Quantitative finance

Financial Engineering for Quants

The mathematics of pricing, hedging and risk, from no-arbitrage pricing and stochastic calculus to derivatives, numerical methods, interest rates, credit, portfolios and execution.

  • Advanced
  • 112 lessons
  • Free
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About this course

Financial engineering uses mathematics to price, hedge and measure the risk of financial contracts. Its central argument is replication, which ties the price of a claim to the cost of a portfolio that pays the same in every state, and its models of random prices rely on probability, stochastic calculus, statistics and numerical methods. Every result holds only under stated assumptions about information, trading and the pricing measure, so the mathematics shows where a model applies and where it stops. This course starts by writing contracts as dated cash flows and proving the law of one price, then covers probability and econometrics, bonds, curves and swaps, discrete arbitrage pricing, stochastic calculus, the Black-Scholes model, volatility and path-dependent options, numerical methods, interest rate models, credit and valuation adjustments, risk measurement, portfolio allocation and statistical research. It ends with execution, calibration and model validation.

Worked examples use constructed numbers rather than market data, and the course teaches mathematics, not investment advice. The videos are coming out one at a time, in course order. Every lesson has its own page with precise definitions, a worked example for each key term and a few exercises with answers, so you can start reading now. The reference sheet collects the definitions, formulas, models and conventions unit by unit, with the assumptions each one needs and links to the lessons that explain them.

The course assumes algebra, functions, exponentials and logarithms, a first course in single-variable calculus that includes integration and differential equations, basic matrix operations and elementary probability. It reviews the multivariable calculus, linear algebra, convex optimization and probability spaces it needs in the Mathematical financial models unit, develops distributions, conditional expectation, estimation, regression and time series in the Probability and financial econometrics unit, and builds stochastic calculus from its foundations. A few lessons also use complex numbers, characteristic functions and the t and chi-squared distributions, and the outline lists them for those lessons. Basic Python helps with numerical experiments, but no software is needed to follow the mathematics.

Other courses on the site are optional background or further study, not prerequisites. Calculus covers single-variable calculus and first-order differential equations, and Multivariable and Vector Calculus develops partial derivatives, multivariable Taylor approximation and multiple integrals in full. Linear Algebra I treats projections, least squares, eigenvalues, the singular value decomposition and pseudoinverses. Introduction to Probability and Statistics go further into distributions, limit theorems, estimation, regression and time series. Differential Equations includes the heat equation and other partial differential equations, Multivariate Statistical Analysis treats covariance matrices and principal components as statistical methods, and Machine Learning goes further into regularization, validation and data leakage.

Good to know first

  • Algebra, functions, exponentials and logarithms
  • Single-variable calculus, including integration
  • A first course in differential equations
  • Basic matrix operations
  • Elementary probability
  • Basic Python is helpful but not required

Course outline

1 of 112 lessons are on YouTube so far. Each lesson is one video. Open a unit to see its lessons, or follow a lesson for its definitions and examples.

  1. Mathematical financial models

    1 of 8 lessons on YouTube
    Open the Mathematical financial models unit page

    Contracts, arbitrage and returns

    1. Financial contracts as mathematical objectsOn YouTube

      Video 1

      financial contract, cash-flow vector, state space, payoff function, positive linear valuation operator, payoff versus present value and sign convention

    2. Arbitrage and the law of one priceComing soon

      Video 2

      arbitrage, law of one price, linear pricing, replication, payoff dominance and short-selling constraint

      Builds on Financial contracts as mathematical objects.

    3. Time value of money and discount functionsComing soon

      Video 3

      time value of money, discount factor, compounding convention, continuous compounding, spot rate and forward rate

      Builds on Financial contracts as mathematical objects.

    4. Returns and self-financing portfolio accountingComing soon

      Video 4

      simple return, gross return, price return, total return, log return, self-financing portfolio and portfolio rebalance

      Builds on Financial contracts as mathematical objects and Time value of money and discount functions.

    Calculus, linear algebra, optimization and probability

    1. Multivariable calculus and Taylor expansionsComing soon

      Video 5

      partial derivative, gradient, Hessian matrix, second-order Taylor expansion, integral remainder, price sensitivity and payoff kink

    2. Linear algebra for covariance and hedgingComing soon

      Video 6

      covariance matrix, positive semidefinite matrix, eigenvalues and eigenvectors, spectral theorem, orthogonal projection, minimum-error hedge and pseudoinverse

    3. Convex optimization and dualityComing soon

      Video 7

      convex function, Lagrangian, weak duality, strong duality, KKT conditions, constraint qualification, Slater's condition and binding constraint

      Builds on Multivariable calculus and Taylor expansions and Linear algebra for covariance and hedging.

    4. Probability spaces and conditional informationComing soon

      Video 8

      probability space, sigma algebra, probability measure, information partition, conditional probability, Bayes' theorem and base rate

  2. Probability and financial econometrics

    8 lessons coming soon
    Open the Probability and financial econometrics unit page

    Distributions, expectation and dependence

    1. Random variables and distribution transformationsComing soon

      Video 9

      random variable, cumulative distribution function, probability density function, change of variables, Jacobian, lognormal distribution and completing the square

      Builds on Probability spaces and conditional information.

