John Hull, Risk Management and Financial Institutions, 4th Edition

P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1

Chapter 10: Volatility (Learning objectives) Define and distinguish between volatility, variance rate, and implied volatility. Describe the power law Explain how various weighting schemes can be used in estimating volatility. Apply the exponentially weighted moving average (EWMA) model to estimate volatility. Describe the generalized autoregressive conditional heteroskedasticity (GARCH (p,q)) model for estimating volatility and its properties. Calculate volatility using the GARCH (1,1) mode Explain mean reversion and how it is captured in the GARCH (1,1) model. Explain the weights in the EWMA and GARCH (1,1) models. Explain how GARCH models perform in volatility forecasting. Describe the volatility term structure and the impact of volatility changes. 2

Define and distinguish between volatility, variance rate, and implied volatility. Volatility of a variable is caused by the changes in the asset s values due to new information reaching the market. A variable s volatility is denoted by sigma, σ, and defined as the standard deviation of the proportional change in its continuously compounding return per unit of time. 3

Define and distinguish between volatility, variance rate, and implied volatility. If is the value of variable at the end of the day then its continuously compounded return per day on day (between end of the previous day 1 and end of the current day ) is given by = Volatility per day, given by is equal to the standard deviation of the continuously compounded return,. If represents the most recent observations on, then volatility can be given as: = ( ) 4

Define and distinguish between volatility, variance rate, and implied volatility (continued) The variance rate is defined as the square of the volatility or standard deviation. The variance rate, just like volatility is defined per unit of time.. The variance rate can be given as: = ( ) 5

Define and distinguish between volatility, variance rate, and implied volatility (continued) The variance rate is defined as the square of the volatility or standard deviation. The variance rate, just like volatility is defined per unit of time.. The variance rate can be given as: = ( ) Simplifying assumptions = 6

Define and distinguish between volatility, variance rate, and implied volatility (continued) Square root rule (SRR) The standard deviation of the return over T days is times the standard deviation of the return over one day. Therefore, it can be said that volatility of the return in time T increases with the square root of time ( ) whereas the variance of this return increases linearly with time(t). = = Square root rule (SRR) assumes i.i.d. returns; that is, assumes no auto-correlation (aka, serial correlation) 7

Define and distinguish between volatility, variance rate, and implied volatility (continued) For example: Assume the daily volatility is equal to 2.0%. We can say that the standard deviation of the continuously compounded return over five days (T = 5) is. % = 4.47%. As five days is considered a short period of time, this is approximately the same as the standard deviation of the proportional change over five days. A one-standard-deviation move would be 80 x 0.0447 = $3.576. If we assume that the change in the asset price is normally distributed, we can be 95% confident that the asset price will be between 80 + 1.96 x 3.576 = $87.01 and 80-1.96 x 3.576 = $72.99 at the end of the five days. 8

Define and distinguish between volatility, variance rate, and implied volatility (continued) Implied Volatility is defined as the volatility which when substituted in a pricing model can be calibrated with the market price of the asset. The most popular index for implied volatility is the VIX index (popularly known as fear index ) calculated by CBOE. It is an index of the implied volatility of 30-day options on the S&P 500 calculated from a wide range of calls and puts. The figure shows the VIX index between January 2004 and October 2016. Usually contained between levels 10 and 20 until 2007, during the credit crisis, it touched a record high of 80 in October and November 2008. 9

Describe the power law The power law is an alternative to the normal distribution: > = > = where and are constants. Example: If it is known that = 3 for a financial variable and if = 10, and the probability that > 10 is known to be 0.05, then we can find out by substituting the other known values in the equation as: 0. = =. Now for any value of, say =, the probability > 20 as > = =. 10

