Double barrier American put option pricing under uncertain volatility model

This paper aims to study the asymptotic behavior of double barrier American-style put option prices under an uncertain volatility model, which degenerates to a single point. We give an approximation of the double barrier American-style option prices with a small volatility interval, expressed by the Black–Scholes–Barenblatt equation. Then, we propose a novel representation for the early exercise boundary of American-style double barrier options in terms of the optimal stopping boundary of a single barrier contract.

A note on the nonlinear Volterra integral equation for the early exercise boundary

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S. In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S) holds for all {S\leq b(t)} with a candidate function {b(t)}, then this {b(t)} must coincide with the American put early exercise boundary {c(t)}. We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function {b(t)} satisfies one particular integral equation (which corresponds to the upper limit {S=b(t)}), then all other integral equations (corresponding to S, {S\leq b(t)}) will be automatically satisfied by the same function {b(t)}.

Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method

This paper proposes an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing. The method works by exploiting and leveraging the information in multiple cross-sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. With this method, less errors are accumulated, and as a result of this, the price estimate is essentially unbiased even for long maturity options. Numerical results demonstrate the improvements from our method and show that these are robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociates the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains can be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method.

Early Exercise Decision in American Options with Dividends, Stochastic Volatility, and Jumps

Using a fast numerical technique, we investigate a large database of investors’ suboptimal nonexercise of short-maturity American call options on dividend-paying stocks listed on the Dow Jones. The correct modeling of the discrete dividend is essential for a correct calculation of the early exercise boundary, as confirmed by theoretical insights. Pricing with stochastic volatility and jumps instead of the Black–Scholes–Merton benchmark cuts the amount lost by investors through suboptimal exercise by one-quarter. The remaining three-quarters are largely unexplained by transaction fees and may be interpreted as an opportunity cost for the investors to monitor optimal exercise.