The valuation of financial derivatives often assumes risk neutrality with respect to the risk-neutral martingale measure, which prevents arbitrage opportunities. However, casual traders may still incur substantial losses when trading at this risk-neutral price, especially when the price has to...

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be...

We show that the VIX Index structurally underestimates model-free implied volatility because its implementation omits extrapolation of the volatility smile in the tails. We use the asymptotic behavior of the volatility surface to construct a correction term that is model-independent and only...