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.     

Convexity of the optimal stopping boundary for the American put option

We show that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model. The methods are adapted from ice-melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations.     

Valuing American options under the CEV model by Laplace-Carson transforms

This paper investigates American option pricing under the constant elasticity of variance (CEV) model. Taking the Laplace-Carson transform (LCT) to the corresponding free-boundary problem enables the determination of the optimal early exercise boundary to be separated from the valuation procedure. Specifically, a functional equation for the LCT of the early exercise boundary is obtained. By applying Gaussian quadrature formulas, an efficient method is developed to compute the early exercise boundary, American option price and Greeks under the CEV model.     

Valuing executive stock options: A quadratic approximation

This paper develops a continuous-time model for valuing executive stock options (ESOs) with features of early exercise, delayed vesting and forfeiture. Applying the quadratic approximation established for valuing American options into ESOs, we obtain an explicit formula for the fair ESO value at its grant date. We show that the approximation formula is consistent with the exact results for two special cases either with no dividend or infinite maturity, and also that the perpetual value for the latter case gives an upper bound of the ESO value. To see the performance of the formula, we numerically examine it with benchmark results generated by a binomial-tree model for some particular cases. Numerical experiments show that there is a complementary relation between the vesting and trading periods with respect to exit rate of ESO holders.     

Prices and sensitivities of Asian options: A survey

Asian options are hard to price both analytically and numerically. Even though they have been the focus of much attention in recent years, there is no single technique which is widely accepted to price Asian options for all choices of market parameters. For hedging purposes, the estimation of the price sensitivities is often as important as the evaluation of the prices themselves. This paper provides a survey of current methods for pricing Asian options and computing their sensitivities to the key input parameters. The methods discussed include: Monte Carlo simulation, the finite difference approach and various quasi analytical approaches and approximations. We discuss practical numerical issues that arise in implementing these methods. The paper compares the accuracy and efficiency of the different approaches and offers some general conclusions.     

The pricing and optimal strategies of callable warrants

In this paper, we introduce a valuation model of callable warrants under a setting of the optimal stopping problem between the holder (investor) and the issuer (firm). A warrant is the right to purchase new shares at a predetermined price. When the new stocks are issued, the value of the stock is diluted. We consider the model taking the dilution into account. After identifying optimal policies for the issuer and the investor, we explore the analytical properties of the optimal exercise and call boundaries for the holder and the issuer, respectively. Furthermore, the value of such a callable warrant and the optimal critical prices are examined numerically using the binomial method.     

Properties of game options

A game option is an American option with the added feature that not only the option holder, but also the option writer, can exercise the option at any time. We characterize the value of a perpetual game option in terms of excessive functions, and we use the connection between excessive functions and concave functions to explicitly determine the value in some examples. Moreover, a condition on the two contract functions is provided under which the value is convex in the underlying diffusion value in the continuation region and increasing in the diffusion coefficient.Mathematics Subject Classification (2000) Primary 91A15, Secondary 60G40, 91B28     

An approximate moving boundary method for American option pricing

We present a method to solve the free-boundary problem that arises in the pricing of classical American options. Such free-boundary problems arise when one attempts to solve optimal-stopping problems set in continuous time. American option pricing is one of the most popular optimal-stopping problems considered in literature. The method presented in this paper primarily shows how one can leverage on a one factor approximation and the moving boundary approach to construct a solution mechanism. The result is an algorithm that has superior runtimes-accuracy balance to other computational methods that are available to solve the free-boundary problems. Exhaustive comparisons to other pricing methods are provided. We also discuss a variant of the proposed algorithm that allows for the computation of only one option price rather than the entire price function, when the requirement is such.     

A variational inequality arising from American installment call options pricing

In this paper we consider a parabolic variational inequality with two free boundaries arising from American continuous-installment call options pricing. We prove the existence and uniqueness of the solution to the problem. Moreover, we obtain the monotonicity and smoothness of two free boundaries and show its numerical solution by the binomial method.     

On the behaviour near expiry for multi-dimensional American options

In this paper we analyse the behaviour, near expiry, of the free boundary appearing in the pricing of multi-dimensional American options in a financial market driven by a general multi-dimensional Ito diffusion. In particular, we prove regularity for the pricing function up to the terminal state and we establish a sufficient criteria for the conclusion that the optimal exercise boundary approaches the terminal state faster than parabolically.     

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.     

Valuing interdependent multi-stage IT investments: A real options approach

In this paper, we use the market asset disclaimer assumption and develop a binomial lattice based real options model to include cash flow interdependencies between multi-stage information technology (IT) investments. Using a simple two-stage IT investment problem with interdependent cash flows, we apply the binomial lattice based real options model to obtain combined valuation of the two-stage IT investment. In addition to investment valuation, our experience with the two-stage IT investment valuation suggests that the binomial lattice based real options model provides a powerful decision aid tool for appropriate timing, delaying and abandoning of the second-stage IT investment.     

A NUMERICAL METHOD FOR DETERMINING THE OPTIMAL EXERCISE PRICE TO AMERICAN OPTIONS

American Options can be exercised prior to the date of expiration,the valuation of American options then constitutes a free boundary value problem.How to determine the free boundary,i.e. the optimal exercise price,is a key problem.In this paper,a nonlinear equation is given.The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.     

The early exercise region for Bermudan options on two underlyings

This paper investigates the early exercise region for Bermudan options on two underlying assets. We present a set of analytical validation results for the early exercise region which can be used as a means of validating pricing techniques. When all strike prices are identical we show the existence of an intersection point such that for any asset price pair below this point early exercise is always optimal. We develop an approximation to this point in the two asset put case. When the strike prices are not all equal, we show that three separate cases exist for the early exercise region. For a Bermudan put on two assets we present these cases and show that there exists a critical point in which the boundaries of the two asset early exercise region bifurcate. Comparisons are drawn between the Bermudan results presented and the corresponding American option results.     

On the numerical feasibility of continuation methods for nonlinear programming problems

In the present paper the radius of convergence of a class of locally convergent nonlinear programming algorithms (containing Robinson's and Wilson's methods) applied to a parametric nonlinear programming problem is estimated. A consequence is the numerical feasibility of globalizations of Robinson's and Wilson's methods by means of continuation techniques.     

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.     

A note on a nonlinear functional equation and its application

This paper treats the following type of nonlinear functional equations