Describe the power law (continued) Real World shows the percentage of days when absolute size of daily exchange rate moves is greater than {1, 2, 3, 4, 5 or 6} standard deviations To illustrate the fourth(green) row: Real-world observation: 0.29% of days are greater than 4 SD If rate moves were normal, then 4 SD = 0.006% = 2*(1 - NORM.S.DIST(4,TRUE)) In the last two columns, LN(4) = 1.386 and LN(0.29%) = -5.843 Real World Normal x Pr(v>x) Model ln(x) ln[pr(v>x)] > 1 SD 25.04% 31.731% 0.000-1.385 > 2 SD 5.27% 4.550% 0.693-2.943 > 3 SD 1.34% 0.270% 1.099-4.313 > 4 SD 0.29% 0.006% 1.386-5.843 > 5 SD 0.08% 0.000% 1.609-7.131 > 6 SD 0.03% 0.000% 1.792-8.112 11

Describe the power law (continued) To test if the power law apples, we plot against >. Here we observe the logarithm of probability of exchange rate increasing by more than standard deviations is approximately linearly dependent on for >3. Regression at x = {3, 4, 5, and 6}, gives best-fit line of: > =.. This shows that K =. = 5.765 and = 5.505. 12

Explain how various weighting schemes can be used in estimating volatility. Hull s example of simple unweighted historical volatility calculation We are given n = 21 days of closing prices for a non-dividend paying stock. The Price Relative is the ratio of prices, S(i)/S(i-i), and Daily Return is the log return, LN[S(i)/S(i-i)] 13

Explain how various weighting schemes can be used in estimating volatility. Note: The average return, µ, is near to zero: 0.00074 The sum of squared returns is equal to 0.004241, and when divided by (m = 20), we get the average squared return which is the variance. Variance is 0.000212. The daily volatility is the square root of the daily variance. Volatility is 1.456% 14

Explain how various weighting schemes can be used in estimating volatility (continued) It is advisable to assign greater weights to more recent data because we need to estimate, the volatility on day. This can be achieved with the model: = where is the amount of weight given to the observation (i) days ago. The alpha (α) weights are positive and chosen such that < when >. This means that lesser weight is given to old observations and more weight is given to newer observations. The weights must sum to one, that is: = 1 15

Explain how various weighting schemes can be used in estimating volatility (continued) This model can be further extended by assuming an additional variable in the model called the long run average variance rate( )which is assigned some weight ( ). This can be given as: = + = + 16

Explain how various weighting schemes can be used in estimating volatility (continued) This is known as the ARCH(m) model, where the estimate of variance is given by long-run average variance and (m) observations. If =, this ARCH(m) model is given by: = + It remains true that the weights must sum to one such that: + =. ARCH(m) GARCH(1,1) EWMA Simple Unweighted This ARCH(m) model reduces to all three of our relevant special cases: When the long-run variance is assigned a non-zero weight and the weights decline exponentially, at a rate of β, the model is a GARCH(1,1) model. If no long-run variance in the model, it is an exponentially weighted moving average (EWMA). EWMA is a special case of GARCH is a special case of ARCH(m). When there is no long-run variance and the weights are equal, we have the simple unweighted volatility that was previously illustrated. 17

Apply the exponentially weighted moving average (EWMA) model to estimate volatility. In the variance equation =, which gives lesser weight to older observations, if the weights decrease exponentially such that =, (where is a constant between 0 and 1) then the variance equation can be represented as: = + ( ) For example: For day (n-1) the volatility estimate, σ(n-1) was 1.0% per day. During the day (n-1) the market variable increased by 2.0%, such that =. =. =. =. 18

Apply the exponentially weighted moving average (EWMA) model If the smoothing parameter (aka, decay rate) lambda,, is 0.90 then the EWMA equation = + (1 ) gives EWMA Example Assumptions Last volatility, σ (n-1) 1.00% Last variance, σ 2 (n-1) 0.00010000 Yesterday's price 10.00 Today's price 10.20 Last return, µ (n-1) 2.000% Exponentially weighted moving average (EWMA) Lambda, λ 0.900 1-lambda 0.100 Updated EWMA volatility estimate σ 2 n = λ*σ 2 (n-1)+(1-λ)*µ 2 (n-1) 0.000130 Updated Volatility 1.140% =.. +.. =. Therefore, =. =. =. % The expected value of = = 0.0001 but its realized value is higher at 0.0004. So the value of our volatility estimate( ) is higher. If the realized value of is lower than its expected value, then the value of our volatility estimate( )would decrease. 19

