Pricing and Hedging Long-Term Options

TL;DR

This paper analyzes the behavior and hedging strategies of long-term options, revealing conditions for exponential price decay or growth and deriving simplified expressions for Greeks using advanced mathematical techniques.

Contribution

It introduces a novel combination of martingale extraction and Malliavin calculus to characterize long-term option prices and sensitivities.

Findings

Option prices exhibit exponential decay or growth under certain conditions.

Ratios of Greeks to option prices simplify in the long-term.

The methods provide new insights into long-term hedging strategies.

Abstract

In this article, we investigate the behavior of long-term options. In many cases, option prices follow an exponential decay (or growth) rate for further maturity dates. We determine under what conditions option prices are characterized by this property. To see this, we use the martingale extraction method through which a pricing operator is transformed into a semigroup operator, which is easier to address. We also explore notions of hedging long-term options. Hedging is an attempt to reduce market risks, and we investigate the price sensitivities (Greeks) with respect to such risks, which are typically repre- sented by variations in the underlying process of an option. We combine the Malliavin calculus with the martingale extraction method to analyze Greeks. We see that the ratios between Greeks and the option price are expressed in a simple form in the long term.