Prudent upper and lower valuations from the literature on arbitrage free two price economies provide risk characteristics driving required returns. The risk characteristics assess the risk of price fluctuations. The difference between the upper and lower prudent valuations can be viewed as a...

We propose a model of volatility tail behavior, in which investors display aversion to both low volatility and high volatility states, and, hence, the derived pricing kernel exhibits an increasing and decreasing region in the volatility dimension. The model features investors who have...

We propose a model of volatility tail behavior, in which the pricing measure dominates the physical measure in both tails of the volatility distribution and, hence, the derived pricing kernel exhibits an increasing and decreasing region in the volatility dimension. The model features investors...

The theory of two price markets of Cherny and Madan (2010) yields closed forms for bid and ask prices. Defining profits as the difference between the mid quote and the risk neutral expectation and capital as difference between the ask and the bid price one obtains precise expressions for these...

It is observed that statistical and risk neutral densities of compound Poisson processes are unconstrained relative to each other. Continuous processes are too constrained and generally not consistent with market data. Pure jump limit laws deliver operational models simultaneously consistent...

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...