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# L19 - Lecture 19 19.1 Interest Rate Models Basic elements...

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Lecture 19 Interest Rate Models 19.1 Basic elements in a bond market Consider a market money account B = { B t } with B 0 = 1, B t = exp( R t 0 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, τ ) } , or { Y ( t, τ ) } , or { f ( t, τ ) } ) 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 fixed-income asset or derivative may have its own maturity time. Let (Ω , 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 0 | 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 (continuous-time) Fundamental Theorem of Asset Pricing, which we have not presented. For 0 t < τ T , three equivalent term structures can be defined as follows: Zero-coupon bond : Let B ( t, τ ) be the time- t price of a bond with maturity τ (called τ -bond) and the par value B ( τ, τ ) = 1. The risk-neutral valuation principle implies

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L19 - Lecture 19 19.1 Interest Rate Models Basic elements...

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