Narrow Results

It is well acknowledged that the European options can be valued by an analytic formula, but situation is quite different for the American options. Mathematically, the Black-Scholes model for the American option pricing is a free boundary problem of partial differential equation. This model is a non-linear problem; it has no closed form solution. Although approximate solutions may be obtained by some numerical methods, but the precision and stability are hard to control since they are largely affected by the singularity at the exercise boundary near expiration date. In this paper, we propose a new numerical method, namely SDA, to solve the pricing problem of the American options. Our new method combines the advantages of the Semi-analytical Method and the Sliced-fixed Boundary Finite Difference Method while overcomes demerits of the two. Using the SDA method, we can resolve the problems resulted from the singularity near the optimal exercise boundary. Numerical experiments show that the SDA method is more accurate and more stable than other numerical methods. In this paper, we focus on the American put options, but the proposed method is also applicable to other types of options.

This paper develops a unified formulation and a new computational methodology for the entire class of the multi-factor Markovian interest rate models. The early exercise premium representation for general American options is derived for all Markovian models. The option cash flow functions are decomposed into fast and slowly varying components. The fast varying components have the same expression for all options within a model. They are calculated analytically. Only the slowly varying components are option specific. Their backward induction for a finite time interval is carried out from Taylor expansion expressions. The small coefficient of the expansion is the product of the variance and the width of the time interval. The option price is calculated by dividing its time horizon into smaller intervals and numerically iterating the Taylor expansion expressions of one time interval. Other new results include: (i) The derivation of a new "almost" Markovian LIBOR market model and its related Markovian short-rate model; (ii) the universal form of the critical boundary near the maturity for the American options in the one-factor Markovian models; and (iii) approximate analytic expressions for the entire critical boundary of the American put stock option. The put price calculated from the boundary has relative precision better than 10^-5.

In this paper, we extend the results of Carmona and Touzi [6] for an optimal multiple stopping problem to a market where the price process is allowed to jump. We also generalize the problem of valuation swing options to the context of a Lévy market. We prove the existence of multiple exercise policies under an additional condition on Snell envelops. This condition emerges naturally in the case of Lévy processes. Then, we give a constructive solution for perpetual put swing options when the price process has no negative jumps. We use the Monte Carlo approximation method based on Malliavin calculus in order to solve the finite horizon case. Numerical results are given in the last two sections. We illustrate the theoretical results of the perpetual case and give the numerical solution for the finite horizon case.

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level.

Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions.

Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.

This paper is concerned with regime-switching American option pricing. We develop new numerical schemes by extending the penalty method approach and by employing the θ-method. With regime-switching, American option prices satisfy a system of m free boundary value problems, where m is the number of regimes considered for the market. An (optimal) early exercise boundary is associated with each regime. Straightforward implementation of the θ-method would result in a system of nonlinear equations requiring a time-consuming iterative procedure at each time step. To avoid such complications, we implement an implicit approach by explicitly treating the nonlinear terms and/or the linear terms from other regimes, resulting in computationally efficient algorithms. We establish an upper bound condition for the time step size and prove that under the condition the implicit schemes satisfy a discrete version of the positivity constraint for American option values. We compare the implicit schemes with a tree model that generalizes the Cox-Ross-Rubinstein (CRR) binomial tree model, and with an analytical approximation solution for two-regime case due to Buffington and Elliott. Numerical examples demonstrate the accuracy and stability of the new implicit schemes.

We investigate in this paper a perpetual prepayment option related to a corporate loan. The short interest rate and default intensity of the firm are supposed to follow Cox–Ingersoll–Ross (CIR) processes. A liquidity term that represents the funding costs of the bank is introduced and modeled as a continuous time discrete state Markov chain. The prepayment option needs specific attention as the payoff itself is a derivative product and thus an implicit function of the parameters of the problem and of the dynamics. We prove verification results that allows to certify the geometry of the exercise region and compute the price of the option. We show moreover that the price is the solution of a constrained minimization problem and propose a numerical algorithm building on this result. The algorithm is implemented in a two-dimensional code and several examples are considered. It is found that the impact of the prepayment option on the loan value is not to be neglected and should be used to assess the risks related to client prepayment. Moreover, the Markov chain liquidity model is seen to describe more accurately clients' prepayment behavior than a model with constant liquidity.

Swing options are a type of exotic financial derivative which generalize American options to allow for multiple early-exercise actions during the contract period. These contracts are widely traded in commodity and energy markets, but are often difficult to value using standard techniques due to their complexity and strong path-dependency. There are numerous interesting varieties of swing options, which differ in terms of their intermediate cash flows, and the constraints (both local and global) which they impose on early-exercise (swing) decisions. We introduce an efficient and general purpose transform-based method for pricing discrete and continuously monitored swing options under exponential Lévy models, which applies to contracts with fixed rights clauses, as well as recovery time delays between exercise. The approach combines dynamic programming with an efficient method for calculating the continuation value between monitoring dates, and applies generally to multiple early-exercise contracts, providing a unified framework for pricing a large class of exotic derivatives. Efficiency and accuracy of the method are supported by a series of numerical experiments which further provide benchmark prices for future research.

