1 Introduction 2 The Binomial Model 3 A More General One Period Model 4 Stochastic Integrals 5 Differential Equations 6 Portfolio Dynamics 7 Arbitrage Pricing 8 Completeness and Hedging 9 Parity Relations and Delta Hedging 10 The Martingale Approach to Arbitrage Theory (For advanced readers) 11 The Mathematics of the Martingale Approach (For advanced readers) 12 Black-Scholes from a Martingale Point of View (For advanced readers) 13 Multidimensional Models: Classical Approach 14 Multidimensional Approach: Martingale Approach (For advanced readers) 15 Incomplete Markets 16 Dividends 17 Currency Derivatives 18 Barrier Options 19 Stochastic Optimal Control 20 Bonds and Interest Rates 21 Short Rate Models 22 Martingale Models for the Short Rate 23 Forward Rate Models 24 Change of Numeraire (For advanced readers) 25 LIBOR and Swap Market Models 26 Forwards and Futures Appendix A Measure and Integration (For advanced readers) Appendix B Probability Theory (For advanced readers) Appendix C Martingales and Stopping Times (For advanced readers) References Index

http://www.gbv.de/dms/hebis-mainz/toc/071897542.pdf

Arbitrage Theory in Continuous Time

http://www.cirano.qc.ca/pdf/publication/2002s-58.pdf

Alternative models for stock price dynamics

A self-contained theory is presented for pricing and hedging LIBOR and swap derivatives by arbitrage. Appropriate payoff homogeneity and mea- surability conditions are identified which guarantee that a given payoff can be attained by a self-financing trading strategy. LIBOR and swap derivatives satisfy this condition, implying they can be priced and hedged with a finite number of zero-coupon bonds, even when there is no instantaneous saving bond. No- tion of locally arbitrage-free price system is introduced and equivalent criteria established. Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage- free model with such volatility specification. The construction is explicit for the lognormal LIBOR and swap "market models", the former following Musiela and Rutkowski (1995). Primary examples of LIBOR and swap derivatives are discussed and appropriate practical models suggested for each.

/pdf/libor-and-swap-market-models-and-measures-4vdeqzr9x4.pdf

LIBOR and swap market models and measures

https://www.nber.org/system/files/working_papers/w2341/w2341.pdf

The Term Structure of Interest Rates

We derive a unified model that gives closed form solutions for caps and floors written on interest rates as well as puts and calls written on zero-coupon bonds. The crucial assumption is that simple interest rates over a fixed finite period that matches the contract, which we want to price, are log-normally distributed. Moreover, this assumption is shown to be consistent with the Heath-Jarrow-Morton model for a specific choice of volatility. CLOSED FORM SOLUTIONS FOR interest rate derivatives, in particular caps, floors, and bond options, have been obtained by a number of authors for Markovian term structure models with normally distributed interest rates or alternatively log-normally distributed bond prices (see, for example, Jamshidian (1989, 1991a); Heath, Jarrow, and Morton (1992); Brace and Musiela (1994); Geman, El Karoui, and Rochet (1995)). These models support Black-Scholes type formulas most frequently used by practitioners for pricing bond options and swaptions. Unfortunately, these models imply negative interest rates with positive probabilities, and hence they are not arbitrage free in an economy with opportunities for riskless and costless storage of money. Briys, Crouhy, and Schobel (1991) apply the Gaussian framework to derive closed form solutions for caps, floors, and European zero-coupon bond options. To exclude the influence of negative forward rates on the pricing of zero-coupon bond options, they introduce an additional boundary condition. As shown by Rady and Sandmann (1994) these pricing formulas are only supported by a term structure model with an absorbing boundary for the forward rate at zero, where the absorbing probability is not negligible, which for a term structure model is a quite problematic assumption.

http://janroman.dhis.org/finance/Interest%20Rates/Caps%20Floors.pdf

Closed Form Solutions for Term Structure Derivatives with Log‐Normal Interest Rates

A class of term structure models with volatility of lognormal type is analyzed in the general HJM framework. The corresponding market forward rates do not explode, and are positive and mean reverting. Pricing of caps and floors is consistent with the Black formulas used in the market. Swaptions are priced with closed formulas that reduce (with an extra assumption) to exactly the Black swaption formulas when yield and volatility are flat. A two-factor version of the model is calibrated to the U.K. market price of caps and swaptions and to the historically estimated correlation between the forward rates.

/pdf/the-market-model-of-interest-rate-dynamics-3o1qbztm3q.pdf

Dariusz G¸atarek

Papers

The Market Model of Interest Rate Dynamics