    2. Expectation conditional expectation and varianceComing soon

      Video 10

      expectation, integrability, conditional expectation, tower property, law of total variance, within-group variance and between-group variance

      Builds on Random variables and distribution transformations.

    3. Multivariate distributions and dependenceComing soon

      Video 11

      joint distribution, covariance, correlation, zero correlation versus independence, jointly Gaussian vector, copula, Sklar's theorem and portfolio variance

      Builds on Linear algebra for covariance and hedging and Expectation conditional expectation and variance. Also uses characteristic functions.

    Estimation and regression

    1. Statistical estimation and likelihoodComing soon

      Video 12

      likelihood function, maximum likelihood estimator, consistency, law of large numbers, central limit theorem, standard error and confidence interval

      Builds on Random variables and distribution transformations and Expectation conditional expectation and variance. Also uses characteristic functions and Student t distribution.

    2. Regression and statistical identificationComing soon

      Video 13

      ordinary least squares, normal equations, exogeneity, statistical identification, omitted variable bias, heteroskedasticity and sandwich covariance

      Builds on Linear algebra for covariance and hedging and Statistical estimation and likelihood.

    Time series, volatility and filtering

    1. Stationarity autocovariance and ARMA modelsComing soon

      Video 14

      weak stationarity, autocovariance, AR(1) process, causal stationary representation, ARMA model, conditional forecast variance and unit root

      Builds on Expectation conditional expectation and variance and Regression and statistical identification. Also uses complex roots of polynomials.

    2. Conditional volatility and GARCHComing soon

      Video 15

      conditional variance, ARCH model, GARCH model, variance persistence, variance forecast, strict stationarity and unconditional variance

      Builds on Stationarity autocovariance and ARMA models.

    3. State space models and the Kalman filterComing soon

      Video 16

      state-space model, state equation, observation equation, latent state, Kalman filter, Kalman gain, innovation covariance and posterior covariance

      Builds on Linear algebra for covariance and hedging, Multivariate distributions and dependence and Regression and statistical identification.

  3. Cash flows instruments and curves

    8 lessons coming soon
    Open the Cash flows instruments and curves unit page

    Bonds, discount curves and rate sensitivities

    1. Bond cash flows and yield mathematicsComing soon

      Video 17

      coupon bond, accrued interest, clean price, full price, yield to maturity, internal rate of return and reinvestment assumption

      Builds on Time value of money and discount functions.

    2. Bootstrapping discount curvesComing soon

      Video 18

      bootstrapping, discount curve, triangular system, curve node, curve interpolation, implied forward rate and curve consistency

      Builds on Time value of money and discount functions and Bond cash flows and yield mathematics.

    3. Duration convexity and key rate sensitivitiesComing soon

      Video 19

      Macaulay duration, modified duration, convexity, parallel yield shift, key rate duration and second-order price approximation

      Builds on Multivariable calculus and Taylor expansions, Bond cash flows and yield mathematics and Bootstrapping discount curves.

    Forwards, futures and swaps

    1. Forwards and cost of carryComing soon

      Video 20

      forward contract, forward price, prepaid forward, cost of carry, known dividends, continuous dividend yield and value of an existing forward

      Builds on Arbitrage and the law of one price, Time value of money and discount functions and Returns and self-financing portfolio accounting.

    2. Futures daily settlement and convexityComing soon

      Video 21

      futures contract, daily settlement, variation margin, futures price, forward-futures difference and convexity adjustment

      Builds on Forwards and cost of carry.

    3. Interest rate swaps and forward rate agreementsComing soon

      Video 22

      forward rate agreement, interest rate swap, fixed leg, floating leg, swap annuity, par swap rate and single-curve telescoping

      Builds on Bootstrapping discount curves and Forwards and cost of carry.

    Currency and commodity forwards

    1. Foreign exchange and covered interest parityComing soon

      Video 23

      exchange-rate quotation, FX forward, covered interest parity, domestic pricing measure, reciprocal quotation and cross-currency basis

      Builds on Time value of money and discount functions and Forwards and cost of carry.

    2. Commodity forwards storage and convenience yieldComing soon

      Video 24

      commodity forward, storage cost, convenience yield, net carrying rate, contango, backwardation and reverse cash-and-carry

      Builds on Forwards and cost of carry.

  4. Discrete arbitrage pricing

    8 lessons coming soon
    Open the Discrete arbitrage pricing unit page

    Replication, state prices and binomial trees

    1. One period replication and state pricesComing soon

      Video 25

      one-period model, replicating portfolio, payoff matrix, state price, pricing kernel and dual pricing system

      Builds on Arbitrage and the law of one price, Linear algebra for covariance and hedging and Probability spaces and conditional information.

