Econometrics of Volatility

Transcription

1 Econometrics of Volatility Siddharth Arora Mathematical Institute, University of Oxford Hilary Term, 2017

2 Course Outline Session I Basics of volatility, estimating volatility Session II ARCH, GARCH modelling approaches Session III Value at Risk (VaR), Technical analysis These lecture notes are based on the material prepared by J Taylor and P McSharry It is good to have an end to journey toward; but it is the journey that matters, in the end E. Hemingway 2

3 Volatility Volatility is a measure of the variability of the price of a financial instrument over time. Historical volatility is calculated from the time series of past market prices. For example, one could calculate the daily returns and then use the standard deviation of these returns as a measure of the historical volatility. Fragility is the quality of things that are vulnerable to volatility - N.N. Taleb 3

4 Annualized volatility In order to create a standard, it is common practice to estimate an annualized measure of volatility. This is a useful approach as the annualized volatility can then be directly compared with the annualized return for that particular instrument. If we are subjected to high volatility, we would expect a high level of return. 4

5 Volatility and direction Volatility does not covey any information about the likely direction of price changes. It focuses on the dispersion of returns since the standard deviation is computed using the squares of the returns. Negative and positive returns have the same effect on the volatility measure. 5

6 Volatility and risk Two instruments with different volatilities may have the same expected return. In this sense, volatility warns us about the size of fluctuations that we might expect. A financial instrument with higher volatility will have larger swings in values over a given period of time and can therefore be said to be more risky. 6

7 Risk and Reward 7

8 Volatility sizing Many investors will wish to construct a portfolio of financial instruments. In order to ensure that each asset in the portfolio represents an equal level of risk, the volatility can be used to infer appropriate weights. This is the motivation behind the 60%-40% split for equities and bonds since equities typically have a higher level of volatility. 8

9 Retirement A rule of thumb for the split between equities and bonds is to invest your age in bonds and the remainder in equities. This way, the overall portfolio becomes less volatile as your approach retirement. When saving for retirement, high volatility results in a wider distribution of possible final portfolio values. 9

10 Trading Traders attempt to time the market in order to perform better than a buy and hold long-term strategy. Volatility presents opportunities to buy assets cheaply and sell when overpriced. Volatility is particularly important for mean-reversion strategies which rely on fluctuations in prices over relatively short periods of time. 10

11 Idealized risk versus return 11

12 Portfolio allocation: reward/risk 12

13 Volatility patterns? Shiller (1987) investigates volatility in industrial production, short term interest rates (commercial paper), price level (producer price index), and housing starts. First, volatility seems to change dramatically through time for typical financial and macroeconomic variables. Second, there seem to be as many patterns of volatility changes as there are variables explored. Volatility shows no reliable uptrend through time 13

14 Causes of volatility Potential causes of volatility include: Mergers and acquisitions, IPOs Analysts reports and credit ratings Quarterly reports Investor sentiment and confidence Booming and busting bubbles Macroeconomic figures National and international economic policy Economic and political crises Natural disasters, wars and terrorism events 14

15 Volatility 15

16 Estimating volatility There are a variety of methods and approaches for estimating volatility. These fall into two broad categories: Implied volatility: refers to the market's assessment of future volatility. Realized volatility (historical volatility): historical deviations from price usually computed using the standard deviation and therefore measuring what actually happened in the past. 16

17 Implied volatility The implied volatility of an option contract is the volatility implied by the market price of the option based on an option pricing model (such as Black-Scholes). For a given pricing model, the implied volatility is that value for volatility which generates a theoretical value for the option equal to the current market price. Implied volatility can be viewed as the market s estimate of risk. 17

18 VIX VIX is the Chicago Board Options Exchange Volatility Index, a popular measure of the implied volatility of S&P 500 index options It is known by some as the fear index as it represents one measure of the market's expectation of volatility over the next 30 day period The formula uses a kernel-smoothed estimator that takes as inputs the current market prices for all out-of-the-money calls and puts for the front month and second month expirations 18

