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A Stochastic Processes Toolkit for Risk Management

Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer, Fares Triki

arXiv:0812.4210v1q-fin.RM

TL;DR

Risk management needs to characterize the statistical features of risk-factor time series, including fat tails and mean reversion. The tutorial surveys stochastic-process models and methods for testing and estimating them from historical data; its examples include evidence of fat tails in FTSE100 returns and mean reversion in a tested series.

  • Problem

    Risk management needs to characterize risk-factor time series, whose statistical features include fat tails and mean reversion.

    Source: §1 · p. 3, §5 · p. 21

  • Method

    The tutorial presents stochastic-process models for these features and describes testing data and estimating model parameters from historical observations.

    Source: Check for mean reversion or stationarity · p. 4, §3 · p. 16

  • Results

    The FTSE100 log-return QQ-plot shows tail quantiles significantly larger than the normal distribution, while an Augmented Dickey-Fuller statistic of -3.7373 rejects a unit root at the 1% significance level for the tested series.

    Source: §2.2 · p. 7, §6 · p. 27

  • Takeaways & Limitations

    The calibrated Exponential Vasicek model is presented as a candidate for historical credit-spread estimation because it is mean reverting, strictly positive, right-skewed, and fat tailed.

    Source: §8 · p. 32

    Vasicek permits negative rates, while Exponential Vasicek can have an explosion problem in pricing models involving bank-account evaluation.

    Source: §7 · p. 29, §8 · p. 32

Abstract

from arXiv · show

In risk management it is desirable to grasp the essential statistical features of a time series representing a risk factor. This tutorial aims to introduce a number of different stochastic processes that can help in grasping the essential features of risk factors describing different asset classes or behaviors. This paper does not aim at being exhaustive, but gives examples and a feeling for practically implementable models allowing for stylised features in the data. The reader may also use these models as building blocks to build more complex models, although for a number of risk management applications the models developed here suffice for the first step in the quantitative analysis. The broad qualitative features addressed here are {fat tails} and {mean reversion}. We give some orientation on the initial choice of a suitable stochastic process and then explain how the process parameters can be estimated based on historical data. Once the process has been calibrated, typically through maximum likelihood estimation, one may simulate the risk factor and build future scenarios for the risky portfolio. On the terminal simulated distribution of the portfolio one may then single out several risk measures, although here we focus on the stochastic processes estimation preceding the simulation of the risk factors Finally, this first survey report focuses on single time series. Correlation or more generally dependence across risk factors, leading to multivariate processes modeling, will be addressed in future work.

1 Introduction

Risk analysis begins by identifying a risk factor’s statistical features, since different asset classes may call for different stochastic processes. This tutorial offers practical models for fat tails and mean reversion, plus guidance on testing, calibration, and use.

1 Introduction

  • Different risk factors may require different processes, so the financial nature of the series can guide whether to consider fat tails, mean reversion, or both.The tutorial presents the analysis as a starting point for risk-management or rating decisions, not a replacement for judgmental analysis or possible stress testing.Source: §1 · p. 3
  • The tutorial surveys implementable processes spanning a basic Brownian model, fat-tail models, mean-reverting models, and models combining both features, formulated in returns or levels.The models may serve as building blocks, and some suffice for a first step in quantitative analysis.Source: §1 · p. 3

Check for mean reversion or stationarity

  • For linear processes with normal shocks, AR features can indicate stationarity and support a mean-reverting starting assumption, but their absence does not rule out broader mean reversion.The report notes that nonlinear processes and non-Gaussian shocks may require broader notions of stationarity and ergodicity, which it does not address.Source: Check for mean reversion or stationarity · pp. 3–4
  • When AR features are absent, the report recommends checking for fat tails and comparing the goodness of fit of candidate models; predictive power can also be tested after calibration.Suggested diagnostics include QQ-plots and sample moments, with more rigorous normality and goodness-of-fit tests also noted.Source: Check for mean reversion or stationarity · p. 4

Prerequisites

  • The discussion assumes familiarity with probability theory, stochastic calculus, Itō’s formula, and basic coding skills.MATLAB code and examples using actual financial time series are provided for each section.Source: Prerequisites · p. 4

2 Modelling with Basic Stochastic Processes: GBM

This section introduces GBM as a basic asset-price model, examines its assumptions and parameter estimation using FTSE100 data, and presents functions for characterizing distributions and their moments. It also explains how moment-generating and characteristic functions simplify sums of independent variables, illustrating the result with normal variables.

