Valuing continuous-installment options

Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.     

Pricing of perpetual American and Bermudan options by binomial tree method

In this paper, we consider the binomial tree method for pricing perpetual American and perpetual Bermudan options. The closed form solutions of these discrete models are solved. Explicit formulas for the optimal exercise boundary of the perpetual American option is obtained. A nonlinear equation that is satisfied by the optimal exercise boundaries of the perpetual Bermudan option is found.      

Pricing American currency options in an exponential Lévy model

In this article the problem of the American option valuation in a Lévy process setting is analysed. The perpetual case is first considered. Without possible discontinuities (i.e. with negative jumps in the call case), known results concerning the currency option value as well as the exercise boundary are obtained with a martingale approach. With possible discontinuities of the underlying process at the exercise boundary (i.e. with positive jumps in the call case), original results are derived by relying on first passage time and overshoot associated with a Lévy process. For finite life American currency calls, the formula derived by Bates or Zhang, in the context of a negative jump size, is tested. It is basically an extension of the one developed by Mac Millan and extended by Barone‐Adesi and Whaley. It is shown that Bates' model generates pretty good results only when the process is continuous at the exercise boundary.     

On modified Mellin transforms, Gauss-Laguerre quadrature, and the valuation of American call options

We extend a framework based on Mellin transforms and show how to modify the approach to value American call options on dividend-paying stocks. We present a new integral equation to determine the price of an American call option and its free boundary using modified Mellin transforms. We also show how to derive the pricing formula for perpetual American call options using the new framework. A result due to Kim (1990) [24] regarding the optimal exercise price at expiry is also recovered. Finally, we apply Gauss-Laguerre quadrature for the purpose of an efficient and accurate numerical valuation.     

Differential quadrature method for pricing American options

In this article, differential quadrature method (DQM), a highly accurate and efficient numerical method for solving nonlinear problems, is used to overcome the difficulty in determining the optimal exercise boundary of American option. The following three parts of the problem in pricing American options are solved. The first part is how to treat the uncertainty of the early exercise boundary, or free boundary in the language of the PDE treatment of the American option, because American options can be exercised before the date of expiration. The second part is how to solve the nonlinear problem, because the problem of pricing American options is nonlinear. And the third part is how to treat the initial value condition with the singularity and the boundary conditions in the DQM. Numerical results for the free boundary of American option obtained by both DQM and finite difference method (FDM) are given and from which it can be seen the computational efficiency is greatly improved by DQM. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 711–725, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10028.     

Optimal stopping in infinite horizon: An eigenfunction expansion approach

We develop an eigenfunction expansion based value iteration algorithm to solve discrete time infinite horizon optimal stopping problems for a rich class of Markov processes that are important in applications. We provide convergence analysis for the value function and the exercise boundary, and derive easily computable error bounds for value iterations. As an application we develop a fast and accurate algorithm for pricing callable perpetual bonds under the CIR short rate model.     

Further calculations for Israeli options

Recently Kifer introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment not less than the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a stochastic saddle point problem associated with Dynkin games. Kyprianou, A.E. (2004) "Some calculations for Israeli options", Fin. Stoch. 8, 73-86 gives two examples of perpetual Israeli options where the value function and optimal strategies may be calculated explicity. In this article, we give a third example of a perpetual Israeli option where the contingent claim is based on the integral of the price process. This time the value function is shown to be the unique solution to a (two sided) free boundary value problem on (0, ∞) which is solved by taking an appropriately rescaled linear combination of Kummer functions. The probabilistic methods we appeal to in this paper centre around the interaction between the analytic boundary conditions in the free boundary problem, Itô's formula with local time and the martingale, supermartingle and submartingale properties associated with the solution to the stochastic saddle point problem.     

Pricing perpetual American options under a stochastic-volatility model with fast mean reversion

In this paper, we present a “correction” to Merton’s (1973) well-known classical case of pricing perpetual American puts by considering the same pricing problem under a general fast mean-reverting SV (stochastic-volatility) model. By using the perturbation method, two analytic formulae are derived for the option price and the optimal exercise price, respectively. Based on the newly obtained formulae, we conduct a quantitative analysis of the impact of the SV term on the price of a perpetual American put option as well as its early exercise strategies. It shows that the presence of a fast mean-reverting SV tends to universally increase the put option price and to defer the optimal time to exercise the option contract, had the underlying been assumed to be falling. It is also noted that such an effect could be quite significant when the option is near the money.     

