Notes on Quantitative Analysis in Finance.

he purpose of these notes is to present the subject of mathematical finance with natural degree of rigor and with clear motivation. The reader may judge the success of such endeavor. I welcome constructive critical notes sent to kaslanidi@yahoo.com.

I maintain the “brutal force” approach to every problem and use only generally applicable techniques. Presentation of such techniques is preferred over enumeration of results. However, such priorities are not the only reason for omission of some results. This is a developing project. I continuously work to make these notes better.

    Sincerely,
       Konstantin Aslanidi,
       September, 2007.

P.S. If some of the math does not display correctly then hit the "Refresh" button.

Table of Contents. A. Notation. I. Basic math. 1. Conditional probability. A. Definition of conditional probability. B. A bomb on a plane. C. Dealing a pair in the "hold' em" poker. D. Monty-Hall problem. E. Two envelopes. F. Two headed coin drawn from a bin of fair coins. G. Randomly unfair coin. H. Recursive Bayesian calculation. I. Birthday problem. J. Backward induction. K. Conditional expectation. Filtration. Flow of information. Stopping time. 2. Normal distribution. A. Definition of normal variable. B. Linear transformation of random variables. C. Multivariate normal distribution. Choleski decomposition. D. Calculus of normal variables. E. Central limit theorem (CLT). 3. Brownian motion. A. Definition of standard Brownian motion. B. Brownian motion passing through gates. C. Reflection principle. D. Brownian motion hitting a barrier. 4. Topics in stochastic analysis. A. Ito integral. B. Ito calculus. a. Example: exponential of a stochastic process. b. Example: integral of t_dW. c. Example: integral of W_dW. d. Example: integral of W_dt. C. Change of measure. a. Definition of the change of measure. b. Most common application of the change of measure. c. Transformation of SDE under the change of measure. D. Girsanov's theorem. a. Change of measure-based verification of Girsanov's theorem statement. b. Direct proof of Girsanov's theorem. E. Forward Kolmogorov's equation. F. Backward Kolmogorov's equation. a. Several dimensions. G. Bellman equation. 5. Poisson process. A. Definition of the Poisson process. B. Distribution of the Poisson process. C. Poisson stopping time. D. Arrival of k-th Poisson jump. Gamma distribution. E. Cox process. 6. Kolmogorov's equations in general setting. A. Example: backward Kolmogorov equation for diffusion. B. Example: backward Kolmogorov equation for Ito process with jump. 7. Hamilton-Jacobi Equations. A. Characteristics. B. Hamilton equations. C. Lagrangian. D. Connection between Hamiltonian and Lagrangian. E. Lagrangian for the heat equation. 8. Convex Analysis. A. Basic concepts of convex analysis. a. Affine sets and hyperplanes. b. Convex sets and cones. c. Convex functions and epigraphs. B. Separation theorems. C. Convex duality. D. Support function. E. Infimal convolution. F. System of inequalities. G. Recession cones and unboundedness. H. Directional derivatives and subgradients. I. The minimum of a convex function. J. Ordinary convex programs and Lagrange multipliers (Kuhn-Tucker coefficients). K. Lagrangian of the ordinary convex problem. L. Bifunctions and duality of ordinary convex programs. 9. Real Variable. A. Operations on sets and logical statements. B. Fundamental inequalities. C. Various types of convergence. a. Uniform convergence and convergence almost surely. Yegorov's theorem. b. Convergence in probability. c. Infinitely often events. Borell-Cantelli's lemma. d. Integration and convergence. e. Convergence in Lp. D. Measure introduced by random variable. E. Partial ordering and the maximal principle. F. Hahn-Banach theorem. 10. Fundamental solutions. Calculus of distributions. A. Space of distributions. B. Fundamental solution. C. Fundamental solution for the heat equation. II. Pricing and Hedging. 1. Basics of derivative pricing I. A. Why Ito process? B. Existence of the risk neutral measure via Girsanov's theorem. C. Existence of the risk neutral measure via backward Kolmogorov's equation. Delta hedging. a. An economy with one risky asset. b. An economy with two risky assets. D. Optimal utility function based interpretation of delta hedging. 2. Change of numeraire. A. Definition of the change of numeraire. B. Useful calculation. C. Transformation of SDE based on change of measure results. D. Transformation of SDE in a two asset situation. E. Transformation of SDE based on term matching. F. Invariant representation for the drift modification. G. Transformation of SDE based on delta hedging. H. Example. Change of numeraire in the Black-Scholes economy. I. Other ways to look at the change of numeraire. 3. Basics of derivative pricing II. A. Option pricing formula for an economy with stochastic riskless rate. B. T-forward measure. C. HJM. 4. Market model. A. Forward LIBOR. B. LIBOR market model. C. Swap rate. D. Swap measure. 5. Topics in Currency Exchange. A. Change of numeraire in the currency markets. B. Invariant form of the SDE transformation formula. C. Delta hedging in the currency markets. D. Example: forward contract to purchase a foreign stock for domestic currency. E. Example: forward currency exchange contract. F. Example: quanto forward contract. G. Example: quanto caplet. H. Example: quanto fixed-for-floating swap. 6. Credit risk. A. Delta hedging in a situation of predictable jump I. B. Delta hedging in a situation of predictable jump II. C. Backward Kolmogorov's equation for a jump diffusion. D. Risk neutral valuation in the predictable jump size situation. E. Examples of credit derivative pricing. a. Credit Default Swap. b. At-the-money CDS coupon. c. Option on CDS. d. Basket Credit derivative. F. Credit correlation. a. Generic Copula. b. Gaussian copula. c. Example: two dimensional Gaussian copula. d. Simplistic Gaussian copula. G. Valuation of CDO tranches. a. Definitions of CDO contract. b. Present values of CDO tranches. c. Distribution of defaulted notional of CDO tranches. 