2.1 The Geometric Brownian Motion

  • GBM models asset prices with constant drift and volatility; its independent Gaussian increments make log returns normal, asset prices log-normal, and simulation exact.The process is Markov: the current price suffices for computing probabilities of future prices.Source: §2 · p. 5, §2.1 · pp. 5–6

2.2 Parameter Estimation

  • FTSE100 log-return tail quantiles exceed the normal comparison, so GBM provides at best a rough approximation to the index.The QQ-plot compares historical return quantiles with standard-normal quantiles.Source: §2.2 · p. 7, Figure 2 · p. 7
  • GBM parameters are calibrated by maximum likelihood, with closed-form estimates available for the mean and variance of Gaussian iid log returns.The log-likelihood is used to avoid numerical problems from very small likelihood products.Source: §2.2 · pp. 7–9
  • Figure 3 is identified as showing the autocorrelation and partial autocorrelation functions for FTSE 100, but its caption reports no lag results or conclusion about GBM independence.Source: Figure 3 · p. 8
  • For GBM drift estimation, only the sample’s initial and final points provide relevant information, and estimating drift precisely is extremely difficult.The paper notes that drift estimation improves considerably for mean-reverting processes.Source: §2.2 · pp. 9–10
  • As the sample size increases, confidence intervals for parameter estimates tighten, and simulated sample-mean and variance distributions closely match their theoretical limits.The paper derives confidence intervals for the mean and variance under normal returns and compares simulated distributions with theoretical ones.Source: §2.2 · p. 10, Figure 4 · p. 11

2.3 Characteristic and Moment Generating Functions

  • Moment generating functions yield moments by differentiation but may not exist for some distributions; characteristic functions exist for every density and are Fourier transforms.The moment generating function is the Laplace transform of the probability density.Source: §2.3 · p. 12

2.4 Properties of Moment Generating and Characteristic Functions

  • Random variables with the same moment generating function have the same probability distribution, and the same property holds for characteristic functions.Source: §2.4 · p. 12
  • For independent variables, moment-generating and characteristic functions turn sums into products, avoiding direct density convolution in return and shock modelling.Working in function space allows multiplication before returning to density space.Source: §2.4 · p. 12
  • The sum of independent normal variables remains normal, with the means and variances adding.For means µ1 and µ2 and variances σ2 1 and σ2 2, the resulting mean is µ1 + µ2 and variance is σ2 1 + σ2 2.Source: §2.4 · p. 13

3 Fat Tails: GARCH Models

GARCH models represent changing, state-dependent volatility, which can produce fat-tailed returns and volatility clustering. NGARCH adds asymmetric responses to positive and negative returns and fits FTSE100 monthly returns better than constant-volatility GBM in the reported comparison.

3 Fat Tails: GARCH Models

  • GARCH’s state-dependent volatility produces a non-Gaussian unconditional return distribution with fat tails, unlike constant-volatility GBM.The report presents GARCH as one of several approaches to modeling fat tails.Source: §3 · pp. 13–14
  • GARCH volatility depends on past information, making high- and low-volatility periods tend to cluster.The model’s volatility is stochastic unconditionally but locally known given information from the previous time step.Source: §3 · pp. 13–14
  • NGARCH adds an asymmetry parameter so positive returns reduce, and negative returns increase, their subsequent impact on conditional variance.With γ = 0, the model reduces to symmetric GARCH(1,1).Source: §3 · p. 14
  • NGARCH parameters are estimated by numerically maximizing likelihood, using a normal density conditional on the preceding return and variance.The model has five parameters, and its conditional variance follows a recursive update.Source: §3 · pp. 16–17
  • NGARCH’s simulated FTSE100 monthly-return quantiles align better with historical returns than the constant-volatility model’s quantiles.The reported QQ-plot comparison shows a significantly better fit than the constant-volatility model.Source: Figure 6 · p. 14, §3 · p. 17

4 Fat Tails: Jump Diffusion Models

The jump-diffusion model extends GBM by adding random jumps whose arrivals follow a Poisson process. Its calibration treats returns as a mixture over possible jump counts; the reported FTSE100 comparison finds a better fit than simple GBM.

