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

Cheng Cai, Tiziano De Angelis, Jan Palczewski

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.

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

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 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.

Paper Structure

This paper contains 21 sections, 21 theorems, 209 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results
  4. Optimal stopping boundary
  5. Smoothness of the value function
  6. Continuity of the stopping boundary and Dynkin's formula
  7. Early exercise premium
  8. Integral equation for the stopping boundary
  9. Hedging portfolio
  10. Numerical analysis
  11. Computational approach
  12. Sensitivity analysis
  13. Monotonicity and Lipschitz continuity of the option value
  14. Properties of the free boundary
  15. Continuous differentiability of the option value
  16. ...and 6 more sections

Key Result

Proposition 3.1

The value function $v$ is finite for all $(t,r,x)\in{\mathcal{O}}$ and it satisfies the following conditions:

Figures (5)

  • Figure 1: Stopping boundary surface $b(t,r)$.
  • Figure 2: Sections of the value function $v(t,r,x)$ through the point $(0, 0.0478, 82.11)$. The dashed line displays the payoff $(K-x)^+$.
  • Figure 3: The $r$-sections of the stopping boundary (left panel) and the value function (right panel) for the mean-reversion parameter $\kappa \in \{0.1, 0.55, 1\}$.
  • Figure 4: The $r$ and $t$-sections of the stopping boundary for the correlation coefficient $\rho \in \{-0.8, 0, 0.8\}$.
  • Figure 5: Effect of the volatility of the stock price $\sigma$. Panels (a) and (b) display the $r$ and $t$-sections of the stopping boundary $b(t,r)$ and Panels (c) and (d) show the $r$ and $x$-sections of the value function $v$ for $\sigma \in \{0.1, 0.3, 0.5\}$.

Theorems & Definitions (49)

  • Remark 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Theorem 3.5
  • proof
  • Proposition 3.7
  • ...and 39 more