Advanced Risk Management I Lecture 7 Non normal returns and historical simulation.

Slides:



Advertisements
Similar presentations
Value-at-Risk: A Risk Estimating Tool for Management
Advertisements

Credit Risk Plus.
Introduction CreditMetrics™ was launched by JP Morgan in 1997.
VAR METHODS. VAR  Portfolio theory: risk should be measure at the level of the portfolio  not single asset  Financial risk management before 1990 was.
TK 6413 / TK 5413 : ISLAMIC RISK MANAGEMENT TOPIC 6: VALUE AT RISK (VaR) 1.
XIV International Conference on Economic and Social Development, 2-5 April 2013, Moscow A new copula approach for high-dimensional real world portfolios.
Historical Simulation, Value-at-Risk, and Expected Shortfall
Advanced Risk Management I Lecture 7. Example In applications one typically takes one year of data and a 1% confidence interval If we assume to count.
Chapter 21 Value at Risk Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012.
Chapter 21 Value at Risk Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012.
Introduction to Algorithmic Trading Strategies Lecture 8 Risk Management Haksun Li
Behavioral Finance and Asset Pricing What effect does psychological bias (irrationality) have on asset demands and asset prices?
RISK VALUATION. Risk can be valued using : Derivatives Valuation –Using valuation method –Value the gain Risk Management Valuation –Using statistical.
Primbs, MS&E 345, Spring The Analysis of Volatility.
Non-Normal Distributions
CHAPTER 10 Overcoming VaR's Limitations. INTRODUCTION While VaR is the single best way to measure risk, it does have several limitations. The most pressing.
Market-Risk Measurement
Discrete-Event Simulation: A First Course Steve Park and Larry Leemis College of William and Mary.
Bootstrap in Finance Esther Ruiz and Maria Rosa Nieto (A. Rodríguez, J. Romo and L. Pascual) Department of Statistics UNIVERSIDAD CARLOS III DE MADRID.
Copyright K.Cuthbertson, D. Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture VaR:
KYIV SCHOOL OF ECONOMICS Financial Econometrics (2nd part): Introduction to Financial Time Series May 2011 Instructor: Maksym Obrizan Lecture notes III.
Market Risk VaR: Historical Simulation Approach
Copula functions Advanced Methods of Risk Management Umberto Cherubini.
Measuring market risk:
Stress testing and Extreme Value Theory By A V Vedpuriswar September 12, 2009.
Correlations and Copulas 1. Measures of Dependence 2 The risk can be split into two parts: the individual risks and the dependence structure between them.
Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull Chapter 18 Value at Risk.
Value at Risk.
Lecture 7: Simulations.
Risk Management and Financial Institutions 2e, Chapter 13, Copyright © John C. Hull 2009 Chapter 13 Market Risk VaR: Model- Building Approach 1.
Advanced Risk Management I Lecture 6 Non-linear portfolios.
Advanced Risk Management I Lecture 5 Value at Risk & co.
Financial Products and Markets Lecture 7. Risk measurement The key problem for the construction of a risk measurement system is then the joint distribution.
Advanced Risk Management I Lecture 1 Market Risk.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 1 Distributions and Copulas for Integrated Risk Management Elements.
Value at Risk Chapter 20 Value at Risk part 1 資管所 陳竑廷.
Downside Risk: Implications for Financial Management Robert Engle NYU Stern School of Business Tilburg April 22, 2004.
Topic 5. Measuring Credit Risk (Loan portfolio)
Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull Value at Risk Chapter 18.
Advanced Risk Management I Lecture 2. Cash flow analysis and mapping Securities in a portfolio are collected and analyzed one by one. Bonds are decomposed.
Chapter 12 Modeling the Yield Curve Dynamics FIXED-INCOME SECURITIES.
Value at Risk Chapter 16. The Question Being Asked in VaR “What loss level is such that we are X % confident it will not be exceeded in N business days?”
Market Risk VaR: Historical Simulation Approach N. Gershun.
1 A non-Parametric Measure of Expected Shortfall (ES) By Kostas Giannopoulos UAE University.
© The MathWorks, Inc. ® ® Monte Carlo Simulations using MATLAB Vincent Leclercq, Application engineer
Measurement of Market Risk. Market Risk Directional risk Relative value risk Price risk Liquidity risk Type of measurements –scenario analysis –statistical.
 Measures the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval  For example: ◦ If the VaR.
Value at Risk Chapter 20 Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008.
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 16.1 Value at Risk Chapter 16.
Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 14.1 Value at Risk Chapter 14.
Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Probabilistic Cash Flow Analysis.
OPTIONS PRICING AND HEDGING WITH GARCH.THE PRICING KERNEL.HULL AND WHITE.THE PLUG-IN ESTIMATOR AND GARCH GAMMA.ENGLE-MUSTAFA – IMPLIED GARCH.DUAN AND EXTENSIONS.ENGLE.
Lecture 8 Stephen G. Hall ARCH and GARCH. REFS A thorough introduction ‘ARCH Models’ Bollerslev T, Engle R F and Nelson D B Handbook of Econometrics vol.
Structural Models. 2 Source: Moody’s-KMV What do we learn from these plots? The volatility of a firm’s assets is a major determinant of its.
1 VaR Models VaR Models for Energy Commodities Parametric VaR Historical Simulation VaR Monte Carlo VaR VaR based on Volatility Adjusted.
Returns and stylized facts. Returns Objective: Use available information to say something about future returns.
Probabilistic Cash Flow Analysis
Types of risk Market risk
The Three Common Approaches for Calculating Value at Risk
5. Volatility, sensitivity and VaR
Financial Risk Management of Insurance Enterprises
Market-Risk Measurement
Centre of Computational Finance and Economic Agents
Ch8 Time Series Modeling
Market Risk VaR: Historical Simulation Approach
Types of risk Market risk
Market Risk VaR: Model-Building Approach
Lecture 2 – Monte Carlo method in finance
VaR Introduction I: Parametric VaR Tom Mills FinPricing
Advanced Risk Management II
Presentation transcript:

