Finanzas cuantitativas

Index

1. PROBABILITY THEORY

   1.1. Probability space
   1.2. Conditional probability and independence
   1.3. Measurable functions and random variables
   1.3.1. Expectation, variance, moment generating function, and characteristic function
   1.3.2. Some probability distributions
   1.3.3. Joint distribution and independence
   1.3.4. Conditional expectation
   1.3.5. Inequalities of random variables
   1.3.6. Stochastic convergence
   1.4. Exercises

2. STOCHASTIC PROCESSES

   2.1. Random walk and binomial processes
   2.2. Brownian motion
   2.3. Martingales
   2.3.1. Increments of Brownian motion
   2.4. Processes associated with Brownian motion
   2.4.1. Brownian motion with drift
   2.4.2. Geometric Brownian motion
   2.4.3. Brownian bridge
   2.4.4. Integrated Brownian motion
   2.5. Poisson process
   2.5.1. Time between occurrences
   2.5.2. Integrated Poisson process
   2.6. Quadratic variation
   2.7. Exercises

3. STOCHASTIC INTEGRAL AND ITÔ'S FORMULA

   3.1. Itô integral
   3.1.1. Properties of the Itô integral
   3.1.2. Integration with respect to a compensated Poisson process
   3.2. Stochastic differentiation: Itô's lemma
   3.2.1. Itô's lemma for Brownian motion
   3.2.2. Itô's lemma for Itô diffusions
   3.3. Multidimensional Itô's lemma
   3.4. Itô's lemma for Poisson processes
   3.5. Itô's lemma for jump-diffusion processes
   3.6. Stochastic integration by parts
   3.7. Martingale representation theorem
   3.8. Girsanov theorem
   3.9. Exercises

4. STOCHASTIC DIFFERENTIAL EQUATIONS

   4.1. Diffusion-type equations
   4.1.1. Classification of a SDE
   4.2. Direct integration method
   4.3. Exact equations - Itô coefficients
   4.4. Weakly linear SDEs and integrating factors
   4.5. Variation of parameters
   4.6. Expected value and variance of the process
   4.7. Exercises

5. STOCHASTIC DIFFERENTIAL EQUATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

   5.1. Generator of an Itô process
   5.2. Feynman-Kac theorem
   5.3. Kolmogorov equations
   5.4. Exercises

6. NUMERICAL METHODS AND PARAMETER ESTIMATION

   6.1. Order of convergence
   6.1.1. Strong convergence
   6.1.2. Weak convergence
   6.2. Simulation of the trajectories of Wt
   6.3. Simulation of the solution of a SDE
   6.3.1. Simulation from the exact solution
   6.3.2. Euler-Maruyama method
   6.3.3. Milstein approximation
   6.3.4. Predictor-corrector method
   6.4. Parameter estimation in SDEs
   6.4.1. Maximum likelihood estimation
   6.4.2. Ordinary least squares estimation
   6.4.3. Pseudo-likelihood methods

7. FINANCIAL APPLICATIONS I (DISCRETE MODELS)

   7.1. One-period market model
   7.1.1. Portfolios and arbitrage
   7.1.2. Contingent assets and valuation
   7.2. Multi-period market model
   7.2.1. Portfolios and arbitrage
   7.2.2. Contingent assets and valuation
   7.2.3. Binomial algorithm
   7.3. Multi-state market model
   7.3.1. Absence of arbitrage
   7.3.2. Equivalent martingale measures and valuation
   7.3.3. Martingale valuation
   7.3.4. Completeness
   7.3.5. Stochastic discount factor
   7.4. Exercises

8. FINANCIAL APPLICATIONS II (CONTINUOUS MODELS)

   8.1. Black-Scholes model
   8.1.1. Contingent assets and arbitrage
   8.1.2. The Black-Scholes PDE
   8.2. Risk-neutral valuation
   8.2.1. The Black-Scholes formula
   8.3. Completeness
   8.4. Parity relationships and Greeks
   8.4.1. Greeks
   8.4.2. Delta and gamma hedging
   8.5. Exercises

9. FINANCIAL APPLICATIONS III (LOCAL VOLATILITY AND STOCHASTIC VOLATILITY MODELS)

   9.1. Local volatility models
   9.1.1. Constant elasticity of variance (CEV) models
   9.1.2. Dupire model
   9.2. Stochastic volatility models
   9.2.1. Hull-White model
   9.2.2. Stochastic Alpha-Beta-Rho (SABR) model
   9.2.3. Scott model
   9.2.4. Stein and Stein model
   9.2.5. Heston model
   9.2.6. Some observations on OU and CIR processes
   9.3. Exercises

10. FINANCIAL APPLICATIONS IV (JUMP DIFFUSION MODEL)

10.1. Merton model
10.1.1. Compound Poisson process
10.1.2. Derivation of the model
10.1.3. Distribution of returns
10.1.4. Valuation of derivatives

11. FINANCIAL APPLICATIONS V (NON-LINEAR PROBLEMS AND BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS)

11.1. Non-linear second-order partial differential equations
11.1.1. Parabolic PDEs
11.1.2. Comparison principle and uniqueness
11.1.3. Super-solutions, sub-solutions, and comparison principle
11.2. Stochastic optimal control and the Hamilton-Jacobi-Bellman PDE
11.2.1. Bellman principle
11.2.2. The HJB PDE
11.2.3. Verification theorem
11.3. Viscous solutions
11.3.1. Feynman-Kac theorem in the viscous sense
11.4. Some non-linear problems in finance
11.4.1. Model with uncertain volatility
11.4.2. Model with transaction costs
11.4.3. Differential interest rates for investment or lending
11.4.4. Passport options

BIBLIOGRAPHY