The American Put Option and Its Critical Stock Price

Abstract

We derive an expression for the critical stock price for the American put. We start by expressing the put price as an integral involving first-passage probabilities. This approach yields intuition for Merton's result for the perpetual put. We then consider the finite-lived case. Using~1! the fact that the put value ceases to depend on time when the critical stock price is reached and~2! the result that an American put equals a European put plus an early-exercise premium, we derive the critical stock price. We approximate the critical-stock-price function to compute accurate put prices.KIM~1990!, JACKA~1991!, AND CARR, JARROW, AND MYNENI~1992! showed that the American put price is equal to the corresponding European put price plus an integral representing the early-exercise premium. However, to use their approach one needs to know the critical stock price, which may be defined in three equivalent ways.1. It is the value of the stock price at which one is indifferent between exercising and not exercising the put. 2. It is the highest value of the stock price for which the value of the put is equal to the exercise price less the stock price. 3. It is the highest value of the stock price at which the put value does not depend on time to maturity.The second definition has ordinarily been used to find the critical stock price, but it requires solving an integral equation numerically. If it is optimal to exercise the put immediately, then its value cannot depend on how much time remains to maturity. Hence, we have the third definition, which, we believe, is new to the literature. We show that the third definition leads to an analytic expression for the critical stock price. Our result is important not only because it provides a better way to get American put prices but also

Pricing American Options under Stochastic Volatility and Stochastic Interest Rates

“…Table 2 reports option values from Bunch and Johnson (2000) ( Table II) and our results based on a 4th order asymptotic expansion (see Appendix B). The accuracy of our approximation is comparable with that of Bunch and Johnson (2000), and is roughly equivalent to a 300-step tree.…”

“…where e y is implicitly defined in (15). That is, the minimum level of moneyness such that the American put is best approximated by its payoff and, therefore, should be exercised immediately.…”

“…Time-to-maturity , asset price S, interest rate r, volatility , and dividend rate  are indicated in the table. Table II of Bunch and Johnson (2000). "True value" is a 10,000-step binomial tree approximation computed in Bunch and Johnson (2000).…”

“…Table II of Bunch and Johnson (2000). "True value" is a 10,000-step binomial tree approximation computed in Bunch and Johnson (2000). Here asset price S = 40, interest rate r = 0.0488 and dividend yield is zero.…”

Option Pricing Under Stochastic Interest Rates

“…Several semi-analytical approximations for American option prices have been proposed in the literature (Barone Adesi and Whaley [1], 1987; Broadie and Detemple [5], 1996; Bunch and Johnson [6], 2000). Although these approaches are fast and accurate, they can not easily be extended beyond the Black-Scholes model.…”