\[
% MathJax has no bm package; redefine \bm in terms of \boldsymbol, which it supports natively
\newcommand{\bm}[1]{\boldsymbol{#1}}
% --- Operators -----------------------------------------------------------
% Expectation operator
\DeclareMathOperator{\E}{E}
% Variance operator
\DeclareMathOperator{\Var}{Var}
% Covariance operator
\DeclareMathOperator{\Cov}{Cov}
% Correlation operator
\DeclareMathOperator{\Corr}{Corr}
% Rank operator (Spearman rank correlation)
\DeclareMathOperator{\Rank}{Rank}
% Skewness operator
\DeclareMathOperator{\Skewness}{Skewness}
% Kurtosis operator
\DeclareMathOperator{\Kurtosis}{Kurtosis}
% Difference/differential operator (upright d per ISO 80000-2)
\newcommand{\Diff}{\mathrm{d}}
% --- Risk Measures -------------------------------------------------------
% Value at Risk
\DeclareMathOperator{\VaR}{VaR}
% Expected Shortfall
\DeclareMathOperator{\ES}{ES}
% Marginal Value at Risk
\DeclareMathOperator{\MVaR}{MVaR}
% Component Value at Risk
\DeclareMathOperator{\CompVaR}{CVaR}
% Incremental Value at Risk
\DeclareMathOperator{\IVaR}{IVaR}
% Component Expected Shortfall
\DeclareMathOperator{\CompES}{CES}
% Weighted sensitivity for risk factor in FRTB SBM bucket
\DeclareMathOperator{\WS}{WS}
% Hedge-benefit ratio for FRTB DRC bucket
\DeclareMathOperator{\HBR}{HBR}
% Stress scenario risk measure for FRTB NMRF capital add-on
\DeclareMathOperator{\SES}{SES}
% Jump-to-default exposure for obligor (FRTB DRC)
\DeclareMathOperator{\JTD}{JTD}
% Loss given default for obligor (FRTB DRC)
\DeclareMathOperator{\LGD}{LGD}
% Default risk weight for obligor (FRTB DRC)
\DeclareMathOperator{\RW}{RW}
% ES ratio function
\newcommand{\ESratio}{\lambda}
% Arbitrary risk measure function (coherence axioms)
\newcommand{\RiskMeasure}{\varphi}
% Asset position (coherence axioms)
\newcommand{\Asset}{A}
% First named asset in examples
\newcommand{\AssetA}{A}
% Second named asset in examples
\newcommand{\AssetB}{B}
% Third named asset in examples
\newcommand{\AssetC}{C}
% Constant (risk measure axioms)
\newcommand{\Constant}{c}
% Bucket-level aggregate sensitivity in FRTB SBM
\newcommand{\BucketAgg}{S}
% Cross-bucket correlation in FRTB SBM
\newcommand{\CrossBucketCorr}{\gamma}
% Notional amount for an instrument (FRTB DRC and RRAO)
\newcommand{\Notional}{\text{Notional}}
% --- Distributions -------------------------------------------------------
% Binomial distribution
\DeclareMathOperator{\Binomial}{Binomial}
% Uniform distribution
\DeclareMathOperator{\Uniform}{Uniform}
% Normal distribution
\newcommand{\NormalDist}{\mathcal{N}}
% Student-t CDF
\newcommand{\StudentCDF}{t}
% Student-t PDF
\newcommand{\StudentPDF}{f}
% --- Returns -------------------------------------------------------------
% Price
\newcommand{\Price}{P}
% Simple (arithmetic) return
\newcommand{\SimpleReturns}{R}
% Log (compound) return
\newcommand{\CompoundReturns}{y}
% Matrix of compound returns
\newcommand{\ReturnMatrix}{\bm{y}}
% Dividend payment
\newcommand{\dividend}{d}
% --- Volatility ----------------------------------------------------------
% Realised variance
\DeclareMathOperator{\RealisedVar}{RV}
% Realised volatility (square root of realised variance)
\DeclareMathOperator{\RealisedVol}{RVol}
% Volatility
\newcommand{\Vol}{\sigma}
% GARCH intercept parameter
\newcommand{\GARCHconst}{\omega}
% ARCH coefficient (weight on past squared returns)
\newcommand{\ARCHcoeff}{\alpha}
% GARCH coefficient (weight on past variance)
\newcommand{\GARCHcoeff}{\beta}
% EWMA decay factor (typically 0.94)
\newcommand{\EWMAdecay}{\lambda}
% Standardized residual / error term
\newcommand{\StdNormal}{\epsilon}
% Leverage parameter in apARCH
\newcommand{\APARCHleverage}{\zeta}
% Power parameter in apARCH
\newcommand{\APARCHpower}{\delta}
% Degrees of freedom (Student-t)
\newcommand{\DOF}{\nu}
% GJR-GARCH leverage parameter
\newcommand{\GJRleverage}{\gamma}
