An Elementary Introduction to Stochastic Interest Rate Modeling
This textbook is written as an accessible introduction to interest rate modeling and related derivatives, which have become increasingly important subjects of interest in financial mathematics. The models considered range from standard short rate to forward rate models and include more advanced topics such as the BGM model and an approach to its calibration. An elementary treatment of the pricing of caps and swaptions under forward measures is also provided, with a focus on explicit calculations and a step-by-step introduction of concepts. Each chapter is accompanied with exercises and their complete solutions, making this book suitable for advanced undergraduate or beginning graduate-level students.
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A Review of Stochastic Calculus
A Review of BlackScholes Pricing
Short Term Interest Rate Models
Forward Rate Modeling
The HeathJarrowMorton HJM Model
The Forward Measure and Derivative Pricing
Curve Fitting and a Two Factor Model
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absence of arbitrage adBt arbitrage price BGM model Black-Scholes formula bond price Bt)teR+ caplets Compute the price conditional expectation defined deﬁnition denote derivatives deterministic function diﬀerent diﬀerential dynamics Exercises Exercise forward curves forward measure forward rate f(t forward rate f(t,T,S forward rate process Gaussian random variable geometric Brownian motion Girsanov theorem Graph hence Hull-White model instantaneous forward rate intentionally left blank interest rate models interest rate process Ito formula Ito's calculus left blank Chapter LIBOR market logP(t,T market data Markov property martingale Nelson-Siegel option payoﬀ portfolio price P(t,T Proposition Protter Recall Relation rsds rt)dt rudu short rate models short rate process short term interest solution standard Brownian motion stochastic differential equation stochastic integral stochastic process swap rate swaption T)dt term interest rate Tk+1 variance Vasicek model volatilities Xsds