7. Incomplete markets. A. Single time period discrete price incomplete market. a. Existence of pricing vector. b. Uniqueness of pricing vector. c. Bid and ask. B. Coherent measure. C. Incomplete market with multiple participants. D. Example: uncertain local volatility. III. Explicit techniques. 1. Black-Scholes formula. A. No drift calculation. B. Calculation with drift. 2. Change of variables for Kolmogorov equation. A. One dimensional Black equation. B. Two dimensional Black equation. 3. Mean reverting equation. 4. Affine SDE. A. Ricatti equation. B. Evaluation of option price. C. Laplace transform. 5. Heston equations. A. Affine equation approach to integration of the Heston equations. B. PDE approach to integration of the Heston equations. 6. Displaced Heston equations. A. Analytical tractability of the displaced Heston equations. B. Displaced Heston equations with the term structure. a. Parameter averaging. b. Parameter averaging applied to the displaced diffusion. 7. Stochastic volatility. A. Recovering implied distribution. B. Local volatility. C. Gyongy's lemma. a. Multidimensional Gyongy's lemma. D. Static hedging of European claim. a. Example: European put-call parity. b. Example: Log contract. E. Variance swap pricing. a. Variance swap pricing for drifting price process. b. Volatility smile formula for fair variance. 8. Markovian projection. A. Markovian projection on displaced diffusion. a. Example of Markovian pojection of a separable process on a displaced diffusion. B. Markovian projection on Heston model. IV. Data Analysis. 1. Topics in Time Series. A. Time series forecasting. B. Updating a linear forecast. C. Kalman filter I. a. Kalman filter computation at t=1. b. Kalman filter computation for general t. c. Calibration of parameters with Kalman filter. D. Kalman filter II. a. The general Kalman filter problem. b. The general Kalman filter solution. c. Convolution of normal distributions. d. Kalman filter calculation for a linear model. e. Kalman filter in the non-linear situation. f. Unscented transformation. i. Unscented approximation of the mean. ii. Unscented approximation of covariance matrix. E. Simultaneous equations. a. Simple linear reduction. b. Simultaneous equations bias. c. Two stage least squares procedure for simultaneous equations. d. General note of applicability. 2. Topics of classical statistics. A. Basic concepts and common notation of classical statistics. B. Chi squared distribution. C. Student's t-distribution. D. Classical estimation theory. a. Sufficient statistics. b. Sufficient statistic for normal sample. c. Maximal likelihood estimation (MLE). d. Asymptotic consistency of the MLE. Fisher's information number. e. Asymptotic efficiency of the MLE. Cramer-Rao low bound. E. Pattern recognition. a. Decision rule based on a loss function. b. Hypothesis testing problem. c. Neyman-Pearson Lemma. 3. Topics in Bayesian statistics. A. Basic idea of bayesian analysis. B. Estimating the mean of a normal distribution with known variance. C. Estimating unknown parameters of a normal distribution. a. The structure of the model with unknown parameters. b. Recursive formula for the posterior joint distribution. c. Marginal distribution of the mean. d. Marginal distribution of the precision. D. Hierarchical analysis of a normal model with known variance. a. The joint posterior distribution of the mean and the hyperparameters. b. The posterior distribution of the mean conditionally on the hyperparameters. c. The marginal posterior distribution of the hyperparameters. i. Distribution of mu conditionally on gamma. ii. Posterior distribution of gamma. iii. Prior distribution for gamma. V. Implementation tools. 1. Finite differences. A. Finite difference basics. a. Definitions and the main convergence theorem. b. Approximations of basic operators. c. Stability of the general evolution equation. d. Spectral analysis of the finite difference Laplacian. B. One dimensional heat equation. a. The finite difference schemes for heat equation. b. Stability of one-dim heat equation schemes. c. Remark on stability of financial problems. d. Lagrangian coordinate technique. e. Factorization procedure for the heat equation. C. Two dimensional heat equation. a. The Peaceman-Rackford (alternating directions) scheme. b. Stability of Peaceman-Rackford. D. General techniques for reduction of dimensionality. a. Stabilization. b. Predictor-corrector. c. Separation of variables for the Krank-Nicolson scheme. E. Time dependent case. 2. Gauss-Hermite Integration. 3. Asymptotic expansions. A. Integrating exponential with a large parameter. 4. Generation of random samples. A. Uniform [0,1] random variable. B. Inverting cumulative distribution. C. The accept/reject procedure. D. Normal distribution. Box-Miller procedure. E. Gibbs sampler. 5. Monte-Carlo. A. Acceleration of convergence. a. Antithetic variables. b. Control variate. c. Importance sampling. d. Stratified sampling. B. Monte-Carlo with optimal control. a. Longstaff-Schwartz technique. i. Normal Equations technique. C. Calculation of sensitivities. a. Pathwise differentiation. b. Calculation of sensitivities for Monte-Carlo with optimal control. VI. Applications. 1. CDFX model. A. Definition of CDFX model. B. The martingale normalization (CDFX). C. Fourier transform (CDFX). D. Calculation of Fourier transform (CDFX). E. Calculation of Premium Leg of CDS. F. Calculation of the protection leg of the CDS. 2. Lattice approach to derivative pricing. A. Basic definitions of the lattice approach. B. Markov Generator for a smooth process. Dyson decomposition. Kolmogorov equations. C. Path-Integral representation of Markov generator. D. Markov generator for a piecewise smooth process. a. Change of measure on lattice between jumps. b. Change of measure at jump points. c. Connection to generic change of numeraire. d. Monotonic processes and martingales. e. Self financing strategies. Arbitrage. Fundamental theorem of finance.