This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 19 Interest Rate Models 19.1 Basic elements in a bond market Consider a market money account B = { B t } with B = 1, B t = exp( R t r u du ), t [0 ,T ], where the short rate { r t } is assumed to be a stochastic process. The bank account B and related term structures ( { B ( t, ) } , or { Y ( t, ) } , or { f ( t, ) } ) in a bond market serve as underlying assets similar to stocks in a stock market. We will introduce several interest rate models and use them to price various derivatives defined on the underlying assets. Note that each fixedincome asset or derivative may have its own maturity time. Let ( , F ,P ) be a probability space equipped with a filtration {F t } t [0 ,T ] . Assume { r t } is an adapted process with P R T  r t  dt < = 1. The filtration {F t } t [0 ,T ] is usually generated by a multidimensional Brownian motion (called shocks or factors) as a source of uncertainty. Since our main purpose is derivative pricing, we assume the existence of a risk neutral measure Q , equivalent to P following the Girsanov transformation (detail skipped), under which the discounted processes of all underlying assets are martingales. This will rule out arbitrage opportunities, according to the (continuoustime) Fundamental Theorem of Asset Pricing, which we have not presented. For t < T , three equivalent term structures can be defined as follows: Zerocoupon bond : Let B ( t, ) be the time t price of a bond with maturity (called bond) and the par value...
View Full
Document
 Spring '08
 Staff

Click to edit the document details