American Options in Financial Quotes

Abstract

We build a mathematical model for the risk involved when a person makes a binding fixed-price offer to buy or sell something that fluctuates in value. This situation often arises in financial markets, where such an offer is called a quote. Quotes involve a risk for the person giving them, and an opportunity for the person receiving them. We investigate two different versions of quotes: offers that cannot be canceled before a certain time has passed ("minimum resting times"), and quotes that are automatically canceled if the price moves past a specified barrier ("last look"). Our model of financial quotes is in many respects similar to option pricing models. Quotes with minimum resting times are in a certain sense similar to American options, and quotes with last look are similar to American barrier options. There are also differences however, for example are the time scales of financial quotes orders of magnitude shorter than the time scales of traditional options. These differences lead us to use different modeling approaches than what is used for traditional option pricing models, in particular we investigate optimal stopping problems for the class of integer-valued Levy processes. We develop both explicit formulas and numerical algorithms.