A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options

This is a complementary study of a recent work by Yoon et al. (2013) [1] [J.-H. Yoon, J.-H. Kim, S.-Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett. 26 (7) (2013)] which excludes a certain level of the elasticity of variance. A second-order correction to the Black–Scholes option price and optimal exercise boundary for a perpetual American put option is made under the stochastic elasticity of variance of a risky asset. Contrary to the case of Yoon et al. (2013) [1], it is given by an explicit closed-form analytic expression so that one can access clearly the sensitivity of the option price and the optimal exercise boundary to changes in model parameters as well as the impact of the presence of a stochastic elasticity term on the option price and the optimal time to exercise.     

A new predictor-corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.     

Penalty methods for the numerical solution of American multi-asset option problems

We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black–Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.     

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.     

Compact finite difference method for American option pricing

A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h   is large (h?0.1)$(h?0.1)$. Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.     

A simple model of deferred callability in defaultable debt

Banks and other financial institutions issue hybrid capital as part of their risk capital. Hybrid capital has no maturity, but, similarly to most corporate debt, includes an embedded issuer’s call option. To obtain acceptance as risk capital, the first possible exercise date of the embedded call is contractually deferred by several years, generating a protection period. We value the call feature as a European option on perpetual defaultable debt. We do this by first modifying the underlying asset process to incorporate a time-dependent bankruptcy level before the expiration of the embedded option. We identify a call option on debt as a fixed number of put options on a modified asset, which is lognormally distributed, as opposed to the market value of debt. To include the possibility of default before the expiration of the option we apply barrier options results. The formulas are quite general and may be used for valuing both embedded and third-party options. All formulas are developed in the seminal and standard Black–Scholes–Merton model and, thus, standard analytical tools such as ‘the greeks’, are immediately available.     

Optimal stopping, free boundary, and American option in a jump-diffusion model

This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price.     

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.     

American Call Options Under Jump‐Diffusion Processes – A Fourier Transform Approach

We consider the American option pricing problem in the case where the underlying asset follows a jump‐diffusion process. We apply the method of Jamshidian to transform the problem of solving a homogeneous integro‐partial differential equation (IPDE) on a region restricted by the early exercise (free) boundary to that of solving an inhomogeneous IPDE on an unrestricted region. We apply the Fourier transform technique to this inhomogeneous IPDE in the case of a call option on a dividend paying underlying to obtain the solution in the form of a pair of linked integral equations for the free boundary and the option price. We also derive new results concerning the limit for the free boundary at expiry. Finally, we present a numerical algorithm for the solution of the linked integral equation system for the American call price, its delta and the early exercise boundary. We use the numerical results to quantify the impact of jumps on American call prices and the early exercise boundary.     

Regularity of the American Put option in the Black–Scholes model with general discrete dividends

We analyze the regularity of the value function and of the optimal exercise boundary of the American Put option when the underlying asset pays a discrete dividend at known times during the lifetime of the option. The ex-dividend asset price process is assumed to follow the Black–Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. This function is assumed to be non-negative, non-decreasing and with growth rate not greater than 1. We prove that the exercise boundary is continuous and that the smooth contact property holds for the value function at any time but the dividend dates. We thus extend and generalize the results obtained in Jourdain and Vellekoop (2011) [10] when the dividend function is also positive and concave. Lastly, we give conditions on the dividend function ensuring that the exercise boundary is locally monotonic in a neighborhood of the corresponding dividend date.     

非饱和土层一维固结问题的解析解

对一有限厚度,处于一维受荷状态,表面为透水透气面,底面为不透水不透气面的非饱和土层,依据Fredlund的非饱和土一维固结理论,由液相及气相的控制方程、Darcy定律及Fick定律,经Laplace变换及Cayley-Hamilton数学方法构造了顶面状态向量与任意深度处状态向量间的传递关系;通过引入初始及边界条件,得到了Laplace变换域内的超孔隙水压力、超孔隙气压力以及土层沉降的解;实现Laplace逆变换,得到了时间域内的解析解;用一典型算例,与差分法结果进行对比,验证了其正确性．     

Asymptotic results for renewal risk models with risky investments

We consider a renewal jump–diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform.