Option traders use a heuristically derived pricing formula which they adapt by fudging and changing the tails and skewness by varying one parameter, the standard deviation of a Gaussian. Such formula is popularly called "Black-Scholes-Merton" owing to an attributed eponymous discovery (though...

Closed form formulae for European barrier options are well known from the literature. This is not the case for American barrier options, for which no closed form formulae have been published. One has therefore had to resort to numerical methods. Using lattice models like a binomial or a...

In this article we show a simple but important relationship between the put-call transformation and the put-call symmetry as well as extend the relationship to also hold for single and double barrier options. These new barrier transformations give new insight in barrier option valuation. Using...

Embedded options in the worlds physical monies, both coin and paper, are introduced. The option value for base metal coins is presented. The various strategies for redemption by the owner and the prevention of redemption by the issuer (central banks) are discussed. The market values of gold...

In the world of modern financial theory, portfolio construction has traditionally operated under at least one of two central assumptions: the constraints are derived from a utility function and/or the multivariate probability distribution of the underlying asset returns is fully known. In...

Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to both sample and population size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially...

We provide a mathematical definition of fragility and antifragility as negative or positive sensitivity to a semi-measure of dispersion and volatility (a variant of negative or positive "vega") and examine the link to nonlinear effects. We integrate model error (and biases) into the fragile or...

Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to sample size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of...