Realized variance option and options on quadratic variation normalized to unit expectation are analyzed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk neutral densities are said to be increasing in the convex order. For Lévy...

We introduce a class of stochastic processes, which we refer to as Lyrebirds. These extend a class of stochastic processes, which have recently been coined as Peacocks, but are more commonly known as processes which are increasing in the convex order. We show how these processes arise naturally...

We extend the Rothschild and Stiglitz (1970, 1971) notion of increasing risk to families of random variables and in this way link their approach to the concept of stochastic processes which are increasing in the convex order. These processes have been introduced in seminal work by Strassen...

Cryptocurrencies have emerged in the last decade as a new asset class unlikely to disappear despite its extraordinary volatility. Futures contracts on Bitcoins were introduced in December 2017 by the Chicago Mercantile Exchange and options are being traded on crypto exchanges. Our goal in this...