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1. introduction 

2. arbitrage, martingales and numerical methods 

2.1 arbitrage and martingales 

2.1.1 basic setup 

2.1.2 equivalent martingale measure 

2.1.3 change of numeraire theorem 

2.1.4 girsanov's theorem and it6's lemma 

2.1.5 application: black-scholes model 

2.1.6 application: foreign-exchange options 

2.2 numerical methods 

2.2.1 derivation of black-scholes partial differentialequation 

2.2.2 feynman-kac formula 

2.2.3 numerical solution of pde's 

2.2.4 monte carlo simulation 

2.2.5 numerical integration 

part ¢¡. spot and forward rate models 

3. spot and forward rate models 

3.1 vasicek methodology 

3.1.1 spot interest rate 

3.1.2 partial differential equation 

3.1.3 calculating prices 

3.1.4 example: ho-lee model 

3.2 heath-jarrow-morton methodology 

3.2.1 forward rates 

3.2.2 equivalent martingale measure 

3.2.3 calculating prices 

3.2.4 example: ho-lee model 

3.3 equivalence of the methodologies 

4. fundamental solutions and the forward-risk-adjustedmeasure 

4.1 forward-risk-adjusted measure 

4.2 fundamental solutions 

4.3 obtaining fundamental solutions 

4.4 example: ho-lee model 

4.4.1 radon-nikodym derivative 

4.4.2 fundamental solutions 

4.5 fundamental solutions for normal models 

5. the hull-white model 

5.1 spot rate process 

5.1.1 partial differential equation 

5.1.2 transformation of variables 

5.2 analytical formulae 

5.2.1 fundamental solutions 

5.2.2 option prices 

5.2.3 prices for other instruments 

5.3 implementation of the model 

5.3.1 fitting the model to the initial term-structure 

5.3.2 transformation of variables 

5.3.3 trinomial tree 

5.4 performance of the algorithm 

5.5 appendix 

¡­¡­ 

part ¢¢. market rate models 

references 

index