Risk neutral variance term structures are characterized by their time elasticities. They are synthesized by space scaling and time changing self decomposable laws at unit time. Monotone time elasticities are modeled using exponential functions while gamma functions permit humps. Results for both...

Risk exposures are seen as instantaneous valuation responses to underlying factors. The primary determinant for option prices being the price of the underlying asset. Nonzero exposures, being zero at zero, are always risky and cannot be nonzero constants. Limit laws model movements in the...

A portfolio diversification index is defined as the ratio of an equivalent number of independent assets to the number of assets. The equivalence is based on either attaining the same diversification benefit or spread reduction. The diversification benefit is the difference in value of a value...

Dynamic contributions to trading are evaluated using covariations between position and price changes a horizon. Other performance measures like Sharpe ratios, Gain loss ratios, Acceptability indices and Drawdowns are also employed. Machine learning strategies based on Gaussian Process Regression...

The risk conscious investor is defined as the maximizer of a conservative valuation or dynamically a nonlinear expectation. Both the static and dynamic problems are addressed using distortions of tail probabilities or distortions of tail measures. The multivariate static problem is solved in the...

Nonlinear martingale theory is used to form lower and upper price processes straddling a martingale. The martingale return is then modeled in terms of risk charges associated with the returns on the straddling lower and upper processes. The move to physically expected returns is made via the...

The Sato process model for option prices is expanded to accomodate credit considerations by incorporating a single jump to default occuring at an independent random time with a Weibull distribution. Explicit formulas for bid and ask prices are derived. Liquidity considerations are captured by...

Two price economy principles motivate measuring risk by the cost of acquiring the opposite of the centered or pure risk position at its upper price. Asymmetry in returns leads to differences in risk charges for short and long positions. Short risk charges dominate long ones when the upper tail...

We contrast two different asset pricing models, where the pricing kernel either (i) increases in the volatility dimension, reflecting investors' aversion to volatility, or (ii) could be non-monotonic in volatility, reflecting heterogeneity in investors' beliefs. The two models yield opposite...

Modeling of uncertainty by probability errs by ignoring the uncertainty in probability. When financial valuation recognizes the uncertainty of probability the best the market may offer is a two price framework of a lower and upper valuation. The martingale theory of asset prices is then replaced...