There has been considerable interest in developing stochastic volatility and jump-diffusion option pricing models, e.g. Hull and White (1987, Journal of Finance, 42, 281–300) and Merton (1976, Journal of Financial Economics, 3, 125–144). These models, however, have some undesirable aspects that arise from introducing some non-traded sources of risks to the models. Furthermore, the models require much analytical complications; thus, if they are applied to American options then it is not easy to acquire practical implications for hedging and optimal exercise strategies. This paper examines the American option prices and optimal exercise strategies where the volatility of the underlying asset changes over time in a deterministic way. The paper considers two simple cases: monotonically increasing and decreasing volatilities. The discussion of these two simple cases gives useful implications for the possibility of early-exercise and optimal exercise strategies.

There is a vast literature on numerical valuation of exotic options using Monte Carlo (MC), binomial and trinomial trees, and finite difference methods. When transition density of the underlying asset or its moments are known in closed form, it can be convenient and more efficient to utilize direct integration methods to calculate the required option price expectations in a backward time-stepping algorithm. This paper presents a simple, robust and efficient algorithm that can be applied for pricing many exotic options by computing the expectations using Gauss–Hermite integration quadrature applied on a cubic spline interpolation. The algorithm is fully explicit but does not suffer the inherent instability of the explicit finite difference counterpart. A "free" bonus of the algorithm is that it already contains the function for fast and accurate interpolation of multiple solutions required by many discretely monitored path dependent options. For illustrations, we present examples of pricing a series of American options with either Bermudan or continuous exercise features, and a series of exotic path-dependent options of target accumulation redemption note (TARN). Results of the new method are compared with MC and finite difference methods, including some of the most advanced or best known finite difference algorithms in the literature. The comparison shows that, despite its simplicity, the new method can rival with some of the best finite difference algorithms in accuracy and at the same time it is significantly faster. Virtually the same algorithm can be applied to price other path-dependent financial contracts such as Asian options and variable annuities.

In this paper, we employ the Least-Squares Monte-Carlo (LSM) algorithm regarding three discretization schemes, namely, the Euler–Maruyama discretization scheme, the Milstein scheme and the Quadratic Exponential (QE) scheme to price the multiple assets American put option under the Heston stochastic volatility model. Some numerical results are presented to demonstrate the effectiveness of the proposed methods.

This paper focuses primarily on pricing an American put option with a fixed term where the price process is geometric mean-reverting. The change of measure is assumed to be incorporated. Monte Carlo simulation was used to calculate the price of the option and the results obtained were analyzed. The option price was found to be $94.42 and the optimal stopping time was approximately one year after the option was sold which means that exercising early is the best for an American put option on a fixed term. Also, the seller of the put option should have sold $0.01 assets and bought $95.51 bonds to get the same payoff as the buyer at the end of one year for it to be a zero-sum game. In the simulation study, the parameters were varied to see the influence it had on the option price and the stopping time and it showed that it either increases or decreases the value of the option price and the optimal stopping time or it remained unchanged.

In this chapter, we (i) use the decision-tree approach to derive binomial option pricing model (OPM) in terms of the method used by Rendleman and Barter (RB, 1979) and Cox et al. (CRR, 1979) and (ii) use Microsoft Excel to show how decision-tree model can be converted to Black–Scholes model when the number period increases to infinity. In addition, we develop binomial tree model for American option and trinomial tree model. The efficiency of binomial and trinomial tree methods is also compared. In sum, this chapter shows how binomial OPM can be converted step by step to Black–Scholes OPM.

In this chapter, we first review the basic theory of normal and log-normal distribution and their relationship, then bivariate and multivariate normal density function are analyzed in detail. Next, we discuss American options in terms of random dividend payment. We then use bivariate normal density function to analyze American options with random dividend payment. Computer programs are used to show how American co-options can be evaluated. Finally, pricing option bounds are analyzed in some detail.

The main purposes of this introduction chapter are (i) to give an overview of the following 109 papers, which discuss investment analysis, portfolio management, and financial derivatives; (ii) to classify these 109 chapters into nine topics; and (iii) to classify the keywords in terms of chapter numbers.

It is well known that both normal and log-normal distributions are important to understand Black & Scholes-type European and American options. Therefore, we first review the basic theory of normal and log-normal distributions and their relationship, then bivariate and multivariate normal density functions are analyzed in detail. Next, we discuss American options in terms of random dividend payment. We then use bivariate normal density function to analyze American options with random dividend payment. Excel programs are used to show how American co-options can be evaluated. Finally, pricing option bounds are analyzed in some detail.

The main aims of this chapter are (i) to use the decision tree approach to derive binomial option pricing model (OPM) in terms of the method used by Rendleman and Barter (RB, 1979) and Cox, Ross, and Rubinstein (CRR, 1979) and (ii) to use Microsoft Excel to show how decision tree model can be converted to Black–Scholes model when the number period increases to infinity. In addition, we develop binomial tree model for American option and trinomial tree model. The efficiency between binomial and trinomial tree is also compared. In sum, this chapter shows how binomial OPM can be converted step by step to Black–Scholes OPM.

In this paper generalized barrier options of American type in discrete time are studied. Instead of a barrier, a domain of knock out type is considered. To find the optimal time of exercising the contract, or stopping a Markov price process, an optimal stopping domain can be constructed. To determine the optimal stopping domain Monte Carlo simulation is used. Probabilities of classification errors when determining the structure of the optimal stopping domain are analyzed.