by both borrower and bank with equal half amount of the dividends. Then the present value of the payoff of the borrower is 23

5668 Z. Zhang et al. exp rt S t + + 2 I t K expθt]. 4.3 2 S + δ nt] exp The value of the stock loan should be the expected present value of the payoff of the borrower. Thus, the stock loan value is given by the definition as below. Definition 4. Assume dividends associated to the stock are gained by both borrower and bank with equal half amount of the dividends, and the stock loan has loan amount K, loan interest rate θ and loan maturity T.Let f denote the value of the stock loan. Then the value of the stock loan is f = E exp rt S t + ] + 2 I t K expθt. 4.4 Theorem 4. Assume a stock loan for the stock model 4. has loan amount K, loan interest rate θ and loan maturity T. Then the value of the stock loan is f = 2 S exp rt + δ nt] exp K expθt] + d. 4.5 Proof As we know, C t has an inverse uncertainty distribution t = t 3. 4.6 S t + 2 I t = 2 S + δ nt] expμt + σ C t is increasing with respect to C t, hence S t + 2 I t has an inverse uncertainty distribution Ψt = 2 S + δ nt] expμt + σ t = 2 S + δ nt] exp. 4.7 Since Jx = exp rtx K expθt] + is an increasing function, it follows from Theorem 2.6 that J S t + 2 I t = exp rts t + 2 I t K expθt] + has an inverse uncertainty distribution Υt = exp rt K expθt] +. 4.8 Therefore, the value of the stock loan is f = 2 S exp rt + δ nt] exp K expθt] + d. 4.9 Example 4. Suppose that the stock price follows the uncertain stock model 4., the parameters are μ =.7, σ =.35 and δ =.5. Assume the riskless interest rate r =.6, the initial stock price S = 4, the loan amount K = 28, loan interest rate θ =.8, the maturity time T =, and the dividend are paid at deterministic times T =.5 and T 2 =. By the formula of Theorem 4., we can calculate out that the value of stock loan is f = 3.6. 5 Conclusions In this paper, we investigated the valuation of stock loan within the framework of uncertainty theory. Based on the assumption that the underlying stock price follows the geometric Liu process, the formulas of price of stock loan for Liu s uncertain stock model and the stock model with periodic dividends proposed by Chen, Liu and Ralescu were derived with the method of uncertain calculus. Compliance with ethical standards Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper. Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors. References Ahearne AG, Griever WL, Warnock FE 24 Information costs and home bias: an analysis of US holdings of foreign equities. J Int Econ 62:33 336 23

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