% GARCH-X external regressor coefficient
\newcommand{\RegressorCoeff}{\xi}
% Mean (first moment)
\newcommand{\Mean}{\mu}
% Lag order in volatility models
\newcommand{\Lag}{L}
% Dummy variable (indicator in regressions)
\newcommand{\DummyVar}{D}
% --- Portfolio -----------------------------------------------------------
% Portfolio weight (scalar)
\newcommand{\weight}{w}
% Portfolio weight vector
\newcommand{\weights}{\bm{w}}
% Covariance matrix
\newcommand{\CovMatrix}{\bm{\Sigma}}
% Correlation matrix
\newcommand{\CorrMatrix}{\bm{C}}
% Correlation coefficient
\newcommand{\correlation}{\rho}
% Number of assets in portfolio
\newcommand{\NumberAssets}{K}
% Portfolio value
\newcommand{\PortfolioValue}{\vartheta}
% Asset beta (CAPM)
\newcommand{\AssetBeta}{\beta}
% Ledoit-Wolf shrinkage intensity
\newcommand{\ShrinkageIntensity}{\delta}
% Sample covariance matrix
\newcommand{\SampleCov}{\bm{S}}
% --- Time Series ---------------------------------------------------------
% Sample size
\newcommand{\SampleSize}{T}
% Generic count (number of lags, payments, etc.)
\newcommand{\Count}{N}
% Estimation window length
\newcommand{\EstWindow}{W_E}
% Testing window length
\newcommand{\TestWindow}{W_T}
% Stress window length
\newcommand{\StressWindow}{W_S}
% AR coefficient
\newcommand{\ARcoeff}{\phi}
% MA coefficient
\newcommand{\MAcoeff}{\psi}
% --- Probability ---------------------------------------------------------
% Probability level (e.g. 0.01 for 1% VaR)
\newcommand{\probability}{p}
% Cumulative sorted scenario weight
\newcommand{\CumScenarioWeight}{\Omega}
% p-quantile of the P and L distribution
\newcommand{\Quantile}{q}
% Quantile function
\newcommand{\QuantileFunction}{Q}
% Probability density function
\newcommand{\PDF}{f}
% Cumulative distribution function
\newcommand{\CDF}{F}
% Standard normal CDF
\newcommand{\NormalCDF}{\Phi}
% Inverse standard normal (quantile function)
\newcommand{\NormalQuantile}{\Phi^{-1}}
% Standard normal density function
\newcommand{\NormalPDF}{\phi}
% Likelihood function
\newcommand{\lik}{\mathcal{L}}
% Log-likelihood
\newcommand{\LogLikelihood}{\ell}
% Parameter vector (MLE estimation)
\newcommand{\ParamSet}{\theta}
% Parameter space
\newcommand{\ParamSpace}{\Theta}
% Fisher information matrix
\newcommand{\FisherInfo}{\mathcal{I}}
% Statistical power (Type II error rate)
\newcommand{\StatPower}{\beta}
% Significance level (CI coverage complement)
\newcommand{\SignifLevel}{\gamma}
% --- Options -------------------------------------------------------------
% Call option label
\newcommand{\CallOption}{\text{call}}
% Put option label
\newcommand{\PutOption}{\text{put}}
% Strike price
\newcommand{\Strike}{X}
% Risk-free interest rate
\newcommand{\RiskFree}{r_f}
% Option maturity (time to expiration)
\newcommand{\OptionMaturity}{\tau}
% Option delta
\newcommand{\OptionDelta}{\Delta}
% Option gamma
\newcommand{\OptionGamma}{\Gamma}
% Option vega
\newcommand{\Vega}{\mathcal{V}}
% Holding period
\newcommand{\HoldingPeriod}{H}
% Simulation profit/loss (signed; negative is a loss)
\newcommand{\ProfitLoss}{\Pi}
% Futures price
\newcommand{\Futures}{F}
% Hedge ratio
\newcommand{\HedgeRatio}{h}
% Units of basic asset held
\newcommand{\StockHolding}{x^b}
% Units of options held
\newcommand{\OptionHolding}{x^o}
% --- Interest Rates ------------------------------------------------------
% Bond convexity
\newcommand{\Convexity}{C}
% Interest rate / yield
\newcommand{\Yields}{r}
% Key rate change, basis points (scalar tenor component); used as \KeyRateChange_k or \KeyRateChange_{t,k}. EXCEPTION to the no-macro-to-macro-indirection rule, decided 2026-08-09: composed from \Yields on purpose, so a future change to the rate glyph propagates here without a follow-up edit. \Yields is defined earlier in generated output (same interest_rates category, TOML order), so expansion order is safe; verify this still holds if either entry is reordered.
