Abstract:

The interest in probability models for financial applications makes mathematical finance into an important motivation for research in probability theory. As an example of results motivated by mathematical finance we present a new formula for Brownian motion that comes up in joint work with Jasper Anderluh about double-sided Parisian options.

A Parisian contract is special type of barrier option. A down-and-in call barrier option with strike K, barrier L and time to maturity T pays only if the asset price S_t hits the barrier before maturity in which case the pay-off is given by (S_T-K)⁺. A down-and-in Parisian option is a barrier option that only knocks in if the asset price is below the barrier during a given unbroken time period of length D. A down-and-in double-sided Parisian option is defined by two barriers L₁ and L₂, L₁<L₂, and two time lengths D₁ and D₂. The option knocks in if either the asset price is below L₁ during an unbroken time period D₁ or above L₂ during an unbroken time period D₂. Modeling the asset price as a geometric Brownian motion we can give a formula for the Laplace transform of the price of the option. If, in this formula, we let L₁ converge to L₂ we find that the probability that Brownian motion stays positive during an unbroken time period of length D₁ before it is negative during an unbroken time period of length D₂ must be equal to \sqrt{D₂}/(\sqrt{D₁}+\sqrt{D₂}). The structure of this formula resembles the solution of the ruin problem, a problem very well studied in the probabilistic literature. A gambler starts playing with an amount of A Dollar against an adversary who plays with a capital of B Dollar. The game is continued till one of the players has lost his capital. Let S_t denote the total profit of the gambler up to and including time t. The gambler wins if the process S_t reaches level B before reaching level -A. The probability that the gambler wins is A/(B+A). In this talk we will discuss for the case of simple Random Walk the probability that a sojourn of length D₁ above the axis occurs before a sojourn of length D₂ below the axis. We will try to explain the reason for the similarity of the formula for the sojourns with the formula for the ruin problem.