In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first...

We propose a novel credit default model that takes into account the impact of macroeconomic information and contagion effect on the defaults of obligors. We use a set-valued Markov chain to model the default process, which is the set of all defaulted obligors in the group. We obtain analytic...

In the Bitcoin futures markets, the dominating contracts are inverse contracts. Unlike standard futures, Bitcoin inverse futures have a non-linear payoff structure, are settled in Bitcoin instead of the fiat currency, and require Bitcoins to be deposited into the margin account during trading.We...