\newcommand{\KeyRateChange}{\Delta \Yields^{\mathrm{bp}}}
% Key rate changes, basis points (vector); used bare or as \KeyRateChanges_t. EXCEPTION to the no-macro-to-macro-indirection rule, decided 2026-08-09: composed from \Yields on purpose, so a future change to the rate glyph propagates here without a follow-up edit. \bm{\Yields} depends on \bm/\boldsymbol correctly expanding a macro argument rather than a bare letter --- confirm this renders correctly the first time either consumer compiles it.
\newcommand{\KeyRateChanges}{\Delta \bm{\Yields}^{\mathrm{bp}}}
% Dollar value of a basis point
\newcommand{\DV}{\text{DV01}}
% Mean reversion speed (interest rate models)
\newcommand{\MeanReversion}{\varkappa}
% Modified duration
\newcommand{\ModDur}{D}
% Macaulay duration
\newcommand{\MacDur}{D_{\text{Mac}}}
% Effective duration
\newcommand{\EffDur}{D_{\text{eff}}}
% DV01 vector
\newcommand{\DVvec}{\bm{d}}
% Long-run mean rate (Vasicek/CIR/Hull-White)
\newcommand{\LongRunRate}{\theta}
% Hull-White time-varying drift
\newcommand{\HullWhiteDrift}{\varphi}
% Cash flow (coupon/principal payment)
\newcommand{\CashFlow}{c}
% --- Extreme Value -------------------------------------------------------
% Tail index (EVT, Pareto-type tails)
\newcommand{\TailIndex}{\iota}
% Shape parameter (xi = 1/iota)
\newcommand{\ShapeParam}{\varsigma}
% Threshold value (EVT)
\newcommand{\Threshold}{u}
% Extremal index (Leadbetter 1983)
\newcommand{\ExtremalIndex}{\theta}
% Multivariate extremal coefficient, theta in [1,d]. theta=1 complete tail dependence; theta=d asymptotic independence. Glyph collides with ExtremalIndex and ParamSet; disambiguated at source by macro name.
\newcommand{\ExtremalCoef}{\theta}
% GEV distribution function
\newcommand{\GEV}{\mathcal{H}}
% GPD distribution function
\newcommand{\GPD}{\mathcal{G}}
% GPD scale parameter
\newcommand{\GPDscale}{\beta}
% GPD scale parameter at a threshold u, distinct from the generic GPD scale
\newcommand{\GPDscaleAtThreshold}{\beta_u}
% Number of threshold exceedances (EVT)
\newcommand{\TailCount}{C}
% Sample maximum (block maxima, EVT)
\newcommand{\SampleMax}{M}
% Kurtosis value (realized, e.g. sample kurtosis)
\newcommand{\KurtosisVal}{\kappa}
% Pareto scaling constant (EVT)
\newcommand{\ParetoConst}{\mathcal{A}}
% Generic constant (EVT asymptotics)
\newcommand{\RemainderConst}{\mathcal{C}}
% Little-o asymptotic notation
\newcommand{\LittleO}{o}
% Moment order (EVT)
\newcommand{\MomentOrder}{\mathscr{m}}
% Doubled sample size (EVT block maxima)
\newcommand{\DoubleSample}{D}
% GEV normalizing location constant
\newcommand{\GEVloc}{a}
% GEV normalizing scale constant
\newcommand{\GEVscale}{b}
% Negative return (loss, EVT context)
\newcommand{\NegativeReturn}{L}
% --- Copulas -------------------------------------------------------------
% Copula function
\newcommand{\Copula}{\mathcal{C}}
% Copula density
\newcommand{\CopulaDensity}{c}
% Generator function (Archimedean copulas)
\newcommand{\generator}{\varphi}
% Copula dependence parameter
\newcommand{\CopulaParam}{\theta}
% Copula marginal parameter set
\newcommand{\MarginalParamSet}{\eta}
% Lower tail dependence coefficient
\newcommand{\LowerTailDep}{\lambda_L}
% Upper tail dependence coefficient
\newcommand{\UpperTailDep}{\lambda_U}
% Marginal CDF (second variable)
\newcommand{\MarginalCDF}{G}
% Joint density function
\newcommand{\JointDensity}{h}
% Marginal density function
\newcommand{\MarginalDensity}{g}
