Hedging via Perpetual Derivatives: Trinomial Option Pricing and Implied Parameter Surface Analysis

TL;DR

This paper develops a trinomial tree model for pricing European options in a market with stocks and perpetual derivatives, analyzing implied parameters and calibration to empirical data.

Contribution

It introduces a general recombining trinomial model incorporating perpetual derivatives and derives relationships between real-world and risk-neutral parameters.

Findings

Implied parameter surfaces for real-world parameters are generated from empirical data.

Calibration methods are discussed for both arithmetic and logarithmic return cases.

The model ensures market completeness with a two risky assets and a European option.

Abstract

We introduce a fairly general, recombining trinomial tree model in the natural world. Market-completeness is ensured by considering a market consisting of two risky assets, a riskless asset, and a European option. The two risky assets consist of a stock and a perpetual derivative of that stock. The option has the stock and its derivative as its underlying. Using a replicating portfolio, we develop prices for European options and generate the unique relationships between the risk-neutral and real-world parameters of the model. We discuss calibration of the model to empirical data in the cases in which the risky asset returns are treated as either arithmetic or logarithmic. From historical price and call option data for select large cap stocks, we develop implied parameter surfaces for the real-world parameters in the model.