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ARCH/GARCH Models ARCH/GARCH Models 1 Risk Management Risk: the quantifiable likelihood of loss or less-than-expected returns. In recent decades the field of financial risk management has undergone explosive development. Risk management has been described as “one of the most important innovations of the 20th century”. But risk management is not something new. 2 Risk Management José (Joseph) De La Vega was a Jewish merchant and poet residing in 17th century Amsterdam. There was a discussion between a lawyer, a trader and a philosopher in his book Confusion of Confusions. Their discussion contains what we now recognize as European options and a description of their use for risk management. 3 Risk Management There are three major risk types: market risk: the risk of a change in the value of a financial position due to changes in the value of the underlying assets. credit risk: the risk of not receiving promised repayments on outstanding investments. operational risk: the risk of losses resulting from inadequate or failed internal processes, people and systems, or from external events. 4 Risk Management VaR (Value at Risk), introduced by JPMorgan in the 1980s, is probably the most widely used risk measure in financial institutions. 5 Risk Management Given some confidence level � ∈ 0,1 . The VaR of our portfolio at the confidence level α is given by the smallest value � such that the probability that the loss � exceeds � is no larger than �. 6 The steps to calculate VaR σ t market Volatility days to be position measure forecasted VAR Level of confidence Report of potential 7 loss The success of VaR Is a result of the method used to estimate the risk The certainty of the report depends upon the type of model used to compute the volatility on which these forecast is based 8 Volatility 9 Volatility 10 11 Modeling Volatility with ARCH & GARCH In 1982, Robert Engle developed the autoregressive conditional heteroskedasticity (ARCH) models to model the time-varying volatility often observed in economical time series data. For this contribution, he won the 2003 Nobel Prize in Economics (*Clive Granger shared the prize for co- integration http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/2003/press.html). ARCH models assume the variance of the current error term or innovation to be a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations. In 1986, his doctoral student Tim Bollerslev developed the generalized ARCH models abbreviated as GARCH. 12 10/13/2000 07/06/2000 13 03/21/2000 12/02/1999 08/25/1999 05/12/1999 02/01/1999 10/21/1998 07/08/1998 03/23/1998 12/11/1997 09/04/1997 05/26/1997 02/13/1997 11/05/1996 whith (1,1) Garch 07/30/1996 04/18/1996 01/10/1996 09/28/1995 Volatility of Merval Index modelling 06/22/1995 03/10/1995 12/01/1994 0 2 8 6 4 10 12 16 14 Time series 14 Time series-ARCH (*Note, Zt can be other white noise, no need to be Gaussian) Let �* be N(0,1). The process �* is an ARCH(q) process if it is stationary and if it satisfies, for all � and some strictly positive-valued process �*, the equations �* = �*�* 5 / / �* = �0 + 2 �3�*43 367 Where �0 > 0 and �3 ≥ 0, � = 1, . , �. Note: �� is usually the error term in a time series regression model! 15 Time series-ARCH ARCH(q) has some useful properties. For simplicity, we will show them in ARCH(1). Without loss of generality, let a ARCH(1) process be represented by / �* = �* �0 + �7�*47 Conditional Mean / / � �* �*47 = � �* �0 + �7�*47 �*47 = � �* �*47 �0 + �7�*47 = 0 Unconditional Mean / / � �* = � �* �0 + �7�*47 = � �* � �0 + �7�*47 = 0 So �* have mean zero 16 Time series-ARCH 17 Time series-ARCH DE �* have unconditional variance given by ��� �* = 74DF Proof 1 (use the law of total variance): ��� �* = � ��� �* �*47 + ��� � �* �*47 / = � �0 + �7�*47 + ��� 0 / = �0 + �7 � �*47 = �0 + �7���(�*47) Because it is stationary, ��� �* = ���(�*47). DE So ��� �* = . 74DF 18 Time series-ARCH Proof 2: Lemma: Law of Iterated Expectations Let Ω7 and Ω/ be two sets of random variables such that Ω7 ⊆ Ω/. Let � be a scalar random variable. Then � � Ω7 = � � � Ω/ Ω7 / / ��� �* �*4/ = � �* �*4/ = � � �* �*47 �*4/ / / / = �0 + �7� �*47 �*4/ = �0 + �0�7 + �7 �*4/ / L / ��� �* �*4L = �0 + �0�7 + �0�7 + �7 �*4L … / / ��� �* = � �* = � � … � �* �*47 … �7 *47 * / = �0 1 + ⋯ + �7 + �7 �0 DE Since�7 < 1, as � → ∞, ��� �* = 74DF 19 Time series-ARCH fatter tails The unconditional distribution of Xt is leptokurtic, it is easy to show. T / �(�* ) 1 − �7 ���� �* = / / = 3 / > 3 [�(�* )] 1 − 3�7 Proof: T T T T � �* = � � �* �*47 = � �* � �* �*47 T / / = � �* � �0 + �7�*47 / / / T = 3(�0 + 2�0�7� �*47 + �7 �(�*47)) Z T DE 7[DF / DE So � �* = 3 Z. Also, � �* = 74DF 74LDF 74DF �(�T) 1 − �/ ���� � = * = 3 7 * [�(�/)]/ 1 − 3�/ * 7 20 Time series-ARCH The unconditional distribution of �* is leptokurtic, T / �(�* ) 1 − �7 ���� �* = / / = 3 / > 3 [�(�* )] 1 − 3�7 The curvature is high in the middle of the distribution and tails are fatter than those of a normal distribution, which is frequently observed in financial markets. 