  We consider a special family of occupation-time derivatives, namely
proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96
(1999)]. We develop new closed-form spectral expansions for pricing such
options under a class of nonlinear volatility diffusion processes which
includes the constant-elasticity-of-variance (CEV) model as an example. In
particular, we derive a general analytically exact expression for the resolvent
kernel (i.e. Green's function) of such processes with killing at an exponential
stopping time (independent of the process) of occupation above or below a fixed
level. Moreover, we succeed in Laplace inverting the resolvent kernel and
thereby derive newly closed-form spectral expansion formulae for the transition
probability density of such processes with killing. The spectral expansion
formulae are rapidly convergent and easy-to-implement as they are based simply
on knowledge of a pair of fundamental solutions for an underlying solvable
diffusion process. We apply the spectral expansion formulae to the pricing of
proportional step options for four specific families of solvable nonlinear
diffusion asset price models that include the CEV diffusion model and three
other multi-parameter state-dependent local volatility confluent hypergeometric
diffusion processes.
