The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 7

Interest Rate Hedging

Bond prices and interest rates are not fixed. We certainly might decide to use models with stochastic interest rates. Up to now, only stock prices (and currency exchange rates) have been modeled as stochastic processes. Before we do this, however, we may ask some simple questions about options such as:

• Given that we have a model of stock options and that the value of stock options is driven more by stock volatility than rates volatility, what might the interest rate hedging strategy look like?
• What is the recommended rates hedge implied by the model we have?

Strictly speaking, the model we have already developed for European options takes the volatility of the martingale—the ratio of stock forward to strike present value (PV)—as an input. This implies that stochastic rates, interest rates, and stock loan rates together with stochastic stock price have already been modeled. We merely need to reinterpret the volatility input. Now, the correct hedge is a combination of stock forward and a zero-coupon bond of maturity equal to option expiration. The following analysis shows how much zero-coupon bond to hedge with together with stock forward.

7.1. EULER’S RELATION

Given a call option trading at some price with a particular strike and stock price, consider the price of a related option with twice the strike price and the “underlying” is two shares of stock. This means that instead of one option we essentially have an inseparable package of two options. ...