2013/02/15 by Giuseppe Campolieti, Campolieti, Giuseppe, Roman N. Makarov +3 · 1 citation
Economics, Econometrics and Finance · #Climate Change Policy and Economics #FOS: Economics and business #Pricing of Securities (q-fin.PR) #Stochastic processes and financial applications #q-fin.PR
paper · pdf · doi:10.48550/arxiv.1302.3771
30 pages, 16 figures, submitted to IJTAF
arxiv created 2013/02/15 · openalex publication_date 2013/02/15 · arxiv updated 2013/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.