    2. Binomial trees and backward inductionComing soon

      Video 26

      binomial tree, local no-arbitrage condition, hedge ratio, backward induction, time-step scaling rule and diffusion limit

      Builds on One period replication and state prices.

    Pricing measures, martingales and the fundamental theorem

    1. Risk neutral probabilities and physical probabilitiesComing soon

      Video 27

      physical measure, risk-neutral measure, equivalent measures, likelihood ratio, cash numeraire and pricing versus forecasting probabilities

      Builds on Probability spaces and conditional information, Expectation conditional expectation and variance and One period replication and state prices.

    2. Martingales and discrete conditional pricingComing soon

      Video 28

      filtration, predictable trading strategy, martingale, discounted price process, conditional discounted pricing and cumulative gain process

      Builds on Expectation conditional expectation and variance, Binomial trees and backward induction and Risk neutral probabilities and physical probabilities.

    3. The fundamental theorem of asset pricingComing soon

      Video 29

      fundamental theorem of asset pricing, no-arbitrage condition, equivalent martingale measure, separating hyperplane, finite market model and full-support condition

      Builds on Convex optimization and duality, One period replication and state prices, Risk neutral probabilities and physical probabilities and Martingales and discrete conditional pricing.

    Incompleteness, early exercise and transaction costs

    1. Market completeness and nonunique pricesComing soon

      Video 30

      attainable claim, market completeness, payoff matrix rank, incomplete market, no-arbitrage price interval and superhedging bound

      Builds on Linear algebra for covariance and hedging and The fundamental theorem of asset pricing.

    2. American options and optimal stoppingComing soon

      Video 31

      American option, admissible exercise time, optimal stopping, Snell envelope, continuation value, exercise boundary and early-exercise premium

      Builds on Martingales and discrete conditional pricing.

    3. Transaction costs and no arbitrage price intervalsComing soon

      Video 32

      bid-ask spread, proportional transaction costs, superreplication, seller and buyer price bounds, covering hedge and shadow price

      Builds on Arbitrage and the law of one price, Market completeness and nonunique prices and American options and optimal stopping.

  5. Stochastic calculus

    8 lessons coming soon
    Open the Stochastic calculus unit page

    Information, Brownian motion and quadratic variation

    1. Stochastic processes filtrations and stopping timesComing soon

      Video 33

      adapted process, predictable process, predictable sigma algebra, stopping time, usual conditions and anticipatory strategy

      Builds on Probability spaces and conditional information, Expectation conditional expectation and variance and Martingales and discrete conditional pricing.

    2. Brownian motion and Gaussian incrementsComing soon

      Video 34

      Brownian motion, independent Gaussian increments, transition density, Brownian covariance, Brownian scaling and conditional threshold probability

      Builds on Random variables and distribution transformations, Multivariate distributions and dependence and Stochastic processes filtrations and stopping times.

    3. Quadratic variation and stochastic limitsComing soon

      Video 35

      quadratic variation, partition mesh, convergence in probability, mean-square convergence, finite variation, Brownian roughness and quadratic covariation

      Builds on Brownian motion and Gaussian increments.

    Itô integrals and stochastic differential equations

    1. The Ito integral and isometryComing soon

      Video 36

      simple predictable integrand, Itô integral, Itô isometry, square-integrable integrand, martingale property of the Itô integral and left-endpoint convention

      Builds on Expectation conditional expectation and variance, Stochastic processes filtrations and stopping times and Quadratic variation and stochastic limits.

    2. Ito lemma and the stochastic chain ruleComing soon

      Video 37

      Itô's lemma, stochastic chain rule, second-order correction, multivariate Itô formula, generator of a diffusion and Itô product rule

      Builds on Multivariable calculus and Taylor expansions, Quadratic variation and stochastic limits and The Ito integral and isometry.

    3. Stochastic differential equations and solution conceptsComing soon

      Video 38

      stochastic differential equation, strong solution, weak solution, existence and uniqueness theorem, global Lipschitz condition, pathwise uniqueness, Ornstein-Uhlenbeck process and Euler step

      Builds on The Ito integral and isometry and Ito lemma and the stochastic chain rule.

    Measure changes and Girsanov's theorem

    1. Exponential martingales and measure changesComing soon

      Video 39

      change of measure, Radon-Nikodym derivative, conditional Bayes formula, stochastic exponential, exponential martingale and Novikov condition

      Builds on Expectation conditional expectation and variance, The Ito integral and isometry and Ito lemma and the stochastic chain rule.

    2. Girsanov theorem and risk neutral dynamicsComing soon

      Video 40

      Girsanov's theorem, drift transformation, market price of risk, risk-neutral dynamics, measure equivalence and Lévy's characterization

      Builds on Stochastic differential equations and solution concepts and Exponential martingales and measure changes.

  6. Continuous time derivative pricing

    8 lessons coming soon
    Open the Continuous time derivative pricing unit page

    The Black-Scholes pricing equation

    1. Geometric Brownian motion and exact distributionsComing soon

      Video 41

      geometric Brownian motion, exact GBM solution, lognormal transition density, power moments and exact sampling

      Builds on Random variables and distribution transformations, Ito lemma and the stochastic chain rule and Stochastic differential equations and solution concepts.