Describe the generalized autoregressive conditional heteroskedasticity (GARCH (p,q)) model for estimating volatility and its properties. GARCH is a general autoregressive conditional heteroskedastic model where Variance is regressing on its past lagged variances (autoregressive) Conditional on most recent (and more weighted) variances such that The variances keep changing over time or they are not constant (heteroskedastic) In updating the variance estimate,, GARCH generalizes the EWMA model average by adding a variance rate,in addition to the and used in the EWMA model. 20

Describe the generalized autoregressive conditional heteroskedasticity (GARCH (p,q)) model for estimating volatility and its properties (continued) The GARCH (1,1) model is given as: = + + where is the weight given to, is the weight corresponding to and is the weight corresponding to.the weights must sum to unity such that + + =. If we indicate the value of with, then the GARCH (1,1) model can be given as: = + + The 1,1 in GARCH (1,1) shows that the is calculated from the most recent(single) observation of and the most recent single estimate of variance rate. The generalized form of this model is GARCH(p,q) where the is determined by regressing on the most recent p squared returns ( )and the most recent q estimates of variance rate. 21

Calculate volatility using the GARCH (1,1) model Consider an GARCH (1,1) model given as: =. +. +.. Here, and are 0.000002, 0.13 and 0.86 respectively. We know + + = 1. In this case, = 1 = 1 0.13 0.86 = 0.01. GARCH Example Assumptions Last volatility, σ (n-1) 1.60% Last variance, σ 2 (n-1) 0.00025600 Yesterday's price 10.00 Today's price 9.90 Last return, µ (n-1) -1.000% GARCH (1,1) Beta, β 0.860 alpha, α 0.130 alpha + beta, α+β 0.9900 omega, ω 0.0000020 gamma, γ = 1-α-β 0.010 α+β+γ = 1.0 1.000 LT variance = ω/(1-α-β) 0.0002000 LT volatility 1.4142% Updated GARCH (1,1) volatiltiy estimate σ 2 n = ω + β*σ 2 (n-1) + α*µ 2 (n-1) 0.000235 Updated Volatility 1.533% Also we know that =. So = =.. = 0.00020; this is the long-run average variance per day. So long-run daily volatility is given by 0.0002 = 0.014 or 1.414%. 22

Calculate volatility using the GARCH (1,1) model =. +. +.. If the estimate of daily volatility on day ( 1)is 1.6% per day and if the market variable decreased by 1.0% so that = 0.016 = 0.000256 and = 0.01 = 0.0001 respectively, then GARCH Example Assumptions Last volatility, σ (n-1) 1.60% Last variance, σ 2 (n-1) 0.00025600 Yesterday's price 10.00 Today's price 9.90 Last return, µ (n-1) -1.000% GARCH (1,1) Beta, β 0.860 alpha, α 0.130 alpha + beta, α+β 0.9900 omega, ω 0.0000020 gamma, γ = 1-α-β 0.010 α+β+γ = 1.0 1.000 LT variance = ω/(1-α-β) 0.0002000 LT volatility 1.4142% Updated GARCH (1,1) volatiltiy estimate σ 2 n = ω + β*σ 2 (n-1) + α*µ 2 (n-1) 0.000235 Updated Volatility 1.533% = + + =. +.. +.. =. The updated volatility estimate is 0.00023516 = 0.0153 or 1.53% per day. 23

Explain how mean reversion is captured in the GARCH (1,1) model Mean reversion refers to the variance rates being pulled back to the long run average or mean levels. GARCH models exhibit mean reversion property whereas EWMA models do not. The weights must sum up to unity in any model. In EWMA model + = 1. In GARCH models since + + = 1, the condition remains such that + < 1 so that = 1 The GARCH (1,1) model forecasts the volatility on day + based on data available at end of day 1 as: [ ] = + + ( ) 24

