Occupation Time Derivatives

Vadim Linetsky - University of Michigan

Path-dependent options have payoffs that depend on the entire price history of the underlying asset from the contract exception to expiration, rather than just on the terminal asset price at maturity of the option. Mathematically, the problem of pricing a path-dependent option can be reduced to a problem of calculating expectations of functionals defined on paths of diffusion processes. The Feynman-Kac approach provides a natural framework for handling path-dependent valuation problems[1].

Barrier options are one of the more popular types of path-dependent options. For example, a down-and-out barrier call is a call option with the barrier provision attached — it is extinguished if the price of the underlying asset hits a certain pre-specified price level B from above during the life of the option. The barrier provision allows investors and corporate hedgers to substantially reduce premiums they have to pay for the option. However, standard barriers have a significant risk management disadvantage — the discrete nature of the option payoff and delta.

Motivated by risk management problems with barrier options, in [2] we propose a user friendly modification of the barrier provision and introduce a new family of path-dependent options: step options. They are parametrized by a finite knock-out rate and can be thought of as gradual knock-out options. The terminal payoff is discounted with knock-out rate based on the occupation time below a pre-specified barrier level. Standard barrier options are then recovered in the limit of an infinitely high knock-out rate. The stochastic model underlying these financial contracts is Brownian motion with killing at finite rate below the barrier level. Step options are representatives of an interesting class of contingent claims with payoffs dependent both on the terminal asset price at expiration as well as occupation time of the price process. By employing the Feynmen-Kac approach, we derive closed-form pricing formulas for a variety of occupation time derivatives.

Based on papers:
[1] Linetsky, V. (1996a): The Path Integral Approach to Financial Modeling and Options Pricing, Forthcoming in Computational Economics.
[2] Linetsky, V. (1996b) : Step Options and Forward Contracts, University of Michigan, IOE Technical Report 96-18.

Scheduled for Session 5.4 Computational Finance