Hitting time and time change

Abstract

This paper determines first‐passage time distributions with a twofold emphasis on the dynamics of the state variables and interest rate uncertainty. Underlyings follow two‐dimensional geometric Brownian motions, Ornstein–Uhlenbeck processes or Poisson jump‐diffusion processes, and boundaries are either fixed or indexed on risk‐free bonds. Forward‐neutral changes of numeraire enable one to derive generic valuation expressions, while changing time allows one to determine closed‐form solutions for geometric Brownian motions and moving barriers. In turn, the latter formulas are used to reduce the variance of Monte Carlo simulations in the case of jump‐diffusion processes, by means of the control variate method.

Keywords:

• digital option
• soft barrier
• forward‐neutral measure
• time change
• jump‐diffusion process