Pricing Step Options under the CEV and other Solvable Diffusion Models

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.Comment: 30 pages, 16 figures, submitted to IJTA

Integral Options in Models with Jumps

We present an explicit solution to the formulated in [17] optimal stopping problem for a geometric compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the smooth fit may break down and then be replaced by the continuous fit. The result can be interpreted as pricing perpetual integral options in a model with jumps.Jump process, stochastic differential equation, optimal stopping problem, integral American option, compound Poisson process, Shiryaev´s process, Girsanov´s theorem, Ito´s formula, integrodifferential free-boundary problem, smooth and continuous fit, hypergeometric functions

Financial Securities Under Nonlinear Diffusion Asset Pricing Model

In this thesis we investigate two pricing models for valuing financial derivatives. Both models are diffusion processes with a linear drift and nonlinear diffusion coefficient. The forward price process of these models is a martingale under an assumed risk-neutral measure and the transition probability densities are given in analytically closed form. Specifically, we study and calibrate two different families of models that are constructed based on a so-called diffusion canonical transformation. One family follows from the Ornstein-Uhlenbeck diffusion (the UOU family) and the other—from the Cox-Ingersoll-Ross process (the Confluent-U family). The first part of the thesis considers single-asset and multi-asset modeling under the ∪O∪ model. By applying a Gaussian copula, a multivariate UOU model is constructed whereby all discounted asset (forward) prices are martingales. We succeed in calibrating the ∪O∪ model to market call option prices for various companies. Moreover, the multivariate ∪O∪ model is calibrated to historical return data and captures the correlations for a pool of 4 assets. In the second part of the thesis we examine the application of the Confluent-U model to the credit risk modeling. An equity-based structural first-passage time default model is constructed based on the Confluent-U model with efficient closed-form (i.e. spectral expansions) formulas for default probabilities. The model robustness is tested by its calibration to the credit default swap (CDS) spreads for companies with various credit ratings. It is shown that the model can be accurately calibrated to the credit spreads with a piecewise default barrier level. Finally, we investigate the linkage between CDS spreads and out-of-the-money put options

Close-Form Pricing of Benchmark Equity Default Swaps Under the CEV Assumption

Equity Default Swaps are new equity derivatives designed as a product for credit investors.Equipped with a novel pricing result, we provide closedform values that give an analytic contribution to the viability of cross-asset trading related to credit risk.Cross-Asset Trading of Credit Risk;Constant-Elasticity-of-Variance (CEV) Diffusion