# slides

**Disclaimer: **The slides linked to this page were created as "expanded bullet points" for parts of the book *Stochastic Methods in Asset Pricing, MIT Press (2017)*, and are not meant to be used as a stand-alone exposition. All slides sets were generated by LaTeX with the Beamer class and are intended for PDF-readers that can process overlays (e.g., Acrobat Reader, or Apple Preview in *presentation mode*).

**The One Period Version of the Binomial Asset Pricing Model (sections 1.1, 1.2)**

Keywords: Arrow-Debreu securities, risk-neutral probabilities, state-price density, stochastic discount factors

**Probability Spaces and Related Structures (section 1.3)**

Keywords: sample space, random events, sigma-fields, sigma-fields generated by collections of sets, measures, probability measures, countable additivity, point-mass measure

**The Coin Toss Space and the Random Walk (section 1.4)**

Keywords: event trees, binomial trees, probability measures on the coin toss space, purely atomic measures, non-atomic measures, completion of the sigma-field, first Borel-Cantelli lemma

**Borel sigma-Fields and Lebesgue Measure (section 1.5)**

Keywords: open sets, topology, convergence, Borel sigma-fields, Lebesgue measure, singular measures, the Cantor set, sets of measure zero, non-atomic measures

**Random Variables and Expectation (sections 2.1, 2.2, 2.3)**

Keywords: random variables, distribution laws, measurable functions, Lebesgue theory of integration

**Applications of Lebesgue's Theory of Integration (sections 2.3, 2.4, 2.5, 2.6)**

Keywords: L^p spaces, Cauchy-Schwartz inequality, Markov's inequality, Chebyshev's inequality, monotone convergence theorem, Fatou's lemma, the dominated convergence theorem, Fubini's theorem, change of variables in Lebesgue's integral, absolute continuity and equivalence of measures, the inverse of a distribution function

**Conditioning and Independence. Gaussian Distribution Laws (sections 3.3, 3.4)**

Keywords: conditional probability, independence between random variables and sigma-fields, covariance and correlation, conditional expectations, multivariate Gaussian distribution, Cholesky factorization

**Convergence of Random Variables (chapter 4)**

Keywords: convergence in probability, Ky Fan's distance, convergence almost-surely, convergence in L^p, uniformly integrable families, sequences of independent and identically distributed random variables, second Borel-Cantelli lemma, P. Levy's equivalence theorem, Kolmogorov's three series theorem, the law of large numbers, the central limit theorem

**The Art of Random Sampling (chapter 5)**

Keywords: layer-cake formulas, antithetic variates, importance sampling, acceptance-rejection method

**Asset Pricing in an Endowment Economy (sections 6.1, 6.2, 6.3, 6.4)**

Keywords: event trees, information structure, endowment economy, economic agents, pure-exchange economy, utility functions, risk aversion, investment-consumption decisions, method of Lagrange, state price density, stochastic discount factors

**Stochastic Processes in Continuous Time and Brownian Motion (sections 8.3, 8.4, 8.5)**

Keywords: Brownian motion, filtrations, stopping times, Gaussian processes, continuous stochastic processes, Kolmogorov's continuity criterion, Wiener space and measure

**Crash Course in Brownian Motion (sections 8.7, 8.8, 8.9, 8.10)**

Keywords: processes of finite variation, quadratic variation of a continuous process, the law of the iterated logarithm, running maximum of Brownian motion, reflection principle

**Crash Course in Continuous Time Martingales (sections 9.1, 9.2, 9.3, 9.4)**

Keywords: Levy processes, maximal inequalities, sample-paths regularity, convergence of martingales and submartingales, optional stopping, positive supermartingales, Doob-Meyer decomposition

**Local Martingales and Semimartingales (sections 9.5, 9.6)**

Keywords: quadratic variation, localization sequence of stopping times, local martingales, change of measure, semimartingales, L^2-bounded martingales, Doob's L^p-inequality

**Stochastic Integration and Ito's formula (sections 10.1, 10.2, 10.3, 10.4)**

Keywords: stochastic integrals, local martingales, continuous semimartingales, Ito's formula

**Applications of Ito's Formula (sections 10.5, 10.6, 10.7)**

Keywords: stochastic integrals with respect to Brownian motion, Ito's formula, Brownian filtrations, predictable representation, Girsanov's theorem, Tanaka's formula, local times

**Stochastic Differential Equations (sections 11.1, 11.2)**

Keywords: solution to stochastic differential equations, types of uniqueness of solutions, infinitesimal generators, martingale problem of Stroock and Varadhan, Markov processes, Tanaka's example