    2. Black Scholes replication and the pricing PDEComing soon

      Video 42

      Black-Scholes PDE, self-financing delta hedge, replication argument, terminal condition, boundary conditions and frictionless market assumptions

      Builds on Returns and self-financing portfolio accounting, Ito lemma and the stochastic chain rule, Girsanov theorem and risk neutral dynamics and Geometric Brownian motion and exact distributions.

    3. Feynman Kac and discounted expectationsComing soon

      Video 43

      Feynman-Kac formula, backward PDE, terminal value problem, discounted conditional expectation, controlled-growth solution class and classical solution

      Builds on Ito lemma and the stochastic chain rule, Stochastic differential equations and solution concepts and Black Scholes replication and the pricing PDE.

    Black-Scholes formulas, Greeks and hedging

    1. The Black Scholes formula and put call parityComing soon

      Video 44

      Black-Scholes formula, European call and put, d1 and d2, standard normal CDF, put-call parity, in-the-money probability and expiry and zero-volatility limits

      Builds on Expectation conditional expectation and variance, Geometric Brownian motion and exact distributions, Black Scholes replication and the pricing PDE and Feynman Kac and discounted expectations.

    2. Greeks and differential identitiesComing soon

      Video 45

      option Greeks, delta, gamma, vega, theta, rho and vega-gamma identity

      Builds on Multivariable calculus and Taylor expansions and The Black Scholes formula and put call parity.

    3. Discrete delta hedging and hedging errorComing soon

      Video 46

      discrete delta hedging, trading frequency, gamma exposure, realized variance, hedging error and gamma-weighted variance comparison

      Builds on Returns and self-financing portfolio accounting, Quadratic variation and stochastic limits and Greeks and differential identities.

    Digital and American options

    1. Digital options and distribution recoveryComing soon

      Video 47

      cash-or-nothing digital option, strike derivative, Breeden-Litzenberger formula, state-price density, call price curvature and probability atom

      Builds on Multivariable calculus and Taylor expansions, Feynman Kac and discounted expectations and The Black Scholes formula and put call parity.

    2. Early exercise in continuous timeComing soon

      Video 48

      continuous-time optimal stopping, variational inequality, free boundary, value matching, smooth pasting and perpetual American put

      Builds on American options and optimal stopping, Black Scholes replication and the pricing PDE and Feynman Kac and discounted expectations.

  7. Volatility and path dependent derivatives

    8 lessons coming soon
    Open the Volatility and path dependent derivatives unit page

    Implied volatility and arbitrage-free surfaces

    1. Implied volatility and inverse pricingComing soon

      Video 49

      implied volatility, price monotonicity in volatility, arbitrage bounds on option prices, inverse sensitivity and bracketed inversion

      Builds on The Black Scholes formula and put call parity and Greeks and differential identities.

    2. Static arbitrage and volatility surfacesComing soon

      Video 50

      static arbitrage, volatility surface, call spread, butterfly spread, strike monotonicity, strike convexity, calendar consistency and arbitrage-free interpolation

      Builds on Digital options and distribution recovery and Implied volatility and inverse pricing.

    Local volatility, stochastic volatility and jumps

    1. Dupire local volatilityComing soon

      Video 51

      local volatility, Dupire equation, call price surface, local variance and ill-posed differentiation

      Builds on Ito lemma and the stochastic chain rule, Black Scholes replication and the pricing PDE, Digital options and distribution recovery and Static arbitrage and volatility surfaces.

    2. Heston stochastic volatilityComing soon

      Video 52

      Heston model, variance diffusion, leverage correlation, Feller condition, affine characteristic function and Riccati equations

      Builds on Multivariate distributions and dependence, Ito lemma and the stochastic chain rule, Stochastic differential equations and solution concepts and Girsanov theorem and risk neutral dynamics. Also uses complex numbers.

    3. Jump diffusion and incomplete hedgingComing soon

      Video 53

      jump diffusion, Poisson jumps, compensated jump process, drift compensation, nonlocal pricing generator and jump residual

      Builds on Market completeness and nonunique prices, Stochastic differential equations and solution concepts and Girsanov theorem and risk neutral dynamics.

    Path-dependent and multiasset options

    1. Barrier options and first passageComing soon

      Video 54

      barrier option, hitting time, reflection principle, knock-in knock-out parity, Brownian bridge crossing probability and barrier monitoring

      Builds on Stochastic processes filtrations and stopping times, Brownian motion and Gaussian increments and Feynman Kac and discounted expectations.

    2. Asian options and running averagesComing soon

      Video 55

      Asian option, arithmetic average, geometric average, path dependence, state augmentation and arithmetic-geometric inequality

      Builds on Expectation conditional expectation and variance, Geometric Brownian motion and exact distributions and Feynman Kac and discounted expectations.

