Probability of multiple crossings of barriers and applications to pricing of exotic options

Abstract

In the modern financial market, financial options are important tools for hedging and investment, and pricing of various options depends on sophisticated quantitative analysis. This paper researches about exotic options, especially barrier options. A barrier option is a path dependent option which is activated or becomes worthless on whether the underlying asset price touches or crosses a barrier or barriers before expiration. Based on probabilitic approach, main purpose of this paper is pricing and analysis various barrier options. First we derive probabilities of multiple crossings of curved barriers for Brownian motion with drift, by repeatedly applying the Girsanov theorem and the reflection principle. Using probabilities of multiple crossings, the price of a standard double barrier option is presented as an infinite sum and it is shown that the convergence speed of infinite series is very rapid. Although the pricing formula of standard double barrier option is same with previous results, the method used to derive the formula is different with [36] and [45]. Moreover our method gives an intuitive interpretation for each term in the infinite series. A window double barrier option is a double barrier option whose monitoring period starts after contract initiation and terminates before the contract expiry. [24] found the valuation formula of the window double barrier option whose barriers are constants. We extend constant barrier assumption to curved barriers and derive the pricing formula of generalized window double barrier option. Although usual papers, including our previous results, assume that the underlying asset is continuously monitored, traded barrier options are monitored only on pre-specified dates, i.e. only a finite number of times. Since it is practically impossible to check the possible but unobservable crossings of the asset price between two successive monitoring times, computation of the price of discrete monitored barrier options is dif...