  In this paper, a time substitution as used by Duru and Kleinert in their
treatment of the hydrogen atom with path integrals is performed to price timer
options under stochastic volatility models. We present general pricing formulas
for both the perpetual timer call options and the finite time-horizon timer
call options. These general results allow us to find closed-form pricing
formulas for both the perpetual and the finite time-horizon timer options under
the 3/2 stochastic volatility model as well as under the Heston stochastic
volatility model. For the treatment of timer option under the 3/2 model we will
rely on the path integral for the Morse potential, with the Heston model we
will rely on the Kratzer potential.
