The American put with finite-time maturity and stochastic interest rate

Cheng Cai; Tiziano De Angelis; Jan Palczewski

doi:10.1111/mafi.12361

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arXiv · 2104.08502

The American put with finite-time maturity and stochastic interest rate

Cheng Cai·Tiziano De Angelis·Jan Palczewski

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Abstract

In this paper we study pricing of American put options on the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a $C^1$ function of the initial time, interest rate and stock price. By means of Ito calculus we rigorously derive the option value's early exercise premium formula and the associated hedging portfolio. We prove the existence of an optimal exercise boundary splitting the state space into continuation and stopping region. The boundary has a parametrisation as a jointly continuous function of time and stock price, and it is the unique solution to an integral equation which we compute numerically. Our results hold for a large class of interest rate models including CIR and Vasicek models. We show a numerical study of the option price and the optimal exercise boundary for Vasicek model.

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Cheng Cai, Tiziano De Angelis, Jan Palczewski. 2021-04-17. The American put with finite-time maturity and stochastic interest rate. https://doi.org/10.1111/mafi.12361

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