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Some optimal stopping problems with nontrivial boundaries for pricing exotic options

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Xin Guo  and
Larry Shepp
Xin Guo*
Affiliation:
IBM
Larry Shepp*
Affiliation:
Rutgers University
*
∗ Postal address: IBM T. J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA. Email address: [email protected]
∗∗ Postal address: Department of Statistics, Rutgers University, Piscataway, NJ 08854, USA.
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Abstract

We solve the following three optimal stopping problems for different kinds of options, based on the Black-Scholes model of stock fluctuations. (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more difficult than the closely related one for the Russian option, and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary differential equation. (ii) A new type of stock option, for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a fixed constant l as a ‘hedge’ on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at any exercise time. We show that the optimal policy depends on the ratio of x/l, where x is the current stock price.

Keywords

Lookback optionsBlack-Scholes modeloptimal stoppingBellman equationfree boundary

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 647 - 658
Copyright
Copyright © by the Applied Probability Trust 2001 

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