Investment Decisions with Two-Factor Uncertainty

This paper considers investment problems in real options with non-homogeneous two-factor uncertainty. We derive some analytical properties of the resulting optimal stopping problem and present a finite difference algorithm to approximate the firm’s value function and optimal exercise boundary. An important message in our paper is that the frequently applied quasi-analytical approach underestimates the impact of uncertainty. This is caused by the fact that the quasi-analytical solution does not satisfy the partial differential equation that governs the value function. As a result, the quasi-analytical approach may wrongly advise to invest in a substantial part of the state space.

Numerical Method for a System of PIDEs Arising in American Contingent Claims under FMLS Model with Jump Diffusion and Regime-Switching Process

This paper investigates a numerical method for solving fractional partial integro-differential equations (FPIDEs) arising in American Contingent Claims, which follow finite moment log-stable process (FMLS) with jump diffusion and regime switching. Mathematically, the prices of American Contingent Claims satisfy a system of d problems with free-boundary values, where d is the number of regimes of the market. In addition, an optimal exercise boundary is needed to setup with each regime. Therefore, a fully implicit scheme based on the penalty term is arranged. In the end, numerical examples are carried out to verify the obtained theoretical results, and the impacts of state variables in our model on the optimal exercise boundary of American Contingent Claims are analyzed.

An Artificial Intelligence Approach to the Valuation of American-Style Derivatives: A Use of Particle Swarm Optimization

In this paper, we evaluate American-style, path-dependent derivatives with an artificial intelligence technique. Specifically, we use swarm intelligence to find the optimal exercise boundary for an American-style derivative. Swarm intelligence is particularly efficient (regarding computation and accuracy) in solving high-dimensional optimization problems and hence, is perfectly suitable for valuing complex American-style derivatives (e.g., multiple-asset, path-dependent) which require a high-dimensional optimal exercise boundary.

CLOSED FORM OPTIMAL EXERCISE BOUNDARY OF THE AMERICAN PUT OPTION

We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the finite maturity American put option. The optimal exercise boundary satisfies the nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary.

An investigation on the existence and uniqueness analysis of the optimal exercise boundary of American put option

In this paper, we discuss the Banach fixed point theorem conditions on the optimal exercise boundary of American put option paying continuously dividend yield, to investigate whether its existence, uniqueness, and convergence are derived. In this respect, we consider the integral representation of the optimal exercise boundary which is extracted as a consequence of the Feynman-Kac formula. In order to prove the above features, we define a nonempty closed set in Banach space and prove that the proposed mapping is contractive and onto. At final, we illustrate the ratio convergence of the mapping on the optimal exercise boundary.

On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model

We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.

PENALTY AMERICAN OPTIONS

We present a new American-style option whereby on the event of exercise before expiry, the holder pays the writer a fee (which will be referred to as a ‘penalty’). The valuation of the option is not straightforward as it involves determining when it is optimal for the holder to exercise the option, leading to a free boundary problem. As most options in the traded markets have short maturities, accurate and fast valuations of such options are important. We derive analytic approximations for the value of the option with short times to expiry (up to [Formula: see text] months) and its optimal exercise boundary. Some properties of the option, such as the put–call relationship, are explored as well. Numerical experiments suggest that our solutions both for the optimal exercise boundary and option value provide very accurate results.

INTEGRAL EQUATION FORMULATION FOR SHOUT OPTIONS

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.