    3. Multiasset derivatives and exchange optionsComing soon

      Video 56

      exchange option, change of numeraire, effective volatility, correlated diffusions, basket option and lognormal moment matching

      Builds on Multivariate distributions and dependence, Ito lemma and the stochastic chain rule and The Black Scholes formula and put call parity.

  8. Numerical methods for finance

    8 lessons coming soon
    Open the Numerical methods for finance unit page

    Root finding, Monte Carlo and discretization

    1. Numerical conditioning and root findingComing soon

      Video 57

      problem conditioning, numerical stability, floating-point cancellation, bisection method, Newton's method, quadratic convergence and Newton cycle

      Builds on Multivariable calculus and Taylor expansions and Convex optimization and duality.

    2. Monte Carlo pricing and sampling errorComing soon

      Video 58

      Monte Carlo estimator, risk-neutral sampling, Monte Carlo standard error, sampling error, Monte Carlo confidence interval, interval coverage and model and discretization error

      Builds on Expectation conditional expectation and variance, Statistical estimation and likelihood, Risk neutral probabilities and physical probabilities and Feynman Kac and discounted expectations.

    3. SDE discretization and weak convergenceComing soon

      Video 59

      Euler-Maruyama scheme, Milstein correction, strong order of convergence, weak order of convergence, payoff regularity and square-root state positivity

      Builds on Quadratic variation and stochastic limits, Stochastic differential equations and solution concepts and Numerical conditioning and root finding.

    Variance reduction and quasi-Monte Carlo

    1. Variance reduction and unbiased estimatorsComing soon

      Video 60

      variance reduction, antithetic variates, control variate, optimal control coefficient, importance sampling and importance weight

      Builds on Expectation conditional expectation and variance, Exponential martingales and measure changes and Monte Carlo pricing and sampling error.

    2. Quasi Monte Carlo and effective dimensionComing soon

      Video 61

      quasi-Monte Carlo, star discrepancy, low-discrepancy points, randomized quasi-Monte Carlo, Brownian bridge construction and effective dimension

      Builds on Multivariate distributions and dependence, Brownian motion and Gaussian increments, Monte Carlo pricing and sampling error and Variance reduction and unbiased estimators.

    PDE, regression and Fourier methods

    1. Finite difference PDE methodsComing soon

      Video 62

      finite difference method, heat equation, spatial grid, explicit scheme, implicit scheme, Crank-Nicolson scheme, stability condition and discrete maximum principle

      Builds on Black Scholes replication and the pricing PDE, Feynman Kac and discounted expectations and Numerical conditioning and root finding.

    2. Least squares Monte Carlo for American optionsComing soon

      Video 63

      least-squares Monte Carlo, continuation regression, exercise policy, Bermudan option, independent policy evaluation, policy lower bound and fitting bias

      Builds on Regression and statistical identification, American options and optimal stopping, Early exercise in continuous time and Monte Carlo pricing and sampling error.

    3. Fourier transform pricingComing soon

      Video 64

      characteristic function, Fourier inversion, damping parameter, moment strip, fast Fourier transform, truncation error and aliasing

      Builds on Random variables and distribution transformations, Feynman Kac and discounted expectations, Jump diffusion and incomplete hedging and Numerical conditioning and root finding. Also uses complex numbers.

  9. Interest rate models and measures

    8 lessons coming soon
    Open the Interest rate models and measures unit page

    Numeraires and short-rate models

    1. Numeraires and forward measuresComing soon

      Video 65

      numeraire, density process, forward measure, bond numeraire and numeraire invariance

      Builds on Martingales and discrete conditional pricing, Exponential martingales and measure changes, Girsanov theorem and risk neutral dynamics and Feynman Kac and discounted expectations.

    2. Vasicek and CIR short rate modelsComing soon

      Video 66

      short-rate model, Vasicek model, Cox-Ingersoll-Ross model, mean reversion, affine bond price, CIR boundary behavior and measure-dependent parameters

      Builds on Stochastic differential equations and solution concepts, Feynman Kac and discounted expectations and Numeraires and forward measures.

    3. Hull White and initial curve fittingComing soon

      Video 67

      Hull-White model, Gaussian short rate, time-dependent drift, initial curve fitting, deterministic shift and conditional bond price

      Builds on Bootstrapping discount curves and Vasicek and CIR short rate models.

    Forward-curve and market models

    1. Heath Jarrow Morton and the drift restrictionComing soon

      Video 68

      Heath-Jarrow-Morton framework, instantaneous forward rate, forward-rate volatility, HJM drift restriction, bond price dynamics and exponential volatility kernel

      Builds on The Ito integral and isometry, Ito lemma and the stochastic chain rule, Numeraires and forward measures and Vasicek and CIR short rate models.

    2. Forward rate market modelsComing soon

      Video 69

      forward rate market model, simple forward rate, own forward measure, terminal measure, terminal-measure drift, forward-rate correlation and backward-looking overnight rate

      Builds on Interest rate swaps and forward rate agreements, Girsanov theorem and risk neutral dynamics, Numeraires and forward measures and Heath Jarrow Morton and the drift restriction.

