Stock Loans

Xia, Jianming; Zhou, Xun Yu

doi:10.1111/j.1467-9965.2006.00305.x

Cited by 64 publications

(93 citation statements)

References 7 publications

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“…Notice that c (and hence EX 1 as well) decreases as γ increases. In the context of stock loans, as discussed in [11,49], the negative discount rate is the difference of the risk-free rate and the loan rate, K is the loan amount, and γ is the dividend rate. All the numerical results given below are generated by MATLAB scripts with double precision on a Windows 7 computer with an Intel Xeon CPU E5−2620, 2.00GHz, 24.0GB RAM.…”

Section: Numerical Resultsmentioning

confidence: 99%

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An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

Leung

Yamazaki

Zhang

2015

Int. J. Theor. Appl. Finan.

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ABSTRACT. We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Lévy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Lévy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr [14], we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Lévy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.

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Section: Numerical Resultsmentioning

confidence: 99%

“…A negative discount rate can accommodate a number of applications, such as stock loans [11,49] as well as real option problems where the investment cost grows faster than the risk-free rate. In these cases, one can interpret that the effective discount rate is negative.…”

Section: Introductionmentioning

confidence: 99%

An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

Leung

Yamazaki

Zhang

2015

Int. J. Theor. Appl. Finan.

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“…It is natural for models with regime switching, for instance. Moreover, the VI approach typically leads to partial differential equations that can be solved numerically.In this paper, we first use variational inequalities to solve the stock loan pricing problem considered in [16]. We overcome the difficulty of possibly infinitely many solutions to the VIs by pinning down the right solution which is identical to the value function.…”

mentioning

confidence: 99%

“…In this paper, we first use variational inequalities to solve the stock loan pricing problem considered in [16]. We overcome the difficulty of possibly infinitely many solutions to the VIs by pinning down the right solution which is identical to the value function.…”

mentioning

confidence: 99%

“…For example, if the stock price goes down, the borrower may forfeit the stock instead of repaying the loan. On the other hand, if the stock price goes up, the borrower can repay the loan and regain the stock.In Xia and Zhou [16], the stock loan valuation is studied using a pure probabilistic approach with a classical geometric Brownian model. They pointed out that the variational inequality (VI) approach cannot be directly applied to stock loan pricing as in American option pricing because the associated VIs may have infinitely many solutions.…”

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Valuation of Stock Loans with Regime Switching

Zhang¹,

Zhou²

2009

SIAM J. Control Optim.

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Abstract. This paper is concerned with stock loan valuation in which the underlying stock price is dictated by geometric Brownian motion with regime switching. The stock loan pricing is quite different from that for standard American options because the associated variational inequalities may have infinitely many solutions. In addition, the optimal stopping time equals infinity with positive probability. Variational inequalities are used to establish values of stock loans and reasonable values of critical parameters such as loan sizes, loan rates and service fees in terms of certain algebraic equations. Numerical examples are included to illustrate the results.Key words. Stock loan, regime switching, variational inequalities, optimal stopping, smooth fit AMS subject classifications. 91B28, 60G35, 34B60Running head: Stock Loan Valuation 1. Introduction. This paper is concerned with valuation of stock loans. A stock loan involves two parties, a borrower and a lender. The borrower owns one share of a stock and obtains a loan from the lender with the share as collateral. The borrower may regain the stock in the future by repaying the lender the principal plus interest or alternatively surrender the stock. Stock loan valuation has been attracting attentions of both academic researchers and lending institutions. When a stock holder prefers not to sell his stock due to either capital gain tax consideration or restrictions on sales of his stock, a stock loan is a viable alternative for raising cash. In addition, the loan can help the stock holder hedge against a market downturn. For example, if the stock price goes down, the borrower may forfeit the stock instead of repaying the loan. On the other hand, if the stock price goes up, the borrower can repay the loan and regain the stock.In Xia and Zhou [16], the stock loan valuation is studied using a pure probabilistic approach with a classical geometric Brownian model. They pointed out that the variational inequality (VI) approach cannot be directly applied to stock loan pricing as in American option pricing because the associated VIs may have infinitely many solutions. In addition, the corresponding optimal stopping time equals infinity with positive probability. Nevertheless, the variational inequality approach is very useful for optimal stopping problems because it is associated with a set of sufficient conditions that are easy to verify. It is natural for models with regime switching, for instance. Moreover, the VI approach typically leads to partial differential equations that can be solved numerically.In this paper, we first use variational inequalities to solve the stock loan pricing problem considered in [16]. We overcome the difficulty of possibly infinitely many solutions to the VIs by pinning down the right solution which is identical to the value function. Then, we carry this approach over to models in which the underlying stock price follows a geometric Brownian motion with regime switching. The model with

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Stabilization of a stock‐loan valuation PDE process using differential flatness theory

Rigatos

Siano

Loia

et al. 2019

Asian Journal of Control

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The article proposes flatness-based control for stabilization of a stock-loan valuation process that is described by a partial differential equation. By applying semi-discretization and the finite differences method, the state-space model of the stock loan has been obtained. It has been proven that the individual rows of this state-space model are nonlinear ODEs which can be viewed as differentially flat subsystems. For the local subsystems, into which the stock-loan PDE is decomposed, it becomes possible to apply boundary feedback control. The controller design proceeds by showing that the state-space model of the stock loan PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the stock loan PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space description. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the stock loan PDE system so as to assure that all its state variables will converge to the desirable setpoints.By showing the feasibility of such a control method it is also proven that through selected modification of the PDE boundary conditions the value of the stock loan can be made to converge and stabilize at specific reference values.

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Stock Loans

Cited by 64 publications

References 7 publications

An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

Valuation of Stock Loans with Regime Switching

Stabilization of a stock‐loan valuation PDE process using differential flatness theory

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