Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a randomwalk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.

There have been many results on the asymptotics of the Painlevé transcendents in recent years, but the asymptotics of the fourth Painlevé transcendent has not been studied much. In this note, we study the general fourth Painlevé equation and develop an asymptotic representation of a group of its solutions as the independent variable approaches infinity along a straight line.

In this short paper, we study the asymptotics for the price of call options for very large strikes and put options for very small strikes. The stock price is assumed to follow the Black–Scholes models. We analyze European, Asian, American, Parisian and perpetual options and conclude that the tail asymptotics for these option types fall into four scenarios.