4 Fat Tails: Jump Diffusion Models

  • Jump arrivals follow a homogeneous Poisson process, with independent, identically distributed lognormal jump sizes independent of the Brownian motion.The Poisson process models the occurrence of rare events.Source: §4 · pp. 17–18
  • Conditional on the jump count, returns are normal; the unconditional return density mixes these conditional densities using Poisson probabilities.For small time steps, the mixture can be approximated by the zero- or one-jump cases.Source: §4 · pp. 19–20
  • The jump model fits FTSE100 monthly returns better than simple GBM in the reported QQ-plot comparison.The parameters are estimated by maximum likelihood before simulated returns are compared with historical returns.Source: §4 · p. 20, Figure 10 · p. 21
  • When μY exceeds one and σY² is small, log-return jumps tend to be markedly positive; adding negative jumps offers a more complex alternative.The report points to its later mean-reverting case for details on negative jumps and the resulting mixture features.Source: §4 · p. 20

5 Fat Tails Variance Gamma (VG) process

The Variance-Gamma process creates fat-tailed returns by time-changing Brownian motion with an independent increasing process, interpreted as random market activity time. The report presents its density and maximum-likelihood calibration, including an EUR/USD example.

5 Fat Tails Variance Gamma (VG) process

  • VG time-changes Brownian motion with an independent subordinator, making process time random and interpretable as market activity time.The subordinator’s expected increment is set equal to the corresponding calendar-time interval.Source: §5 · p. 21
  • With matched mean and variance, VG and GBM log-returns differ in higher-order moments.The matching conditions are (¯µ + ¯θ) = µ − σ²/2 and ¯σ² + ¯θ²ν = σ².Source: §5 · p. 22

5.1 Probability Density Function of the VG Process

  • VG returns have a density expressible as an infinite mixture of Gaussian densities weighted by Gamma densities.This follows by integrating out the Gamma-process subordinator.Source: §5.1 · p. 22

5.2 Calibration of the VG Process

  • Maximum-likelihood estimation calibrates the VG parameters from independent log-returns; the EUR/USD example finds VG more suitable than GBM.The report attributes the difference between the arithmetic Brownian motion and VG examples chiefly to the time-change parameter ν, with other parameters roughly similar.Source: §5.2 · pp. 23–25

6 The Mean Reverting Behaviour

The report treats mean reversion in linear Gaussian processes as a stationarity question, using AR(1) behavior and tests such as ADF. In its cleaned spread-index example, the ADF test rejects a unit root at the 1% level.

6 The Mean Reverting Behaviour

  • The tutorial says its introduced mean-reverting processes derive that behavior from an AR(1) feature in their discretized SDEs.It recommends checking the validity of the mean-reversion assumption before fitting a specific process.Source: §6 · p. 25
  • For AR(1), |α| < 1 is necessary and sufficient for stationarity, so testing stationarity tests mean reversion in this model.At α = 1, the process is a random walk with constant drift.Source: §6 · p. 25
  • The example uses a Merrill Lynch spread index as a proxy because CDS data for GLOBOXX are unavailable before 2004.The report says the proxy index appears mean reverting and relates spread mean reversion to the economic cycle.Source: §6 · p. 26
  • The ADF statistic of −3.7373 rejects the unit-root null at the 1% significance level for the tested spread series.The series was cleaned by removing innovations exceeding three times volatility before testing.Source: §6 · p. 27

7 Mean Reversion: The Vasicek Model

The Vasicek model is a mean-reverting Gaussian process whose long-run mean is θ and whose variance remains bounded. The section describes its discrete-time calibration and notes that negative-rate outcomes limit its suitability for positive quantities.

7 Mean Reversion: The Vasicek Model

  • A bad-quote outlier can create a sharp up-and-down movement that is interpreted as strong mean reversion and can influence the final estimate.Before outlier cleaning, the ADF statistic was -5.0226, which the report says confirms the outliers' effect on amplifying detected mean reversion.Source: §7 · p. 27
  • The Vasicek model assigns nonzero probability to negative rates, an unrealistic feature when modeling positive quantities such as interest rates or credit spreads.Source: §7 · p. 29
  • As time increases, the Vasicek process mean approaches θ while its variance remains bounded, yielding a stationary Gaussian long-run distribution.The process is formulated as an AR(1), which is stationary and mean reverting when 0 < b < 1.Source: §7 · p. 29
  • OLS regression of the discrete process on its lag estimates c, b, and δ, and these estimates are used to recover α, θ, and σ.The regression provides maximum-likelihood estimates for c, b, and δ.Source: §7 · p. 29

8 Mean Reversion: The Exponential Vasicek Model

The Exponential Vasicek model transforms a Vasicek process to obtain positive values and can produce fat-tailed outcomes. The section describes its calibration on log spreads and notes both its tail behavior and pricing limitations.