21 Time series-ARCH For ARCH(1), we can rewrite it as / / / / �* = �* �* = �0 + �7�*47 + �* where / / �* = �* �* − 1 T �(�* ) is finite, then it is an AR(1) process for / �* . / / Another perspective: �(�* �*47 = �0 + �7�*47 The above is simply the optimal forecast of / �* if it follows an AR(1) process 22 Time series-EWMA Before introducing GARCH, we discuss the EWMA (exponentially weighted moving average) model / / / �* = ��*47 + 1 − � �*47 where � is a constant between 0 and 1. The EWMA approach has the attractive feature that relatively little data need to be stored. 23 Time series-EWMA / We substitute for �*47,and then keep doing it for � steps ^ / 347 / ^ / �* = 1 − � 2 � �*43 + � �*4^ 367 ^ / For large �, the term � �*4^is sufficiently small to be ignored, so it decreases exponentially. 24 Time series-EWMA The EWMA approach is designed to track changes in the volatility. The value of � governs how responsive the estimate of the daily volatility is to the most recent daily percentage change. For example, the RiskMetrics database, which was invented and then published by JPMorgan in 1994, uses the EWMA model with � = 0.94 for updating daily volatility estimates. 25 Time series-GARCH The GARCH processes are generalized ARCH processes in / the sense that the squared volatility �* is allowed to depend on previous squared volatilities, as well as previous squared values of the process. 26 Time series-GARCH (*Note, Zt can be other white noise, no need to be Gaussian) Let �* be N(0,1). The process �* is a GARCH(p, q) process if it is stationary and if it satisfies, for all � and some strictly positive-valued process �*, the equations �* = �*�* 5 c / / / �* = �0 + 2 �3�*43 + 2 �b �*4b 367 b67 Where �0 > 0 and �3 ≥ 0, � = 1, . , �, �b ≥ 0, � = 1, . , �. Note: �� is usually the error term in a time series regression model! 27 Time series-GARCH GARCH models also have some important properties. Like ARCH, we show them in GARCH(1,1). The GARCH(1, 1) process is a covariance-stationary white noise process if and only if �1 + � < 1 . The variance of the covariance-stationary process is given by �0/(1 − �1 − �). In GARCH(1,1), the distribution of �* is also mostly leptokurtic – but can be normal. / 1 − �1 + � ���� �* = 3 / / ≥ 3 1 − �1 + � − 2�7 28 Time series-GARCH We can rewrite the GARCH(1,1) as / / / / �* = �* �* = �0 + �1 + � �*47 − ��*47 + �* where / / �* = �* �* − 1 T 2 �(�* ) is finite, then it is an ARMA(1,1) process for Xt . 29 Time series-GARCH The equation for GARCH(1,1) can be rewritten as / / / �* = ��h + �0 + �1�*47 + ��*47 where � + �1 + � = 1. The EWMA model is a special case of GARCH(1,1) where � = 0, �1 = 1 − �, � = � 30 Time series-GARCH The GARCH (1,1) model recognizes that over time the variance tends to get pulled back to a long-run average level of �h . / Assume we have known �* = �, if � > �h, then this expectation is negative 2 2 E[σ t+1 −V | It ] = E[γ VL +α1Xt + βV −V | It ] 2 < E[γ V +α1Xt + βV −V | It ] 2 = E[−α1V +α1Xt | It ] = −α1V +α1V = 0 31 Time series-GARCH If � < �h, then this expectation is positive 2 2 E[σ t+1 −V | It ] = E[γ VL +α1Xt + βV −V | It ] 2 > E[γ V +α1Xt + βV −V | It ] 2 = E[−α1V +α1Xt | It ] = −α1V +α1V = 0 This is called mean reversion. 32 Time series-(ARMA-GARCH) In the real world, the return processes maybe stationary, so we combine the ARMA model and the GARCH model, where we use ARMA to fit the mean and GARCH to fit the variance. For example, ARMA(1,1)-GARCH(1,1) Xt = µ +φXt−1 + εt +θεt−1 εt = σ t Zt σ 2 = α +α ε 2 + βσ 2 t 0 1 t−1 t−1 33 Time Series in R Data from Starbucks Corporation (SBUX) 34 Program Preparation Packages: >require(quantmod): specify, build, trade and analyze quantitative financial trading strategies >require(forecast): methods and tools for displaying and analyzing univariate time series forecasts >require(urca): unit root and cointegration tests encountered in applied econometric analysis are implemented >require(tseries): package for time series analysis and computational finance >require(fGarch): environment for teaching ‘Financial Engineering and Computational Finance’ 35 Introduction >getSymbols('SBUX') >chartSeries(SBUX,subset='2009::2013') 36 Method of Modeling >ret=na.omit(diff(log(SBUX$SBUX.Close))) >plot(r, main='Time plot of the daily logged return of SBUX') 37 KPSS test KPSS tests are used for testing a null hypothesis that an observable time series is stationary around a deterministic trend.
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