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20 Trading volatility The VIX is a useful index and can be used for making trading decisions. But suppose we have a view about future volatility, can we trade it directly? The answer is yes and this opens up a new type of asset class. One approach is to use options. 20

21 Estimating volatility Volatility clustering Implied volatility Stochastic volatility Exponential smoothing GARCH Leverage and asymmetry Nonlinear volatility models 21

22 Volatility Volatility usually refers to the standard deviation of the logarithmic returns of a financial instrument with a specific time horizon It is frequently employed as a proxy for the risk associated with a particular instrument As the definition is dependent on the particular time horizon, volatility is often expressed in annualised terms 22

23 S&P 500 Index 23

24 Volatility of a random walk For a financial instrument whose price follows a random walk, the volatility increases as time increases This implies that as time increases, there is an increasing probability that the instrument's price will be farther away from the initial price Under the random walk model, volatility increases with the square-root of time Intuitively, volatility should not be expected to increase linearly with time because some fluctuations will cancel each other out 24

25 High frequency trading Volatility is sometimes seen as a negative characteristic since it quantifies uncertainty and risk For high frequency traders, volatile markets are attractive since shorting on the peaks and buying on the lows can generate positive returns The greater the volatility, the greater are the opportunities for such trading strategies This approach is in contrast to the long term investment view of buy and hold In today's markets, it is also possible to trade volatility directly, through the use of derivative securities such as options and variance swaps 25

26 Forecasting volatility In the following we look at time series analysis techniques as a means of forecasting volatility. We start from simple estimators and eventually increase the complexity of the models. Both linear and nonlinear model specifications will be considered. 26

27 Forecast performance Performance of VIX (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep Volatility is measured as the standard deviation of S&P500 one-day returns over a month's period. 27

28 Traditional volatility forecasting The simplest model or forecast for volatility is given by the simple moving average over a window of the previous n periods: This provides a benchmark volatility forecast against which to test other volatility models 28

29 Exponentially weighted moving average An exponentially weighted moving average of volatility is given by where 0<α<1. RiskMetrics (1996) suggest using α = 0.06 for daily returns which corresponds to a EWMA window of 1/α or approximately 17 days. 29

30 S&P 500 EWMA volatility 30

31 Daily price information The information captured on a daily basis about each financial instrument consists of the open, high, low and close prices. An open-high-low-close chart (also OHLC chart, or simply bar chart) illustrates price movements over time. The closing price is usually associated with the largest volume and is therefore taken to be most representative for price discovery. 31

32 Japanese candle sticks Japanese candle sticks convey more information than OHLC charts. They display the absolute values of the open, high, low, and closing price for a given period. However they also show how those prices are relative to the prior periods' prices, so one can tell by looking at one bar if the price action is higher or lower than the prior one. 32

33 The range The range is the difference between the high and low price on any given day. The moving average, for example over ten days, of the range can be used as a measure of volatility. As this uses intraday price information, one might imagine that it would offer a superior estimate of volatility. 33

34 True range J. Welles Wilder (1978) developed the indicators known as the true range and average true range as measures of volatility. The true range accounting for the gap (sharp change in price, up or down, with no trading in between) and more accurately measures the daily volatility than was possible by using the simple range calculation. 34

35 True range estimate True range is the largest value found by solving the following three equations: (1) TR t = H t L t (2) TR t = H t C t-1 (absolute values) (3) TR t = L t C t-1 (absolute values) TR represents the true range H t represents today's high L t represents today's low C t-1 represents yesterday's close If the market has gapped higher, equation (2) will accurately show the volatility of the day as measured from the high to the previous close. Subtracting the previous close from the day's low, as done in equation (3), will account for days that open with a gap down 35

36 Average true range The average true range (ATR) is an exponential moving average of the true range. Wilder used a 14-day ATR to explain the concept. Traders can use shorter or longer timeframes based on their trading preferences. Longer timeframes will be slower and will likely lead to fewer trading signals, while shorter timeframes will increase trading activity. 36

37 Parkinson (1980) Instead of using closing prices this estimator uses the high and low prices: One drawback of this estimator is that it assumes continuous trading, hence it underestimates the volatility as potential movements when the market is shut are ignored. 37

38 Garman and Klass (1980) This is an extension of the Parkinson estimator which includes opening and closing prices (if opening prices are not available, the close from the previous day can be used instead). As overnight jumps are ignored this estimator underestimates the volatility 38

39 Rogers and Satchell (1991) Previous volatility estimators assume the average return (or drift) is zero. Securities which have a drift, or non-zero mean, require a more sophisticated measure of volatility. The Rogers-Satchell estimator is able to properly measure volatility for such securities: It does not, however, handle jumps, and therefore it underestimates the volatility 39

40 Garman-Klass-Yang-Zhang Yang-Zhang modified the Garman-Klass volatility measure in order to let it handle jumps: The measure does assume a zero drift, hence it will overestimate the volatility if a security has a non-zero mean return. 40

41 Yang and Zhang (2000) In 2000 Yang-Zhang created a volatility measure that handles both opening jumps and drift. It is the sum of the overnight volatility (close-to-open volatility) and a weighted average of the Rogers-Satchell volatility and the open-to-close volatility: 41

42 Summary of volatility estimators The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic - Ronald Fisher 42

43 Modelling volatility Conventional forecasting methods produce estimates for expected values of a variable In finance and economics, need also to forecast variance, e.g. option pricing, risk analysis, portfolio optimisation, trading strategies In traditional time series modelling, ε t is assumed to have constant variance (homoskedastic) y t+k = a o + a 1 X t + ε t y t+k = b o + b 1 y t + ε t However, many economic and financial series exhibit periods of unusually large volatility followed by periods of relative tranquillity 43

44 Dollar/Sterling exchange rate Heteroskedasticity D(DO LLAR) 44

45 Autoregressive conditional heteroskedasticity - Introduction Volatility clustering is often evident in financial returns - large returns more likely to be followed by large returns of either sign than by small returns This suggests that, instead of causal modelling, univariate time series modelling may be useful Autoregressive conditional heteroskedasticity (ARCH) is a univariate modelling approach to volatility forecasting 45

46 Autoregressive conditional heteroskedasticity (ARCH) Engle (1982) introduced the ARCH model which describes the current variance as a linear function of previous squared returns Consider returns ε t = σ t z t with z t ~NID(0,1) where σ t 2 = α 0 + q 2 α i ε t i i=1 where α 0 > 0, and α i 0 i > 0 ARCH can be estimated using ordinary least squares 46

47 Modelling Methodology Test for ARCH effects Identify model to be tentatively entertained Estimate the parameters of the tentative model Diagnostic checking. Is the model adequate? No Yes Use the model for forecasting JWT

48 GARCH Bollerslev (1986) introduced generalized autoregressive conditional heteroskedasticity (GARCH): GARCH(p,q) has p GARCH terms and q ARCH terms: σ t 2 = α 0 + q 2 α i ε t i + p 2 β j σ t j i=1 j=1 q The parameter constraint i=1 p α i + j=1 β i < 1 is often employed Characteristic of GARCH is that variance persists at new levels for several periods (it tends to evolve 48

49 GARCH and ARMA As σ t 2 = E[ε t 2 ], we can write ε t 2 = σ t 2 + ϑ t, where ϑ t is white noise Substituting for σ t 2 in the GARCH model gives: m=max(p,q) p ε t 2 = α (α i +β i )ε t i + ϑ t β j ϑ t j i=1 j=1 This demonstrates that in GARCH(p,q), ε t 2, follows an ARMA(p,q) process ACF and PACF of squared residuals can thus be used to identify the order of the GARCH(p,q) process in a similar way to ARIMA modelling 49

50 GARCH(1,1) GARCH(1,1) is generally sufficient to describe the volatility of most financial instruments: σ t = α 0 + αε t 1 + βσ t 1 Large values of β tend to generate persistent volatility with shocks taking a long time to decay Large values of α increase the sensitivity such that the volatility reacts quickly to shocks 50

51 IGARCH Integrated generalized autoregressive conditional heteroskedasticity (IGARCH) is a restricted version of the GARCH model: q i=1 α i + p j=1 β i = 1 This constraint produces a unit root in the ARMA process whereby σ t 2 is not mean reverting In other words, σ t 2 is non-stationary and will behave like a random walk 51

52 IGARCH(1,1) IGARCH(1,1) can be written as: σ t = α 0 + αε t 1 + (1 α)σ t 1 If the intercept, α 0, is set to zero, IGARCH(1,1) is equivalent to EWMA 52

53 News impact curve Engle & Ng (1993) suggested comparing GARCH models by investigating their news impact curves The news impact curve quantifies the implied relationship between the past return, ε t-1, and the current volatility estimate, σ t All past conditional volatilities are held constant by setting them equal to the unconditional volatility 53

54 News impact curve There is evidence of asymmetry in stock price behaviour downward movements in market (negative shocks) seem to increase volatility more than upward movements (positive shocks) of same size 54

55 GJRGARCH Glosten, Jagannathan & Runkle (1993) proposed GJRGARCH, an asymmetric GARCH model. GJRGARCH(1,1) is σ t 2 = α ε t 1 > α 1 ε t ε t 1 > 0 γ 1 ε t 1 + β 1 σ t 1 1[ ] is an indicator function, equal 1 if the argument is true and zero otherwise The asymmetry arises when α 1 > γ 1, corresponding to the leverage effect 55

56 New impact curves 56

57 EGARCH EGARCH is another specification for modelling asymmetric dependence of volatility on past returns Negative coefficients were frequently found for GARCH models. This is a problem as it can produce negative variance. An alternative to imposing non-negativity constraints is a log GARCH model ln(σ 2 2 t ) = α 0 + α 1 ln(ε t 1 ) + β 1 ln(σ t 1 2 ) σ t 2 is derived by taking exponential of both sides which is strictly positive 57

58 Model identification and estimation In GARCH(p,q), ε t 2, follows an ARMA(p,q) process m=max(p,q) p ε t 2 = α (α i +β i )ε t i + ϑ t β j ϑ t j i=1 j=1 ACF and PACF of the squared residuals can then be used to identify m and p (and therefore q). E.g. ARMA(2,0) in squared residuals suggests GARCH(0,2) (i.e. ARCH (2)) 58

59 Model estimation Consider a model: y t = a 1 y t 1 + ε t + b 1 ε t 1 ε t NID(0, σ t ) σ t = α 0 + α 1 ε t 1 + β 1 σ t 1 Maximum likelihood is used to estimate the model parameters A 2-stage approach estimating ARMA parameters first and then using squared residuals to estimate GARCH parameters is weak. (Heteroskedasticity of residuals causes problems for ARMA estimation.) 59

60 GARCH(1,1) volatility forecasts DLOG(FTA) *volatility -1.96*volatility 60

61 Model evaluation Check t-stats of parameter estimates SBC and AIC assess fit of both ARIMA and GARCH parts of the model GARCH modelling only aims to fit a model to heteroskedasticity, it does not aim to eliminate it. So residuals will still exhibit heteroskedasticity. However, standardized residuals, ε i σ i, should be white noise. Check correlogram of ε i σ i and correlogram of ( ε i σ i ) 2 for remaining ARCH effects Lagrange multiplier (LM) tests can also be applied to standardized residuals. Histogram should be closer to normal than that of residuals considered earlier. 61

62 Summary In finance and economics, we often need to estimate variances. Volatility clustering is often evident in returns, so time series modelling may be useful. Autocorrelation function of squared residuals will indicate whether error is an ARCH process. Three stages of ARCH modelling: Identification time series plot, ACF, PACF Estimation maximum likelihood Diagnostic evaluation standardized residuals: correlogram of ε i σ i and correlogram of ( ε i σ i ) 2, LM test 62

