Interest Rate Problems: Implied Volatility of Options on Bonds and Forward Rates, and Optimal Times to Buy and Sell a Home

Abstract

In this Thesis, we examine problems in financial mathematics whose characteristics are significantly influenced by the dynamics of the interest rate.In the first part of the Thesis, we derive an explicit asymptotic approximation for the implied volatilities of Call options written on bonds and forward rates assuming the instantaneous short-rates of interest are described by Affine Term-Structure (ATS) models and Quadratic Term-Structure (QTS) models, respectively. For specific short-rate models, we perform numerical experiments in order to gauge the accuracy of our approximation. In the second part of the Thesis, we derive the optimal stopping times to buy and sell a home. We begin by assuming that home prices are set by a representative home-buyer, who can afford to pay only a fixed cash-flow per unit time for housing. The cash-flow is a fraction of their salary, which grows at a rate that is proportional to the risk-free rate of interest. The mortgage rate paid by the home-buyer is fixed at the time of purchase and equal to the risk-free rate of interest plus a positive constant. In this setting, we consider an investor who wishes to buy and then sell a home in order to maximize his discounted expected profit. This leads to a nested optimal stopping problem, which can be solved using nonnegative concave majorant approach. Additionally, we provide a detailed analytic and numerical study of the case in which the risk-free rate of interest is modeled by a Cox-Ingersoll-Ross (CIR) process.