Unit 1 of 14
Mathematical financial models
- 8 lessons
- Videos 1 to 8
Financial engineering starts by writing a contract as dated cash flows across states. This unit builds that language and the calculus, linear algebra, optimization and probability the course uses.
A payoff is kept separate from its present value, and a positive linear valuation operator prices cash flows in a finite model. The law of one price follows from a precise definition of arbitrage, discount factors give spot and forward rates under stated compounding conventions, and self-financing accounting follows wealth through dividends and rebalancing. Taylor expansions with remainder, covariance matrices and projections, convex duality and finite probability spaces with Bayes' theorem complete the toolkit. Continuous-time results come in later units.
Every lesson is one video. Each lesson page has the definitions and an example for every key term.
Contracts, arbitrage and returns
Financial contracts as mathematical objects
How a contract becomes dated, state-dependent cash flows on a state space with a sign convention, how a payoff function differs from a present value, and how state prices give a positive linear valuation operator.
Arbitrage and the law of one price
What an arbitrage is in a finite one-period market, why no arbitrage implies the law of one price and linear pricing by replication, how payoff dominance orders prices, and what a short-selling constraint changes.
Video 2
Time value of money and discount functions
How discount factors measure the time value of money, how simple, periodic and continuous compounding conventions report them as spot rates, and how forward rates follow from ratios of discount factors.
Video 3
Returns and self-financing portfolio accounting
Defines simple, gross, price, total and log returns, explains why log returns add over time only while wealth stays positive, and derives dividend-inclusive self-financing accounting through a rebalance.
Video 4
Calculus, linear algebra, optimization and probability
Multivariable calculus and Taylor expansions
Defines partial derivatives, gradients and Hessians of a value function, derives the second-order Taylor expansion with its integral remainder, and shows why price sensitivities fail at a payoff kink.
Video 5
Linear algebra for covariance and hedging
Proves that covariance matrices are positive semidefinite, uses eigenvalues and the spectral theorem to find directions of risk, and solves minimum-error hedges by orthogonal projection and the pseudoinverse.
Video 6
Convex optimization and duality
How the Lagrangian of a convex problem gives lower bounds by weak duality, why KKT conditions certify a global minimum, when Slater's condition gives strong duality, and what a binding constraint's multiplier measures.
Video 7
Probability spaces and conditional information
How a probability space, its sigma algebra and an information partition describe what can be observed, and how conditional probability and Bayes' theorem update a default probability given its base rate.
Video 8