Explain how mean reversion is captured in the GARCH (1,1) model (continued) When + < 1 and when (t) increases the last term in the above equation becomes smaller. So the expected future variance rate will tend toward the long-run average variance rate. The variance rate in a GARCH model therefore exhibits mean reversion with a reversion level of (reversion toward the level) and a reversion rate of 1 or. This means that the forecast for the future variance rates tends towards the mean reverting level as we look more and more towards the future. Also, higher refers to greater mean reversion rate. For this forecast to remain stable, the condition + < must be satisfied. Otherwise, if + > 1,it means that the weight given to long term average variance rate is negative ( < 1). This suggests a condition of mean fleeing instead of mean reverting. 25

Explain how mean reversion is captured in the GARCH (1,1) model (continued) Example: Consider Hull s given assumptions: Alpha, α = 0.083394 Beta, β = 0.9101160 Omega, ω = 0.0000013465 Therefore, + = 0.9935 and = 0.0002075 such that the implied long term volatility is 0.0002075 = 1.4404%. Our estimate of current variance rate per day is given as 0.00030 such that current volatility per day as 0.00030 = 1.732% per day. In 10 days the expected variance rate will be = 0.0002075 + 0.9935 0.00030 0.0002075 = 0.00029417 So, the expected volatility per day is 0.00029417 = 1.715%, which is still pretty far from the long term volatility of 1.4404%. 26

Explain how mean reversion is captured in the GARCH (1,1) model (continued) Instead of 10 days, say we forecast the expected variance rate in 100 days. In this case, = 0.0002075 + 0.9935 0.00030 0.0002075 = 0.00025572 So, the expected volatility per day is 0.00025572 = 1.599%, which appears to be closer to the long term volatility of 1.4404%. This example illustrates that the further we look into the future, the more the future variance rate will tend towards the mean reverting level of. 27

Explain the weights in the EWMA and GARCH (1,1) models. The EWMA model is given as: = + The estimate of the volatility for day (made at end of day 1)is calculated from (the estimate that was made at the end of the day 2 of the volatility for day 1) and (the most recent daily percentage change). To analyze why this equation represents exponentially decreasing weights, we substitute with [ + (1 ) ] to get: = [ + (1 ) ] + (1 ) which can be rearranged as: = 1 + + 28

Explain the weights in the EWMA and GARCH (1,1) models (continued) Similarly, can be substituted with [ + (1 ) ] to get: = 1 + + + This process can be continued (m) times and the equation can be generalized, as in: = 1 + When m becomes large, the second term ( )becomes so small that it is ignored and therefore the above equation becomes similar to the equation = that gives lesser weight to older observations with = (1 ). Thus it can be observed that the weights for decline at a rate as we move back in time or we can say that each weight is times its previous weight. 29

Explain the weights in the EWMA and GARCH (1,1) models (continued) To understand how the GARCH model adds weights exponentially, consider the GARCH (1,1) model = + + Substituting with + + we get = + + + + This can be rewritten as: = + + + + Likewise, substituting with + + we get = + + + + + + Continuing the same way, we can say that weight applied to is. 30

Explain how GARCH models perform in volatility forecasting. The GARCH (1,1) model estimates variance rate at the end of day day as: = + + 1 for We know that the weights should sum up to unity. So + + = 1 which means that = 1. Substituting this value for in the GARCH (1,1) model we get = (1 ) + + Reducing from both sides of the equation and after rearranging we get = + ( ) 31

Explain how GARCH models perform in volatility forecasting (continued) Therefore, a forecast on day + in the future can be given as: = + ( ) The expected value of is. If expected value is represented by, then = + Using this equation repeatedly gives: [ ] = + ( ) or [ ] = + + ( ) 32

Describe the volatility term structure and the impact of volatility changes. Volatility term structure describes the relationship between the volatility of options and their maturities. GARCH model estimates a downward sloping term structure when the current volatility is above the long-term volatility, and an upward sloping curve when the current volatility is below the long-term volatility On day n, if ( )is defined as the expected variance rate = [ ] And we know that the GARCH (1,1) forecast yields the equation: = + +. If =, then = + (0) 33