63 Risk 63

64 Value at Risk (VaR) VaR measures the risk of loss on a specific portfolio of financial assets. For a particular portfolio, probability and time horizon, VaR is defined as a threshold value such that the probability that the loss on the portfolio over the given time horizon exceeds this value For example: If a portfolio of stocks has a one-day 5% VaR of $1 million, there is a 0.05 probability that the portfolio will fall in value by more than $1 million over a one day period Informally, a loss of $1 million or more on this portfolio is expected on 1 day out of 20 days 64

65 Value at Risk (VaR) Source: 65

66 VaR historical Source: investopedia.com 66

67 VaR historical 5% VaR: with 95% confidence, we expect that our worst daily loss will not exceed 4%. If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%). 1% VaR: with 99% confidence, we expect that the worst daily loss will not exceed 7%. If we invest $100, we are 99% confident that our worst daily loss will not exceed $7. 67

68 VaR normal distribution Source: investopedia.com 68

69 Utility functions In finance, utility reflects the satisfaction gained from obtaining a quantity of money The utility function often used in portfolio construction is: u x = 1 exp( kx) where k is a risk-aversion constant and x isthe amount of money Other utility function include the power function u x = x a and the log utility, u x = log(x) 69

70 Utility functions 70

71 Portfolio risk When constructing portfolios, risk aversion is usually viewed as the additional marginal reward an investor requires to accept additional risk The risk is typically measured as the standard deviation of the time series of returns, σ Other measures of risk are the number and duration of drawdown periods when the returns are negative 71

72 Portfolio optimization The aim is to construct an optimal portfolio which displays the lowest possible level of risk for a specified level of return Risk is usually measured by the standard deviation of the returns Limitations include: (i) assumption that asset returns are (jointly) normally distributed random variables and (ii) risk can be measured by the standard deviation 72

73 Finance: risk and reward 73

74 Investment portfolio 74

75 Sharpe ratio The Sharpe ratio is a measure of the excess return (or risk premium) per unit of risk in an investment asset or a trading strategy: SR = μ σ where μ is the mean and σis the standard deviation of the return time series Higher Sharpe ratio is preferable 75

76 Risk management An estimate of the conditional distribution of financial returns is important for risk management, as well as option pricing and portfolio management VaR has become the standard approach to assessing market risk. It amounts to estimating conditional tail quantiles It is a risk to love. What if it doesn t turn out? Ah, but what if it does P. McWilliams 76

77 VaR methods Three categories of VaR methods: 1.Nonparametric - historical simulation 2.Parametric - volatility forecast with assumed distribution, e.g. GARCH with t-dist. 3.Semiparametric - extreme value theory - CAViaR 77

78 Extreme value theory Extreme value theory is a branch of statistics dealing with extreme deviations from the median of probability distribution The peak over threshold (POT) approach investigates cases where the variable exceeds a given threshold Gumbel (1958) showed that for any well- behaved (continuous and has an inverse) distribution, the distribution of extremes can be described by a Gumbel distribution with cumulative density function F(x) = exp(-exp(-x)) 78

79 Catastrophe modelling Risk = Hazard x Exposure x Vulnerability Under certain assumptions, extreme value theory or simulation can be used to perform extrapolation in order to assess the probability of extreme losses. Loss from Katrina was $90 billion 79

80 Technical Analysis Technical analysts attempt to identify patterns in historical financial market data (both price and volume) that can be exploited to generate positive returns Using transformations of financial time series it may be possible to forecast price movements The idea is that large gains from successful trades exceed more numerous but smaller losing trades By controlling for risk, it should be possible generate positive average returns in the long-term 80