8 Mean Reversion: The Exponential Vasicek Model

  • Transforming a Vasicek process by exponentiation produces a positive process, addressing positivity while allowing fatter tails.The report contrasts this with squaring, which also ensures positivity but yields thinner tails than exponentiation.Source: §8 · p. 31
  • The Exponential Vasicek process is mean reverting, stationary in the long run, strictly positive, right-skewed, and fat-tailed for large positive values.Source: §8 · p. 32
  • For pricing applications, the model can have an explosion problem in the bank account, and bond or survival-probability valuation requires numerical methods.Source: §8 · pp. 32–33
  • Its log-spread series is calibrated by OLS on lagged values, with the fitted c, b, and δ used to infer α, θ, and σ.The example uses the EMU Corporate A rated 5-7Y index; reported estimates include α = 4.9701 and θ = 3.8307.Source: §8 · p. 33
  • In 50,000 simulated paths, an Exponential Vasicek path can occasionally rise above its 95th-percentile level.The simulation starts from the index's first observation in the first week of 1998.Source: Figure 18 · p. 32, §8 · p. 33

9 Mean Reversion: The CIR Model

The CIR model is a mean-reverting process for positive quantities, calibrated by maximum likelihood. The section describes simulation choices and contrasts its thinner tails and pricing tractability with the Exponential Vasicek model.

9 Mean Reversion: The CIR Model

  • CIR remains strictly positive when σ^2 ≤ 2αθ, a condition that makes the origin inaccessible.Source: §9 · p. 33
  • The CIR process can be simulated recursively, but a large time step risks approaching zero and producing numerical problems or complex values; drawing from its exact conditional distribution is more precise but more computationally expensive.The conditional distribution is noncentral chi-squared.Source: §9 · pp. 33, 35
  • CIR parameters are calibrated by maximum likelihood, initialized using the Vasicek AR(1) coefficient and the process's first two moments.The long-term mean is θ and the long-term variance is θσ^2/(2α).Source: §9 · p. 35
  • Compared with Exponential Vasicek, CIR has less fat tails but avoids the associated bank-account explosion; its closed-form bond and survival-probability solutions can make it preferable for pricing.The report also describes Exponential Vasicek as easier to estimate and simulate, with fatter tails.Source: §9 · p. 36

10 Mean Reversion and Fat Tails together

The section asks whether a process can combine short-run extreme movements with long-run mean reversion. It introduces a Vasicek diffusion with jumps and examines its distributional and mean-reverting properties.

10 Mean Reversion and Fat Tails together

  • The jump term is centered to give zero-mean shocks, while the resulting long-term mean can depend on the jump sizes and arrival intensity.The report considers Gaussian jump sizes and notes that other distributions are possible.Source: §10 · p. 37
  • The Vasicek jump model is a Gaussian mixture with fatter tails than basic Vasicek, allowing more extreme movements.For small time steps, the mixture weights reflect the arrival intensity of jumps.Source: §10 · p. 39
  • Its mean tends toward θY while its variance remains bounded, and removing jumps recovers the basic Vasicek mean and variance.The limiting variance is (σ2 + λ(µ2Y + σ2Y))/(2α).Source: §10 · p. 39
  • For markedly positive or negative jumps, the model recommends jump sizes with a non-near-zero mean and relatively small variance; a second process can model jumps of the opposite sign.The two-sided extension has a transition density that is a mixture of three Gaussian distributions.Source: §10 · p. 39
  • Figures 21–22 accompany the reported goodness of fit; Vasicek with positive-mean Gaussian jumps can be a weaker-tail alternative to exponential Vasicek, similar to CIR.Adjusting jump parameters or exponentiating the process can produce tails fatter than exponential Vasicek when needed.Source: Figure 21 · p. 38, §10 · p. 40, Figure 22 · p. 40

Notation

The notation section defines probability, distribution, process, estimation, and model-specific symbols used throughout the report.

Notation

  • The glossary defines expectation, variance, stochastic differential equations, distribution functions, quantiles, and Gaussian and Gamma distributions.It also lists Poisson probabilities and standard normal cumulative probabilities.Source: Notation · p. 41

Statistical Estimation

  • Statistical estimation notation identifies maximum likelihood estimation, ordinary least squares, parameters, likelihood, and log-likelihood.Source: Statistical Estimation · p. 41

Geometric Brownian Motion (GBM)

Fat Tails: GBM with Jumps, Garch Models

Time, discretisation, Maturities, Time Steps

Variance Gamma Models

  • Variance Gamma notation distinguishes calendar-time drift from market-activity-time drift and defines the activity-time transform and volatility.Source: Variance Gamma Models · p. 42

Vasicek and CIR Mean Reverting Models

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