**Solutions to Stochastic Differential Equations (sections 11.3, 11.4, 11.5)**

Keywords: strong solutions, linear stochastic differential equations, Brownian bridge, Ornstein-Uhlenbeck process, mean-reverting Brownian motion, Cox-Ingersol-Ross process, constant elasticity of variance process, exponential Ornstein-Uhlenbeck process

**Feynman-Kac Formula and Fokker-Planck Equation (chapter 12)**

Keywords: Feynman-Kac formula, Fokker-Planck equation, Kolmogorov's forward equation

**Crash Course in Continuous Time Finance – I (sections 13.1, 13.2)**

Keywords: excess returns, realized volatility, self-financing trading strategies, wealth process, wealth dynamics

**Crash Course in Continuous Time Finance – II (sections 13.3, 13.4)**

Keywords: equivalent local martingale measures, admissible trading strategies, first and second fundamental theorems of asset pricing

**European-Style Contingent Claims (sections 14.1, 14.2)**

Keywords: pricing by arbitrage, European options, Malliavin calculus, replication, lower and upper hedging strategies, lower and upper hedging prices, risk-neutral valuation, arbitrage-free pricing, market completion, Black-Scholes-Merton formula

**The Martingale Solution to Merton's Problem (section 14.3)**

Keywords: convex duality, Fenchel-Legendre transform, static optimization, method of Lagrange

**American-Style Contingent Claims (section 14.4)**

Keywords: pricing by arbitrage, American options, upper and lower hedging strategies, upper and lower hedging price, hedging, arbitrage-free price, Snell envelope, market completion

**Put-Call Symmetry (section 14.5)**

Keywords: foreign currency exchange options

**Exchange Options and Dupire's Formula (sections 14.6, 14.8)**

Keywords: exchange options, stochastic volatility, implied volatility, local volatility, Dupire's formula

**Stochastic Volatility Models (section 14.7)**

Keywords: implied volatility, volatility smile, stochastic volatility, Hull and White model, Stein and Stein model, Heston model

**Stochastic Processes with Jumps (section 15.1)**

Keywords: optional, predictable, and progressive sigma-fields, increasing and finite variation processes with jumps, predictable projection, semimartingales with jumps, canonical decomposition of local martingales and semimartingales with jumps

**Stochastic Calculus with Jumps (sections 15.2, 15.3)**

Keywords: stochastic integration with jumps, quadratic variation with jumps, Ito's formula with jumps, Girsanov's theorem with jumps

**Random Measures (section 16.1)**

Keywords: counting measures, Poisson random measures, Poisson point process, compound Poisson process

**Brief Introduction to Levy Processes (sections 16.2, 16.3)**

Keywords: Levy process, Bochner-Khinchin theorem, infinitely divisible distribution laws, Levy-Khinchin formula, Levy-Ito decomposition, stochastic integration with respect to Levy processes, Levy-Ito isometry, Ito's formula for Levy processes

**Levy Processes in Asset Pricing (sections 16.4, 16.5, 16.6, 16.7)**

Keywords: stochastic exponents, Dolean-Dade exponents, Levy processes, change of measure involving Levy processes, removal of the drift with jumps, Levy-Ito diffusion processes, Black-Scholes-Merton formula with jumps

**Some History: the Soft Landing on the Moon on July 20, 1969 (section 17.1)**

Keywords: optimal control, controlled dynamical systems

**Principle of Dynamic Programming and HJB equation (sections 17.2, 17.3)**

Keywords: principle of dynamic programming, HJB equation, verification theorem, Merton's problem, optimal exercise of American options

**Merton's Problem with Intertemporal Consumption and Jumps (section 18.1)**

Keywords: Merton's problem with finite and infinite time horizon, Merton's problem with logarithmic and power utility functions

**Merton's Problem with Intertemporal Consumption and Rebalancing Costs (section 18.2)**

Keywords: Merton's problem, transaction costs, solvency cone

**Real Options and American Style Options (sections 18.3, 18.4)**

Keywords: real options, investment rights, the value of flexibility, investment decisions, perpetual American options, the early exercise premium, the optimal exercise boundary, numerical program for American options

**Debt, Equity, Dividend Policy, and the Modigliani-Miller Proposition (section 18.5)**

Keywords: the value of the enterprise, Merton's model of corporate debt, the value of corporate debt, corporate budgeting, Modigliani-Miller proposition