81 Evidence for technical analysis Neftci (1991) showed that some of the rules employed in technical analysis generate welldefined techniques of forecasting, but even well-defined rules were shown to be useless in prediction if the economic time series is Gaussian. However, if the processes under consideration are nonlinear, then the rules might capture some information. Tests showed that this may indeed be the case for the moving average rule. Brock et al. (1992) analysed numerous technical trading rules using 90 years of daily stock prices from the Dow Jones Industrial Average up to 1987 and found that they all outperformed the market. Lo et al. (2000) took a systematic and automatic approach to technical pattern recognition, applying nonparametric kernel regression to a large number of US stocks from 1962 to By comparing the unconditional empirical distribution of daily stock returns to the conditional distribution (conditioned on specific technical indicators) they found that several technical indicators provided incremental information and may have some practical value. 81

82 Relative strength index Wilder (1978) proposed the relative strength index (RSI) to measure price strength by comparing upward and downward price changes: RSI t = 100 s t u s t u + s t d where s t u and s t d are EWMA estimates of u t = max(p t p t 1, 0) and d t = max(p t 1 p t, 0) Wilder recommended a smoothing factor of α=1/14 and considered a security to be overbought when RSI>70 and oversold when RSI<30 Cutler s RSI uses a simple moving average instead of EWMA 82

83 RSI example 83

84 Moving average convergence/divergence (MACD) MACD is a trend-following indicator and measures the difference between a fast and slow exponential moving average (EWMA) of prices: MACD t = EWMA 12 t p EWMA 26 t p s t = EWMA 9 t MACD MACD can be traded in a number of ways: Buy (sell) when MACD t moves above (below) s t Buy (sell) when MACD t moves above (below) zero Divergence between price and MACD levels 84

85 MACD example Source: Wikipedia 85

86 Stochastic oscillators Lane (1950) introduced the stochastic oscillator to measure momentum by comparing the closing price to the price range over a specific period (usually n=14 days): s fast = 100 p close p low p high p low The rational is that prices tend to close near their past highs in bull markets, and near their lows in bear markets. Trading signals can be generated when the fast stochastic oscillator crosses its moving average, the slow stochastic oscillator A pair of fast and slow oscillators may be formed; the slow oscillator can be derived from the fast one using a simple moving average with n=3 days 86

87 Trading stochastic oscillators Buy when s fast crosses up through s slow and sell when s fast crosses down through s slow An alternative trading strategy is to use the oscillators directly based on their level as was the case of RSI; s > 80 implies overbought and s < 20 implies oversold Oscillators may be slowed down further during periods of high volatility by taking additional moving averages (or averages with longer periods) This reduces fluctuations, the number of crossovers and hence transaction costs 87

88 Stochastic oscillators example Source: Wikipedia 88

89 Bollinger bands Bollinger bands provide a conditional measure of the highness or lowness of the price relative to previous trades The bands are defined using an n-period simple moving average, μ t for the centre and ±kσ t for the bands where σ t is the n-period standard deviation (typical values are n=20 and k =2) Trading strategies: Buy when the price touches the lower band and exit when price reaches the centre Buy when the price moves above the upper band or sell when the price goes below the lower band Sell options when bands are historically far apart or buy options when the bands are historically close together 89

90 Bollinger bands example Source: bollingerbands.com/ 90

91 Technical analysis summary Technical analysis embraces a variety of methods Good practise is to trade only if several technique give same signal Use technical analysis for short term and fundamental analysis for medium and long-term Finance academics are mostly sceptical but some studies are encouraging, e.g. Neftci (1991), Brock et al. (1992), and Lo et al. (2000) 91

92 Remember, remember Volatility Implied and Realized Volatility Estimators close to close (C), Parkinson (HL), Garman-Klass (OHLC), Rogers-Satchell (OHLC), GYKZ (OHLC), Yang-Zhang (OHLC) Volatility Modelling ARCH and GARCH. Identification (time series plot, ACF, PACF useful), Estimation (MLE), Diagnosis (standardized residuals) Risk Assessment, Value at Risk (VaR) Technical indicators Relative Strength Index, Moving Average Convergence/Divergence, Stochastic Oscillator and Bollinger Bands Don't be satisfied with stories, how things have gone with others. Unfold your own myth - Rumi 92

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