When the pricing kernel is U-shaped, then expected returns of claims with payout on the upside are negative for strikes beyond a threshold, determined by the slope of the U-shaped kernel in its increasing region, and have negative partial derivative with respect to strike in the increasing...

This paper studies the structure of stock market crashes, rallies, their jump arrival rates, and extremes. Large market moves are characterized in a pure-jump modeling framework. Based on both raw and devolatized returns, it is shown empirically that crashes are more severe in intensity than...

Exponentials of squared returns in Gaussian densities, with their consequently thin tails, are replaced by the absolute return to form Laplacian and exponentially tilted Laplacian densities at unit time. Scaling provides densities at other maturities. Stochastic processes with these marginals...

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The...

It is argued that the growth in the breadth of option strikes traded after the financial crisis of 2008 poses difficulties for the use of Fourier inversion methodologies in volatility surface calibration. Continuous time Markov chain approximations are proposed as an alternative. They are shown...

This paper extends the known results on the equivalence between market completeness and the uniqueness of martingale measures for finite asset economies, to the infinite asset case. Our arguments employ results from the theory of linear operators between locally convex topological vector spaces....

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that...

Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered...

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that...

Stochastic processes arising in the description of the risk-neutral evolution of equity prices are reviewed. Starting with Brownian motion, I review extensions to Lévy and Sato processes. These processes have independent increments; the former are homogeneous in time, whereas the latter are...