Fair valuation of L\'evy-type drawdown-drawup contracts with general insured and penalty functions

TL;DR

This paper develops a mathematical framework for valuing equity-linked contracts based on drawdown and drawup events driven by spectrally negative L\'evy processes, including fair premiums and optimal stopping strategies.

Contribution

It introduces a comprehensive approach to valuing complex drawdown-drawup contracts with cancellable features using L\'evy process fluctuation theory and optimal stopping methods.

Findings

Derived formulas for fair premiums of basic contracts.

Established optimal stopping rules for cancellable contracts.

Extended valuation methods to general insured and penalty functions.

Abstract

In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative L\'evy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity $p$ until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium $p$ for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of L\'evy processes and rely on a theory of optimal stopping.