Valuation Of Stock Loans Research Articles

Regime Classification and Stock Loan Valuation

For traditional perpetual American put options under regime-switching models with positive risk-free interest rates, optimal stopping usually can occur in any regime. Nonetheless, if the risk-free interest rates are allowed to equal zero (the interest rate may drop to zero sometimes in reality), there may exist “continuation regimes” within which optimal stopping can never occur, that is, within which stopping is never optimal. A natural problem is “regime classification,” that is, determination of all continuation regimes. In “Regime Classification and Stock Loan Valuation,” Ning Cai and Wei Zhang develop a unified, fixed point approach to solving this regime classification problem under general regime-switching exponential Levy models with any finite numbers of regimes and general Levy types. Applying this result, they also provide a unified framework for the valuation of infinite maturity stock loans under general regime-switching exponential Levy models.

Valuation of stock loan under uncertain stock model with floating interest rate

Stock loan is a special loan with stocks as collateral, which offers the borrower the right to redeem the stocks at any time prior to the loan maturity by repaying the bank the principal and the loan interest. In this paper, we investigate the valuation of stock loan under an uncertain stock model with floating interest rate. The pricing formulas of standard stock loan and capped stock loan for the stock model are derived by using the method of uncertain calculus. Subsequently, some numerical algorithms are designed to calculate the prices of stock loans based on the pricing formulas. Finally, some numerical experiments are presented to study the relationship between stock loan price and some parameters.

Stock loan valuation based on the Finite Moment Log-Stable process

The empirical test suggests that the log-return series of stock price in US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of stock loan can be described by a space-fractional partial differential equation with time-varying free boundary condition. Firstly, a penalty term is introduced to change the original problem to be defined on a fixed domain, and then a fully-implicit difference scheme has been developed. Secondly, based on the fully-implicit 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. Thirdly, the numerical experiments are carried out to demonstrate differences of stock loan model under the FMLS and the standard normal distribution. Optimal redemption strategy of stock loan has been achieved. Furthermore the impact of key parameters in our model on the stock loan evaluation are analyzed, and some reasonable explanation are given.

Valuation of stock loan under uncertain mean-reverting stock model

Valuation of stock loan under uncertain mean-reverting stock model

Valuation of stock loan under uncertain environment

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.

Stock loan valuation under a stochastic interest rate model

Stock loans are loans collateralized by stocks. They are modern financial products designed for investors with large equity positions. Mathematically, stock loans can be regarded as American call options with a time-dependent strike price. This study is the first in the literature that considers the valuation of stock loans in a stochastic interest rate framework. Based on portfolio analysis, a partial differential equation (PDE) governing the value of stock loans is derived. A set of appropriate boundary conditions, particularly in the interest rate direction, are also proposed to close the pricing system. A sound justification is provided for the proposed boundary conditions mathematically as well as financially. To solve the proposed nonlinear PDE system, a predictor–corrector finite difference method is adopted. Moreover, an alternating direction implicit (ADI) method is used to improve the computational efficiency. Numerical results suggest that the current method is reliable and the stochastic interest rate leads to a higher optimal exercise price of the stock loan in comparison with that calculated from the Black–Scholes model.

Valuation of stock loans under a Markov chain model

In recent years the use of Markov chain models to model stock price movement has received increased attention among researchers. Markov chain models combine the discrete movements of a binomial tree model while retaining the Markovian properties of Brownian motion, thus allowing the best properties of both of these models. In this paper, the authors consider a Markov chain model in which the underlying market is solely determined by a two-state Markov chain. Such a Markov chain model is strikingly simple and yet appears capable of capturing various market movements. By proper selection of parameters, the Markov chain model can produce sample paths that are very similar to or very distinct from a classical Brownian motion, as the authors demonstrate in this paper. This paper studies the stock loan valuation, or the value of a loan in which a risky share of stock is used as collateral, under such a model. Dynamic programming equations in terms of variational inequalities are used to capture the dynamics of the problem. These equations are solved in closed-form. Explicit optimal solutions are obtained. Numerical examples are also reported to illustrate the results.

Semi-analytic valuation of stock loans with finite maturity

In this paper we study stock loans of finite maturity with different dividend distributions semi-analytically using the analytical approximation method in Zhu (2006). Stock loan partial differential equations (PDEs) are established under Black–Scholes framework. Laplace transform method is used to solve the PDEs. Optimal exit price and stock loan value are obtained in Laplace space. Values in the original time space are recovered by numerical Laplace inversion. To demonstrate the efficiency and accuracy of our semi-analytic method several examples are presented, the results are compared with those calculated using existing methods. We also present a calculation of fair service fee charged by the lender for different loan parameters.

Valuation of stock loans with jump risk

A stock loan is a special loan with stocks as collateral, which offers the borrowers the right to redeem the stocks on or before the maturity (Xia and Zhou, 2007; Dai and Xu, 2011). We investigate pricing problems of both infinite- and finite-maturity stock loans under a hyper-exponential jump diffusion model. In the infinite-maturity case, we derive closed-form formulas for stock loan prices and deltas by solving the related optimal stopping problem explicitly. Moreover, we obtain a sufficient and necessary condition under which the optimal stopping time is finite with probability one. In the finite-maturity case, we provide analytical approximations to both stock loan prices and deltas by solving an ordinary integro-differential equation as well as a complicated non-linear system. Numerical experiments demonstrate that the approximation methods for both prices and deltas are accurate, fast, and easy to implement.

Valuation of stock loans using exponential phase-type Lévy models

Stock loans are collateral loans with stocks used as the collateral. This paper is concerned with a stock loan valuation problem in which the underlying stock price is modeled as an exponential phase-type Lévy model. The valuation problem is formulated as the optimal stopping problem of a perpetual American option with a time-varying exercise price. When a transformation is applied to the perpetual American option, it becomes a perpetual American call option in an economy with a negative interest rate, thus causing standard Wiener–Hopf techniques to fail. We solve this optimal stopping problem using a variational inequality approach.

Stochastic volatility asymptotics of stock loans: Valuation and optimal stopping

A stock loan, or equity security lending service, is a loan which uses stocks as collateral. The borrower has the right to repay the principal with interest and regain the stock, or make no repayment and surrender the stock. Therefore, the valuation of stock loan is an optimal stopping problem related to a perpetual American option with a negative effective interest rate. The negative effective interest rate makes standard techniques for perpetual American option pricing failure. Using a fast mean-reverting stochastic volatility model, we applied a perturbation technique to the free-boundary value problem for the stock loan price. An analytical pricing formula and optimal exercise boundary are derived by means of asymptotic expansion.

Stock loan valuation under a regime-switching model with mean-reverting and finite maturity

Stock loans are business contracts between borrowers and lenders in which the borrower uses shares of stock as collateral for the loan. Since the value of the collateral is subject to wide and frequent price swings, valuing such a transaction behaves more like an option pricing problem than a debt valuation problem. This paper will list, prove, and analyze formulas for stock loan valuation with finite horizon under various stock models, including classical geometric Brownian motion, mean-reverting, and two-state regime-switching with both mean-reverting and geometric Brownian motion states. Numerical examples are reported to illustrate the results.

Valuation of Stock Loans with Regime Switching

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.