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Basics of Quantitative Finance

IntermediatePremium7 sections · 30 lessons · 67 questionsLog in to track progress

Quantitative finance is a small number of ideas applied relentlessly. Prices are random variables, risk is the second moment, and nearly every price is an expectation under a measure chosen so that no free money exists. This course is those ideas, worked with numbers.

It begins with expectation, variance and covariance in a financial setting, then the arithmetic of returns and the time value of money. The middle is the part that does the work: the law of one price, replication, forward pricing, put-call parity and where risk-neutral valuation comes from. Portfolio mathematics follows, including the efficient frontier and why its curvature is entirely a fact about correlation. Stochastic modelling follows, from random walks to geometric Brownian motion and Monte Carlo, and then the assumptions all of it rests on and how each one fails in practice. The last section is the mathematics underneath: matrices and least squares, eigenvalues and principal component analysis, and the calculus of sensitivities, optima and mean reversion.

Comfort with probability at the level of Fundamentals of Probability and Statistics is assumed. The final section refreshes the linear algebra and calculus the rest of the course uses.

Random Variables and DistributionsThe distributions quantitative finance actually uses, which financial quantity each one fits, and what the choice commits you to.Free previewExpectation, Linearity, and Conditional ExpectationThe three properties of expectation that do the most work in pricing, hedging and portfolio construction.Free previewVariance, Covariance, and CorrelationRisk measures for a single asset and for the interaction between assets, which is where portfolio risk actually comes from.Free previewLaw of Large Numbers and Central Limit TheoremFinance averages in two directions at once, across time and across holdings, and the two limit theorems behave quite differently in each.
Arithmetic vs Log ReturnsTwo definitions, when each is correct, and the mistake that comes from using the wrong one.Lognormal Returns and CompoundingWhy prices are modelled as lognormal, and the volatility drag that separates expected price from typical outcome.Time Value of Money and Discount FactorsMoving cash flows through time, the conventions that trip people up, and why the discount rate does most of the work.Real-World vs Risk-Neutral ExpectationTwo probability measures, two jobs, and the error that comes from using one for the other's task.
Law of One Price and ReplicationThe single idea underneath all derivatives pricing, and the assumptions that let it work.Forward and Futures PricingDeriving the forward price from a replication argument, and the income term that flips the curve.Put-Call Parity and Synthetic PositionsA model-free relationship between calls, puts and the underlying, and the synthetics it lets you build.Risk-Neutral Valuation IntuitionWhere the risk-neutral probabilities come from, built up from a one-step example.
Mean-Variance OptimizationThe foundational portfolio framework, and why its naive application produces portfolios nobody would hold.Efficient Frontier and Capital Market LineThe frontier of best available portfolios, and the straight line that a risk-free asset draws through it.Diversification and Correlation MatrixHow much diversification is available, why it stops, and why it fails when you need it.Sharpe Ratio and Risk-Adjusted MetricsThe industry's standard performance measure, its scaling property, and the strategies it flatters.Risk Parity and Stress TestingWhere a portfolio's risk really comes from: risk contributions, the risk-parity portfolio worked for stocks and bonds, what it assumes, and how to design stress tests that find the losses a covariance matrix hides.Factor ModelsExplaining many returns with a few shared drivers: the model and its covariance, the three ways factors are built, what alpha means after exposures, and the burden of proof on a new factor.
Random Walks and Brownian MotionThe building block of continuous-time finance, and the strange properties that make it work.Ito Process and Geometric Brownian MotionThe extra term that makes stochastic calculus different, and where it comes from.Lognormal Asset ModelingWhat the lognormal assumption buys, what it costs, and how the market prices the difference.Monte Carlo Simulation and Application to PricingSimulating paths to price what formulas cannot, and the practical details that decide whether it works.
Stationarity, Ergodicity, and Path DependenceThree assumptions every model makes silently, and the ways markets break each of them.Overfitting, Misspecification, and Model RiskThree distinct ways a model fails, and the defences that actually work against each.The Role of Drift vs Volatility in MarketsNoise dominates the short run and drift dominates the long run, with the crossover further out than most people assume.
Vectors, Matrices and What a Matrix DoesPortfolios as vectors and covariance as a matrix: the quadratic form behind every portfolio variance, inverses and rank, and why some correlation matrices cannot exist.Linear Systems and Least SquaresWhen a linear system has one solution, none or many; least squares as a projection, worked by hand; why software solves rather than inverts; and what collinearity does to coefficients.Eigenvalues and EigenvectorsThe directions a matrix only stretches: what they mean for a covariance matrix, how to find them for a 2 by 2 by hand, and how they test and repair a matrix that is not a valid covariance.Principal Component AnalysisRotating correlated returns into uncorrelated components ordered by variance: the steps, the market factor it finds in equities, when to standardise, and how to tell a component from noise.Calculus for QuantsThe calculus that appears in quant work every day: derivatives as sensitivities, Taylor expansions for quick approximations, optimisation with and without constraints, and the differential equation behind mean reversion.

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