1. Fundamental Theorem

• The Theorem

• Martingale Representation

• Synthetic Probability Measure

• Martingale Property

• Pricing Kernel

2. Girsanov Theorem

3. Risk Premium in the Fundamental Theorem

4. Arbitrage-Free Dynamics

[Appendix]  

• Measure Change and Girsanov Theorem [PDF] (Ch 12, 16, 17)

1. Monte Carlo

2. Pricing American Option

3. Tree Calibration 1: Stock Price

4. Tree Calibration 2: Interest Rate

5. Numerical Solution for PDE: Finite Difference Method

[Appendix]  

• Ito Integral and Monte Carlo Simulation [PDF]

• American Option Pricing [PDF]

• Calibrating Geometric Brownian Process [PDF] (Ch. 8, 13)

• Calibration of Short Rate Process [PDF] (Ch. 11, 13)

• Lognormal Distribution [PDF]

• Theoretical Weakness of Lognormality [PDF]

[Appendix]  

• Brownian Motion [PDF]

• Ito Integral and Lognormal Representation [PDF]

• Feynman-Kac Theorem and Others [PDF]

• Kolmogorov Backward Equation [PDF]

• Stochastic Multiplication of PDE Solutions [PDF]

• Derivation of Black-Scholes Formula [PDF]

1. Foreign Exchange Rate Process

2. Foreign Stock Process

3. Quantos

• Quanto Options

• Quanto Forwards

[Appendix]  

• Moments of dS, dW and Multivariate Ito's Lemma [PDF]

• Trivial Maths [PDF]

[Appendix]  

• Measure Change and Girsanov Theorem [PDF] (Ch 12, 16, 17)

• Reflection Principle 1 [PDF]

• Reflection Principle 2 [PDF]

• Reflection Principle with Drift [PDF]

• Reflection Principle for Stock Price [PDF]

• Down Barrier Options [PDF]