Advanced Risk Management I Lecture 7 Non normal returns and historical simulation

Non normality of returns The assumption of normality of of returns is typically not borne out by the data. The reason is evidence of –Asimmetry –Leptokurtosis Other casual evidence on non-normality –People make a living on that, so it must exist –If nornal distribution of retruns were normal the crash of 1987 would have a probability of 10 –160, almost zero…

Why non-normal? Leverage… One possible reason for non normality, particularly for equity and corporate bonds, is leverage. Take equity, of a firm whose asset value is V and debt is B. Limited liability implies that at maturity Equity = max(V(T) – B, 0) Notice that if at some time t the call option (equity) is at the money, the return is not normal.

Why not normal? Volatility Saying that a distribution is not normal amounts to saying that volatility is constant. Non normality may mean that variance either –Does not exist –It is a stochastic variable

Dynamic volatility The most usual approach to non normality amounts to assuming that the volatility changes in time. The famous example is represented by GARCH models h t =  +  shock 2 t-1 +  h t -1

Arch/Garch extensions In standard Arch/Garch models it is assumed that conditional distribution is normal, i.e. H(.) is the normal distribution In more advanced applications one may assume that H be nott normally distributed either. For example, it is assumed that it be Student-t or GED (generalised error distribution). Alternatively, one can assume non parametric conditonal distribution (semi-parametric Garch)

Volatility asymmetry A flow of GARCH model is that the response of the return to an exogenous shock is the same no matter what the sign of the shock. Possible solutions consist in –distinguishing the sign in the dynamic equation of volatility. Threshold-GARCH (TGARCH) h t =  +  shock 2 t-1 +  D shock 2 t-1 +  h t -1 D = 1 if shock is positive and zero otherwise. –modelling the log of volatility (EGARCH) log(h t ) =  + g (shock t-1 / h t -1 ) +  log( h t -1 ) with g(x) =  x +  (  x  - E(  x  )).

High frequency data For some markets high frequency data is available (transaction data or tick-by-tick). –Pros: possibility to analyze the price dynamics on very small time intervals –Cons: data may be noisy because of microstructure of financial markets. “Realised variance”: using intra-day statistics to represent variance, instead of the daily variation.

Subordinated stochastic processes Consider the sequence of log-variation of prices in a given price interval. The cumulated return R = r 1 + r 2 +… r i + …+ r N is a variable that depends on the stoochastic processes a) log-returns r i. b) the number of transactions N. R is a subordinated stochastic process and N is the subordinator. Clark (1973) shows that R is a fat-tail process. Volatility increases when the number of transactions increases, and it is then correlated with volumes.