    Rate options, smiles and collateral curves

    1. Caps floors and swaptionsComing soon

      Video 70

      caplet, floorlet, Black's formula, Bachelier formula, swaption, annuity measure and caplet-floorlet parity

      Builds on Interest rate swaps and forward rate agreements, The Black Scholes formula and put call parity, Numeraires and forward measures and Forward rate market models.

    2. SABR and smile asymptoticsComing soon

      Video 71

      SABR model, elasticity parameter, volatility of volatility, implied volatility smile, Hagan formula and at-the-money limit

      Builds on Implied volatility and inverse pricing, Heston stochastic volatility and Caps floors and swaptions.

    3. Collateral discounting and multiple curvesComing soon

      Video 72

      collateral discounting, OIS discounting, projection curve, tenor basis, multi-curve valuation and compounded overnight rate

      Builds on Bootstrapping discount curves, Interest rate swaps and forward rate agreements, Foreign exchange and covered interest parity and Numeraires and forward measures.

  10. Credit derivatives and valuation adjustments

    8 lessons coming soon
    Open the Credit derivatives and valuation adjustments unit page

    Default models and credit default swaps

    1. Default intensity survival and recoveryComing soon

      Video 73

      default time, hazard rate, survival probability, default intensity, recovery convention and Cox construction

      Builds on Random variables and distribution transformations, Risk neutral probabilities and physical probabilities and Girsanov theorem and risk neutral dynamics.

    2. Structural credit and the Merton modelComing soon

      Video 74

      Merton model, structural credit model, firm asset value, default boundary, equity as a call option, debt valuation and risk-neutral default probability

      Builds on Bond cash flows and yield mathematics, The Black Scholes formula and put call parity and Default intensity survival and recovery.

    3. Credit default swap valuationComing soon

      Video 75

      credit default swap, premium leg, protection leg, accrual on default and par spread

      Builds on Bootstrapping discount curves, Interest rate swaps and forward rate agreements and Default intensity survival and recovery.

    Default dependence and structured credit

    1. Default dependence and credit portfoliosComing soon

      Video 76

      default indicator, joint default law, latent Gaussian factor, Gaussian copula, tail dependence and portfolio loss distribution

      Builds on Multivariate distributions and dependence and Default intensity survival and recovery.

    2. Tranche loss and structured creditComing soon

      Video 77

      tranche, attachment point, detachment point, loss waterfall, tranche loss, expected tranche loss and integrated tail

      Builds on Default dependence and credit portfolios.

    Counterparty exposure and valuation adjustments

    1. Exposure netting and collateral dynamicsComing soon

      Video 78

      signed portfolio value, positive exposure, netting set, netting inequality, signed collateral balance and margin period of risk

      Builds on Interest rate swaps and forward rate agreements, Monte Carlo pricing and sampling error, Collateral discounting and multiple curves and Default intensity survival and recovery.

    2. Credit valuation adjustment and wrong way riskComing soon

      Video 79

      credit valuation adjustment, unilateral CVA, loss at default, closeout value, wrong-way risk and exposure-default covariance

      Builds on Expectation conditional expectation and variance, Default intensity survival and recovery and Exposure netting and collateral dynamics.

    3. Funding capital and margin valuation adjustmentsComing soon

      Video 80

      funding valuation adjustment, debt valuation adjustment, margin valuation adjustment, capital valuation adjustment, XVA and double counting

      Builds on Collateral discounting and multiple curves, Exposure netting and collateral dynamics and Credit valuation adjustment and wrong way risk.

  11. Risk measurement and model uncertainty

    8 lessons coming soon
    Open the Risk measurement and model uncertainty unit page

    Attribution and tail risk measures

    1. Profit and loss attribution and sensitivitiesComing soon

      Video 81

      profit and loss attribution, Taylor expansion of PnL, carry, market move, attribution residual and attribution convention

      Builds on Returns and self-financing portfolio accounting, Multivariable calculus and Taylor expansions, Duration convexity and key rate sensitivities and Greeks and differential identities.

    2. Value at risk and quantile estimationComing soon

      Video 82

      value at risk, loss distribution, empirical quantile convention, historical VaR, failure of subadditivity and quantile estimation uncertainty

      Builds on Random variables and distribution transformations, Statistical estimation and likelihood and Profit and loss attribution and sensitivities.

    3. Expected shortfall and coherent riskComing soon

      Video 83

      expected shortfall, quantile integral, threshold representation, coherent risk measure, dual envelope representation and tail estimation

      Builds on Expectation conditional expectation and variance and Value at risk and quantile estimation.

    4. Extreme value theory and tail modelsComing soon

      Video 84

      extreme value theory, maximum domain of attraction, generalized Pareto distribution, threshold excesses, Pickands-Balkema-de Haan theorem, tail index, high quantile extrapolation and threshold stability

      Builds on Random variables and distribution transformations, Statistical estimation and likelihood, Value at risk and quantile estimation and Expected shortfall and coherent risk.

