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Valuation of non-recourse stock loan using an integral equation approach
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We present an innovative decomposition approach for computing the price and the hedging parameters of American knockout options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight into the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.
A stock loan is a loan, secured by a stock, which gives the borrower the right to redeem the stock at any time before or on the loan maturity. The way of dividends distribution has a significant effect on the pricing of stock loans and the optimal redeeming strategy adopted by the borrower. We present the pricing models of the finite maturity stock loans subject to various ways of dividend distribution. Because closed-form price formulas are generally not available, we provide a thorough analysis to examine the optimal redeeming strategy. Numerical results are presented as well.
In this paper we study a kind of financial product, stock loans, in which there is a capped limit for the stock price when it exceeds a predetermined barrier. Loans with two types of cap are analyzed: constant caps and caps with a constant growth rate. We build the pricing models of the contracts by analyzing the form of the optimal stopping time and derive the formulas of the value functions. We present the numerical results and make an analysis of fair values of related parameters. A comparison between capped stock loans and uncapped ones is also given. We find that capped loans have their own advantages such as more flexibility and lower cost.
This paper surveys some of the literature on American option pricing, in particular the representations of McKean (1965), Kim (1990) and Carr, Jarrow and Myneni (1992). It is proposed that the approach regarding the problem as a free boundary value problem, and solving this via incomplete Fourier transforms, is the most robust for fur- ther developments involving more complex payo structures, and higher dimensional problems such as multi-asset American options. Some comparison of dierent numerical solution methods is also provided.
We present an approximation method for pricing and hedging American options written on a dividend-paying asset. This method
is based on Kim (1990) equations. We demonstrate that a simple approximation of the Kim integral equations by quadrature formulas
leads to an efficient and accurate numerical procedure. This approximation is accompanied by the Newton–Raphson iteration
procedure in order to compute the optimal exercise boundary at each time point. The proposed sequence of approximations converges
monotonically, convergence is fast and accuracy is high, even for long maturity options. We compare numerically our results
with other competing approaches by different authors. JEL classification codes: G12, G13, C63.
In this paper, an integral equation approach is adopted to price American-style down-andout
calls. Instead of using the probability theory as used in the literature, we use the continuous
Fourier sine transform to solve the partial differential equation system governing
the option prices. As a way of validating our approach, we show that the ‘‘early exercise
premium representation’’ for American-style down-and-out calls without rebate can be rederived
by using our approach. We then examine the case that time-dependent rebates are
included in the contract of American-style down-and-out calls. As a result, a more general
integral representation for the price of an American-style down-and-out call, with the presence
of an extra term associated with the rebate, is obtained. Our numerical method based
on the newly-derived integral representation appears to be efficient in computing the price
and the hedging parameters for American-style down-and-out calls with rebates. In addition,
significant effects of rebates on the option prices and the optimal exercise boundaries
are illustrated through selected numerical results.
In this paper, we formulate margin call stock loans in finite maturity as American down-and-out calls with rebate and time-dependent strike. The option problem is solved semi-analytically based on the approach in Zhu (2006). An explicit equation for optimal exit price and a pricing formula for loan value are obtained in Laplace space. Final results are obtained by numerical inversion. Examples are provided to show the dependency of the optimal exit price and margin call stock loan value on various parameters.
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.
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 Levy 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.
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.
This paper introduces a mathematical model for a currently popular financial product called a stock loan. Quantitative analysis is carried out to establish explicitly the value of such a loan, as well as the ranges of fair values of the loan size and interest, and the fee for providing such a service.
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.
Key wordsMean reversion-regime switching-stock loan
This paper works out fair values of the stock loan model with automatic termination clause, cap and margin. This stock loan is treated as a generalized perpetual American option with possibly negative interest rate and some constraints. Since it helps a bank to control the risk, the banks charge lower service fees compared to stock loans without any constraints. The automatic termination clause, cap and margin are in fact a stop order set by the bank. Mathematically, it is a kind of optimal stopping problem arising from the pricing of financial products which is first revealed. We aim at establishing explicitly the value of such a loan and ranges of fair values of key parameters : this loan size, interest rate, cap, margin and fee for providing such a service and quantity of this automatic termination clause and the relationships among these parameters as well as the optimal exercise times. We present numerical results and make analysis about the model parameters and how they impact on value of stock loan.
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 difierent from that for standard American options because the associated variational inequalities may have inflnitely many solutions. In addition, the optimal stopping time equals inflnity 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.
A stock loan is a contract whereby a stockholder uses shares as collateral to borrow money from a bank or financial institution. In Xia and Zhou (20079.
Xia , J. and
Zhou , X. Y. 2007. Stock loans. Mathematical Finance, 17(2): 307–317. [CrossRef], [Web of Science ®]View all references, Stock loans, Mathematical Finance, 17(2), pp. 307–317), this contract is modelled as a perpetual American option with a time-varying strike and analysed in detail within a risk-neutral framework. In this paper, we extend the valuation of such loans to an incomplete market setting, which takes into account the natural trading restrictions faced by the client. When the maturity of the loan is infinite, we use a time-homogeneous utility maximization problem to obtain an exact formula for the value of the loan fee to be charged by the bank. For loans of finite maturity, we characterize the fee using variational inequality techniques. In both cases, we show analytically how the fee varies with the model parameters and illustrate the results numerically.