Stochastic clock The fact that the number of transactions induces non normality of returns suggest the possibility to use a variable that, changing the pace of time, could restore normality. This variable is called stochastic clock. The technique of time change is nowadays one of the most used tools in mathematical finance.

Lévy processes Not only prices are recorded at time intervals that are not continuous. Price changes are also discrete, and change by tick movements of finite dimension. Fot this reason, a possible model of representing prices is by pure jumps. Mixed stochastic processes (diffusive and jumps) are known as Levy processes. Examples of Levy processes: Variance-Gamma models, CGMY (Carr-Geman-Madan-Yor) models.

Copula functions A function z = C(u,v) is called copula iff z, u and v are [0,1] C(0,v) = C(u,0) = 0, C(1,v) = v, C(u,1) = u C(u 2, v 2 ) – C(u 1, v 2 ) – C (u 2, v 1 ) – C (u 1, v 1 )  0 for all values u 2 > u 1 and v 2 > v 1 Sklar theorem: every joint diistribution can be written as a copula function taking marginal distributions as arguments and whatever copula function taking probabilities as arguments gives a joint distribution

Copula function: examples Two risks A and B with joint probability H(A,B) and marginal probability H a (A) and H b (B) H(A,B) = C(H a, H b ), and C is a copula function. Cases: 1) C ind (H a, H b ) = H a H b, independent risks 2) C max (H a, H b ) =min(H a,H b ) perfect positive dependence 3) C min (H a, H b ) =max(H a + H b –1,0) perfect negative dependence Imperfect dependence (Fréchet limits) max(H a + H b –1,0)  C(H a, H b )  min(H a,H b )

Risk measurement wth fat tails Addressing non-normality of returns calls for the solution of three problems –Compression data techniques –Choice of the information source –Choice of the model to be used to substitute the normal distribution.

Data compression First option: re-evaluation of the current portfolio on historical data and estimation and simulation of the distribution of losses Second option: estimation of the sensitivity to the most relevant risk factors of the assets and the portfolios. Third option: traditional technical techniques (principal components and factor models)

Distribution of the returns First option: choosing a new model, or a class of new models of distributions Second option: simulating the distribution using historical data Third option: determining extreme scenarios for the distribution.

Classical historical simulation Re-valuation of the portfolio on historical data –Every set of historical data represents a possible market scenario P&L computation under each scenario Sorting scenarios by dimension of losses –Empircal P& L distribution Quantile computation of the empirical distribution –I.e.. on 100 data the worst represents the 1% VaR

Histogram FIAT

Classical historical simulation Problems –Data may fail to be not i.i.d. –In particular, the distribution of future returns may vary with market conditions –High and low volatility periods may cluster (volatility clustering) Effects –Under or over-valuation of VaR.

Volatility clustering

Filtered historical simulation Barone-Adesi and Giannopoulos Barone-Adesi and Giannopoulos proposed a modification of the algorithm based on a filtering process of data. Filteres historical simulation –Re-valuation of the portfolio on historical data –Estimation of a Garch model on this series –Use of the estimate to filter data –Use of bootstrap techniques to simulate the evolution of returns and volatility

Filtered historical simulation: algorithm Step 1. Re-valuation of the portfolio on historical data, and P&L comnputation Step 2. Specification and estimation of a GARCH model. i.e.

Data filtering Step 3. Computing and saving the time series of residuals  t, for t = 0, 1, …,T Step 4. Computing and saving the time series of volatilities  t, for t = 1, …,T + 1 Step 5. Computing of the time seris of filtered innovation z t =  t /  t for t = 1, …,T

Bootstrap algorithm Step 6. Extract n filtered residuals from the time series z t = z(1), z(2), …,z(n) –n represents the unwinding period Step 7. Set the simulated return for time T + 1 equal to R T+1 = z(1)  T+1 =  T+1 Step 8. Compute volatility  T+2.

Step 9. Repeat step 7 and 8 computing R T+i = z(i)  T+i =  T+i for i = 2, …,n – 1 Step 10. Compute and save R T+n = z(n)  T+n =  T+n R T+1 + R T+2 + … + R T+i … + R T+n …first iteration

…repeart NITER times Step 11. Repeat steps from 6 to 10 a number NITER (i.e. 1000) of iterations. Step 12. Sort the scenarios by loss dimension Step 13. Compute the empirical quantile

Applications This methodology was applied to margin determination of the London Clearing House In a companion paper Barone-Adesi, Engle and Mancini, apply the same methodology to pricing options.