% Joint distribution function
\newcommand{\JointCDF}{H}
% Gaussian copula correlation parameter
\newcommand{\CopulaCorr}{\rho}
% --- Multivariate --------------------------------------------------------
% Diagonal matrix of conditional volatilities
\newcommand{\DiagVolD}{\bm{D}}
% Idiosyncratic variance matrix (diagonal, factor models)
\newcommand{\IdioVarMatrix}{\bm{\Psi}}
% Covariance matrix entry (two indices)
\newcommand{\CovElement}{\sigma}
% Auxiliary matrix in DCC dynamics
\newcommand{\DCCauxQ}{\bm{Q}}
% BEKK constant matrix
\newcommand{\BEKKconst}{\bm{\Omega}}
% BEKK ARCH parameter matrix
\newcommand{\BEKKarch}{\bm{A}}
% BEKK GARCH parameter matrix
\newcommand{\BEKKgarch}{\bm{B}}
% DCC rescaling diagonal matrix
\newcommand{\DCCrescaleZ}{\bm{Z}}
% DCC auxiliary matrix element
\newcommand{\DCCelement}{q}
% Factor loadings matrix (loadings/eigenvectors of the covariance matrix)
\newcommand{\FactorLoadings}{\bm{\Lambda}}
% Factor subscript label
\newcommand{\factor}{\text{factor}}
% PCA eigenvalue
\newcommand{\Eigenvalue}{\lambda}
% Number of factors in factor model
\newcommand{\NumberFactors}{m}
% Factor return vector
\newcommand{\FactorReturn}{\bm{g}}
% PCA factor score (scalar; not an eigenvector)
\newcommand{\PCAFactor}{F}
% BEKK cross-covariance coefficient
\newcommand{\BEKKcross}{\delta}
% Factor loading element
\newcommand{\FactorLoading}{\Lambda}
% DCC correlation persistence parameter
\newcommand{\DCCxi}{\xi}
% DCC news coefficient (weight on recent shocks)
\newcommand{\DCCzeta}{\zeta}
% --- Simulation ----------------------------------------------------------
% Number of simulation paths
\newcommand{\NumberSims}{B}
% Cholesky factor of covariance matrix
\newcommand{\Cholesky}{\bm{L}}
% Uniform random number
\newcommand{\UniformDraw}{\tilde{u}}
% --- Backtesting ---------------------------------------------------------
% Violation ratio (observed/expected violations)
\DeclareMathOperator{\ViolRatio}{VR}
% Stressed Value at Risk
\DeclareMathOperator{\StressedVaR}{SVaR}
% QLIKE loss function for variance forecasts
\DeclareMathOperator{\QLIKE}{QLIKE}
% Mean squared error (forecast loss function)
\DeclareMathOperator{\MSE}{MSE}
% Mean absolute error (forecast loss function)
\DeclareMathOperator{\MAE}{MAE}
% Markov transition probability
\newcommand{\TransProb}{\pi}
% Exception indicator: 1 if y_t < -VaR_t
\newcommand{\ExceptionInd}{\eta}
% Exception count
\newcommand{\ExceptionCount}{\upsilon}
% Probability Integral Transform
\newcommand{\ProbIntegralTransform}{\hat{u}}
% Quantile score (pinball loss function)
\newcommand{\QuantileScore}{\mathcal{S}}
% Kolmogorov-Smirnov test statistic
\newcommand{\KSstat}{D}
% Test statistic (generic base letter)
\newcommand{\TestStat}{J}
% --- General -------------------------------------------------------------
% Regression intercept
\newcommand{\RegressionIntercept}{a}
% Regression slope
\newcommand{\RegressionSlope}{\beta}
% Risk factor
\newcommand{\RiskFactor}{x}
% Forecast horizon
\newcommand{\ForecastHorizon}{h}
% Block length (bootstrap)
\newcommand{\BlockLength}{\ell}
% Capital tau variant
\newcommand{\DeliveryTime}{\Upsilon}
% Today's calendar time in years (option pricing input; distinct from trading-date index t)
\newcommand{\CalendarTime}{t^*}
% Probability measure
\newcommand{\ProbMeasure}{\mathbb{P}}
% Risk-neutral measure
\newcommand{\RiskNeutral}{\mathbb{Q}}
% Pricing function (Black-Scholes, bond pricing, etc.)
