Class Notes

These are the lecture notes on stochastic processes for finance. Each chapter is available in HTML and as a PDF for printing.

Probability

Probability Basics Sets, outcomes and events, random variables and their moments, the mean and variance of linear combinations, the bounds on correlation, and probability mass functions.

Conditioning and Independence Conditional probability and independence, then conditional expectation given a random variable and given an information set, leading to filtrations as a model of information revealed over time.

Continuous Distributions The uniform, exponential, gamma, normal, chi-squared, and lognormal distributions and their moments, the inverse transform method for simulating them, the bivariate normal distribution for correlated returns, and the Student t distribution as a model of heavy tails.

Generating Functions Probability generating functions, moment generating functions, and characteristic functions, how each one turns sums of random variables into products, and how they deliver the Poisson limit and the central limit theorem.

Discrete-Time Processes

Ruin Probabilities A random walk stopped at an upper or lower barrier, with the probability of ruin and the expected duration found by solving difference equations, for fair and biased coins, and checked by simulation.

Martingales in Discrete Time Martingales and betting strategies, why doubling the stake after every loss cannot beat a fair game, and the optional stopping theorem, which recovers the ruin probabilities and expected durations with much less work.

Continuous-Time Processes

Poisson Processes Counting events in continuous time: waiting and interarrival times, the martingales of the Poisson process, superposition and thinning, compound Poisson processes, and applications to default and credit spreads and to the ruin of an insurance company.

Brownian Motion Brownian motion as the limit of a scaled random walk, its martingales and the probability of hitting a barrier, and the quadratic variation and infinite total variation of its paths.

Stochastic Calculus The Itô integral as the gains from trading, Itô processes, Itô’s formula as the chain rule of stochastic calculus, correlated Brownian motions and the product rule, and Girsanov’s theorem for changing the drift by changing the probability measure.

Asset Pricing in Continuous Time

Modeling Stock Prices Geometric Brownian motion as a model of stock prices, solved with Itô’s formula, the lognormal distribution and moments of the price, the gap between expected return and growth rate, and the dynamics of functions of the price.

Discount Factors in Continuous Time The stochastic discount factor carried from discrete to continuous time, the risk-free rate as minus its drift, risk premia as minus the covariance of returns with it, and the consumption-based model as an example.

Derivatives Pricing in Continuous Time The risk-neutral measure built from the SDF with Girsanov’s theorem, bonds, futures, and forwards, the Black-Scholes formula through a change of numeraire, and the partial differential equation that every derivative price satisfies.

Pricing with Jumps The Merton jump-diffusion and its heavy-tailed returns, Itô’s formula with jumps, changing the jump intensity, credit spreads with a default risk premium, Merton’s option formula and the volatility skew, and pricing options from characteristic functions.