Put Options with Linear Investment for Hull-White Interest Rates

Andrzej Korzeniowski,Niloofar Ghorbani

doi:10.4236/jmf.2021.111007

Andrzej Korzeniowski, Niloofar Ghorbani

Open Access

https://doi.org/10.4236/jmf.2021.111007

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Abstract

We derive a Put Option price associated with selling strategy of the underlying security in a random interest rate environment. This extends Put Option pricing under linear investment strategy from the Black-Scholes setting to Hull-White stochastic interest rate model. As an application, Call Option price for the linear investment strategy in the Hull-White model is established. Our results address recent emergence of developing dynamic investment strategies for the purpose of reducing the investor risk exposure associated with European-type options.

Highlights

A steady growth of financial derivatives market over the past decades led to various generalizations of the classical Black-Scholes model
This extends Put Option pricing under linear investment strategy from the Black-Scholes setting to Hull-White stochastic interest rate model
We will use the above formula for the stock price that satisfies a stochastic differential equation (SDE) with drift depending on the random interest rate, whose SDE follows the Hull-White model

Summary

Introduction

A steady growth of financial derivatives market over the past decades led to various generalizations of the classical Black-Scholes model. Ghorbani and Korzeniowski [5] obtained the Call Option price with investment strategy for the Cox-Ingersoll-Ross (CIR) interest rates model via path-integral representation based on n-dimensional Ornstein-Uhlenbeck process. Unlike in the classical Black-Scholes model where the investor buys options and has no position in the underlying stock throughout the option time horizon, the dynamic investment strategy requires the investor to continuously trade the stock, whereby lowering the investor risk which is manifested by the lower option price. This paper is concerned with Put Option hedging by linear investment strategy under the Hull-White stochastic interest rates model. Β is the minimum value of the stock investment proportion It was found in [2] that the Put Option value VT based on the linear investment with parameters α , β , strike price K reads as follows:. We will use the above formula for the stock price that satisfies a stochastic differential equation (SDE) with drift depending on the random interest rate, whose SDE follows the Hull-White model

The Market Model

Hull-White under T-Forward Measure

Put Option Price

Application to Call Option

Conclusion