    Stress testing and liquidity risk

    1. Stress testing and scenario aggregationComing soon

      Video 85

      stress testing, scenario vector, joint stress scenario, nonlinear revaluation, reverse stress test and scenario aggregation

      Builds on Multivariate distributions and dependence, Monte Carlo pricing and sampling error, Profit and loss attribution and sensitivities and Expected shortfall and coherent risk.

    2. Liquidity risk and closeout horizonsComing soon

      Video 86

      liquidity risk, spread cost, liquidation impact cost, inventory variance, liquidation horizon and liquidity adjustment

      Builds on Futures daily settlement and convexity, Profit and loss attribution and sensitivities and Stress testing and scenario aggregation.

    Backtesting and model uncertainty

    1. Risk backtesting and statistical testsComing soon

      Video 87

      VaR backtesting, exception indicator, binomial exception count, unconditional coverage test, independence test and conditional coverage test

      Builds on Statistical estimation and likelihood, Stationarity autocovariance and ARMA models, Value at risk and quantile estimation and Expected shortfall and coherent risk. Also uses chi-squared distribution.

    2. Model uncertainty and robust valuationComing soon

      Video 88

      parameter uncertainty, model ambiguity, valuation envelope, monotone parameter bound, dual norm bound and robust valuation bounds

      Builds on Convex optimization and duality, Statistical estimation and likelihood, Greeks and differential identities, Monte Carlo pricing and sampling error and Stress testing and scenario aggregation.

  12. Portfolio mathematics and allocation

    8 lessons coming soon
    Open the Portfolio mathematics and allocation unit page

    Mean-variance optimization and covariance estimation

    1. Mean variance optimization and efficient frontiersComing soon

      Video 89

      portfolio moments, budget constraint, minimum-variance problem, efficient frontier, quadratic program and active constraint

      Builds on Linear algebra for covariance and hedging, Convex optimization and duality and Multivariate distributions and dependence.

    2. Factor models and covariance estimationComing soon

      Video 90

      factor model, factor loading, residual covariance, sample covariance matrix, covariance shrinkage, principal components and Rayleigh quotient

      Builds on Linear algebra for covariance and hedging, Multivariate distributions and dependence, Regression and statistical identification and Mean variance optimization and efficient frontiers.

    Views, estimation risk and risk parity

    1. Black Litterman and Bayesian portfolio viewsComing soon

      Video 91

      Black-Litterman model, equilibrium prior, view matrix, view confidence, posterior mean and predictive covariance

      Builds on Expectation conditional expectation and variance, Statistical estimation and likelihood, Mean variance optimization and efficient frontiers and Factor models and covariance estimation.

    2. Robust optimization and estimation riskComing soon

      Video 92

      robust optimization, mean uncertainty set, worst-case objective, estimation error, worst-case penalty and regularization effect

      Builds on Convex optimization and duality, Model uncertainty and robust valuation, Mean variance optimization and efficient frontiers and Factor models and covariance estimation.

    3. Risk contributions and risk parityComing soon

      Video 93

      risk contribution, Euler allocation, marginal risk, component risk, risk budget, risk parity and equal risk contribution

      Builds on Multivariable calculus and Taylor expansions and Mean variance optimization and efficient frontiers.

    Growth, dynamic allocation and liabilities

    1. Expected utility and Kelly growthComing soon

      Video 94

      expected utility, log utility, Kelly criterion, Kelly fraction, expected log growth and positive-wealth constraint

      Builds on Random variables and distribution transformations, Expectation conditional expectation and variance and Mean variance optimization and efficient frontiers.

    2. Dynamic allocation and stochastic controlComing soon

      Video 95

      stochastic control, Bellman principle, Hamilton-Jacobi-Bellman equation, CRRA utility, Merton allocation, verification argument and admissible control

      Builds on Convex optimization and duality, Ito lemma and the stochastic chain rule, Stochastic differential equations and solution concepts and Expected utility and Kelly growth.

    3. Liability hedging and immunizationComing soon

      Video 96

      immunization, asset-liability surplus, duration matching, normalized convexity, nonparallel curve shift and key rate hedge

      Builds on Duration convexity and key rate sensitivities, Interest rate swaps and forward rate agreements and Mean variance optimization and efficient frontiers.

  13. Statistical quant research

    8 lessons coming soon
    Open the Statistical quant research unit page

    Data, prediction and learning

    1. Point in time financial data and samplingComing soon

      Video 97

      point-in-time data, information set, timestamp, data revision, survivorship bias and corporate action adjustment

      Builds on Returns and self-financing portfolio accounting, Probability spaces and conditional information and Stationarity autocovariance and ARMA models.

    2. Financial prediction and feature constructionComing soon

      Video 98

      prediction target, admissible feature, feature availability, label horizon, squared-error prediction and conditional mean predictor

      Builds on Regression and statistical identification, Stationarity autocovariance and ARMA models and Point in time financial data and sampling.