\newcommand{\PricingFn}{V}
% Indicator function
\newcommand{\Indicator}{\bm{1}}
% Vector of ones
\newcommand{\OnesVector}{\bm{1}}
% Risk factor sensitivity (first-order P&L)
\newcommand{\Sensitivity}{\delta}
% Second-order sensitivity (convexity/gamma)
\newcommand{\ConvexitySens}{\gamma}
% Output floor percentage (Basel)
\newcommand{\OutputFloor}{\alpha}
% --- Subscript Labels ----------------------------------------------------
% Subscript label: annualized
\newcommand{\Annual}{a}
% Subscript label: implied
\newcommand{\Implied}{I}
% Subscript label: portfolio
\newcommand{\Portfolio}{\pi}
% --- Text Abbreviations --------------------------------------------
% S&P 500 index
\newcommand{\SP}{\text{S\&P-500}}
% Student-t distribution (text)
\newcommand{\St}{\text{Student-t}}
% Geopolitical Risk index (Caldara and Iacoviello)
\newcommand{\GPR}{\text{GPR}}
\]
Version 5
These notes accompany Financial Risk Forecasting written by Jon Danielsson at the London School of Economics. You can find his blog and other writing on the topic of risk on his main webpage ModelsandRisk.org and his academic research on RiskResearch.org.
The book focuses on practical quantitative methods for forecasting market risk. This notebook provides implementations in three languages — R, Python and Julia — shown side by side so readers can follow along in whichever language they prefer. Each code example appears in tabbed panels that stay synchronised as you move through the chapters. The companion website contains additional code examples, including some in Matlab.
Financial institutions need to forecast and manage risk, and the authorities mandate risk management through frameworks like the Basel Capital Accords. Most major financial centres have adopted Basel-style regulations. The aspect relevant to this notebook is trading book risk. We do not discuss financial regulations here. See Chapter 13 on regulations for that material.
These notes do not derive the mathematics of risk forecasting, and do not teach programming from scratch. Better sources exist for both. The accompanying textbook and lecture slides carry the mathematics and statistics, the Programming part of this notebook reviews the language basics needed later, and Chapter 29 links to training material for R, Python and Julia.
What these notes do is put those pieces together — the specifications, the code and the data — and produce market risk forecasts from them.
These notes are written in Quarto. Thanks to Alvaro Aguirre, Jia Rong Fan and Qinxian Wu for assisting with this notebook.
Comments on these notes can be sent here.
Notation
The model chapters lean on a compact set of symbols, and they are used consistently from here to the end. The table below is the reference for them, and is worth a glance before the volatility and risk parts even if the rest of this page is skipped.
| \(\CompoundReturns_t\) |
Log return at time \(t\): \(\CompoundReturns_t = \log \Price_t - \log \Price_{t-1}\) |
| \(\SimpleReturns_t\) |
Simple return (used only when explicitly needed): \(\SimpleReturns_t = \Price_t/\Price_{t-1} - 1\) |
| \(\Vol_t\) |
Conditional volatility forecast for time \(t\) |
| \(\probability\) |
Tail probability (e.g., \(\probability = 0.01\) for 1% VaR); confidence level is \(1-\probability\) |
| \(\VaR(\probability)\) |
Value-at-Risk at probability \(\probability\), expressed as a positive loss magnitude |
| \(\ES(\probability)\) |
Expected Shortfall at probability \(\probability\), expressed as a positive loss magnitude |
| \(^\top\) |
Transpose of a vector or matrix |
In code, the parameter container par holds par$probability (R), par['probability'] (Python) or par["probability"] (Julia) for the tail probability.