    3. Machine learning losses regularization and calibrationComing soon

      Video 99

      empirical risk, ridge estimator, probability calibration, Brier loss, proper loss, distribution shift and covariate shift

      Builds on Convex optimization and duality, Statistical estimation and likelihood, Regression and statistical identification and Financial prediction and feature construction.

    Validation, multiple testing and cointegration

    1. Time series validation and overlapping labelsComing soon

      Video 100

      walk-forward validation, chronological split, overlapping labels, purging, embargo and nested validation

      Builds on Stationarity autocovariance and ARMA models, Financial prediction and feature construction and Machine learning losses regularization and calibration.

    2. Multiple testing and backtest overfittingComing soon

      Video 101

      multiple testing, selection bias, trial accounting, family-wise error rate, Bonferroni correction, false discovery rate, Benjamini-Hochberg procedure and backtest overfitting

      Builds on Statistical estimation and likelihood and Time series validation and overlapping labels.

    3. Cointegration and statistical arbitrageComing soon

      Video 102

      integration order, cointegration, common trend, stationary spread, error correction and statistical arbitrage

      Builds on Regression and statistical identification, Stationarity autocovariance and ARMA models and Time series validation and overlapping labels.

    Signals, costs and capacity

    1. Signal evaluation and performance attributionComing soon

      Video 103

      information coefficient, strategy return, Sharpe ratio, serial dependence, Sharpe ratio aggregation and factor attribution

      Builds on Multivariate distributions and dependence, Statistical estimation and likelihood, Stationarity autocovariance and ARMA models, Factor models and covariance estimation, Time series validation and overlapping labels and Multiple testing and backtest overfitting.

    2. Turnover costs and strategy capacityComing soon

      Video 104

      turnover, trading cost, impact curve, net return, strategy capacity, break-even capacity and profit-maximizing capital

      Builds on Mean variance optimization and efficient frontiers, Financial prediction and feature construction and Signal evaluation and performance attribution.

  14. Execution calibration and mathematical production

    8 lessons coming soon
    Open the Execution calibration and mathematical production unit page

    Order books, execution and market making

    1. Limit order books and queueing modelsComing soon

      Video 105

      limit order book, queue priority, arrival intensity, fill probability, Poisson process, birth-death process and Erlang distribution

      Builds on Probability spaces and conditional information, Stochastic processes filtrations and stopping times and Brownian motion and Gaussian increments.

    2. Order flow adverse selection and price impactComing soon

      Video 106

      signed order flow, adverse selection, informational spread, permanent impact, temporary impact and price response

      Builds on Regression and statistical identification, Stationarity autocovariance and ARMA models and Limit order books and queueing models.

    3. Transaction cost analysis and execution benchmarksComing soon

      Video 107

      transaction cost analysis, implementation shortfall, paper portfolio, arrival price, VWAP benchmark, opportunity cost and benchmark bias

      Builds on Returns and self-financing portfolio accounting, Turnover costs and strategy capacity and Order flow adverse selection and price impact.

    4. Optimal execution and the Almgren Chriss modelComing soon

      Video 108

      Almgren-Chriss model, inventory trajectory, impact cost, risk penalty, Euler-Lagrange equation, optimal execution schedule and execution urgency

      Builds on Convex optimization and duality, Mean variance optimization and efficient frontiers, Turnover costs and strategy capacity and Order flow adverse selection and price impact.

    5. Market making and inventory controlComing soon

      Video 109

      market making, inventory risk, certainty equivalent, reservation price, bid and ask quotes, inventory penalty and exponential fill intensity

      Builds on Stochastic processes filtrations and stopping times, Ito lemma and the stochastic chain rule, Dynamic allocation and stochastic control, Limit order books and queueing models and Order flow adverse selection and price impact.

    Differentiation, calibration and validation

    1. Differentiable pricing and automatic differentiationComing soon

      Video 110

      automatic differentiation, computational graph, reverse-mode adjoint, forward-mode differentiation, pathwise derivative, interchange of derivative and expectation and discontinuous digital payoff

      Builds on Multivariable calculus and Taylor expansions, Greeks and differential identities and Numerical conditioning and root finding.

    2. Calibration inverse problems and identifiabilityComing soon

      Video 111

      model calibration, inverse problem, weighted calibration loss, parameter identifiability, ridge selection, price Jacobian and out-of-sample hedge

      Builds on Convex optimization and duality, Statistical estimation and likelihood, Bootstrapping discount curves, Implied volatility and inverse pricing, Heston stochastic volatility, Numerical conditioning and root finding, Collateral discounting and multiple curves and Differentiable pricing and automatic differentiation.

    3. Model validation reproducibility and monitoringComing soon

      Video 112

      model validation, financial invariant, numerical convergence test, drift monitoring, reproducibility and model limitations

      Builds on Monte Carlo pricing and sampling error, SDE discretization and weak convergence, Finite difference PDE methods, Risk backtesting and statistical tests, Point in time financial data and sampling, Time series validation and overlapping labels and Calibration inverse problems and identifiability.