Describe the volatility term structure and the impact of volatility changes (continued) Now the average variance rate per day between today and time T can be integrated as: 1 = + 1 [ (0) ] This ( )is the estimate of instantaneous variance rate in t days. If the life of the option is longer, then this rate will be closer to the long-run average variance rate. If is considered as the volatility per annum, then assuming 252 trading days in a year, ( ) is 252 times the average variance rate per day. So, ( ) = 252 + 0 34

Describe the volatility term structure and the impact of volatility changes (continued) Impact of volatility changes The equation for ( ) can be rearranged as: ( ) = 252 + 1 0 252 If (0)changes by a small amount, say (0), the approximate change in is: 1 (0) ( ) (0) 35

Describe the volatility term structure and the impact of volatility changes (continued) Spreadsheet Snapshot: this exhibit repeats (summarizes) the previous four examples of volatility calculations. 36

Chapter 11: Correlation and Copulas Define correlation and covariance, and differentiate between correlation and dependence. Calculate covariance based using the EWMA and GARCH (1,1) models. Apply the consistency condition to covariance. Describe the procedure of generating samples from a bivariate normal distribution. Describe the properties of correlations between normally distributed variables when using a one-factor model. Define copula, describe the key properties of copula and copula correlation. Explain one tail dependence. Describe Gaussian copula, Student t-copula, multivariate copula and one factor copula. 37

Define correlation and covariance, and differentiate between correlation and dependence. The correlation coefficient is calculated using the following formula: = ( ) Where:. = denotes expected value.= denotes standard deviation The mathematical relationship between correlation and covariance can be illustrated by the formula below: = 38

Define correlation and covariance, and differentiate between correlation and dependence (continued) Correlation and dependence both refer to a statistical relationship between two variables, but dependence is more general. Correlation is a special case of dependence. The key relationship given by cov(x,y) = ρ(x,y)*σ(x)*σ(y) implies: If the correlation, ρ(x,y), is equal to zero, then the covariance is also equal to zero, and If the covariance is zero, then the correlation is zero. Correlation implies dependence but dependence does not imply correlation; If variables are independent, their correlation (and covariance) is zero; but zero correlation does not imply independence. 39

Define correlation and covariance, and differentiate between correlation and dependence (continued) The charts below illustrate the correlation coefficient, denoted by (r), in four cases. In the bottom panel, notice that relationships are clearly dependent yet the correlation is zero. 40

Define correlation and covariance, and differentiate between correlation and dependence (continued) ρ(x,y) = Correlation(X,Y) = covariance(x,y)/[σ(x)*σ(y)] 41

Calculate covariance based using the EWMA and GARCH (1,1) models. The formula for updating covariance using EWMA is: = + ( ) Example of updating correlation with EWMA approach (Hull Ex 11.1) Scenario: Suppose that λ = 0.95 and that the estimate of the correlation between two variables X and Y on day (n-1) is 0.60. Suppose further that the estimate of the volatilities for X and Y on day (n-1) are 1.0% and 2.0%, respectively. Suppose that the percentage changes in X and Y on day (n-1) are 0.5% and 2.5%, respectively. 42

Calculate covariance based using the EWMA and GARCH (1,1) models (continued) Using an EWMA approach, we can update the previous correlation as follows: COV(X,Y) on day (n-1) = 0.60 * 1.0%*2.0% = 0.000120 Updated variance(x): σ(x)^2(n) = 0.95*1.0%^2 + 0.05*0.5%^2 = 0.00009625; updated σ(x) = SQRT(0.00009625) = 0.98% Updated variance(y): σ(y)^2(n) = 0.95*2.0%^2 + 0.05*2.5%^2 = 0.00041125; updated σ(y) = SQRT(0.00041125) = 2.03% Updated covariance(x,y) = 0.95*0.000120 + 0.05*0.5%*2.5% = 0.0001203 Updated ρ(x,y) = 0.0001203 / (0.98% * 2.03%) = 0.6044 43

Calculate covariance based using the EWMA and GARCH (1,1) models (continued) The formula for updating covariance using GARCH is: = + + Example of updating correlation with GARCH(1,1) approach (Hull EOC Question #6) Suppose that the current daily volatilities of asset X and asset Y are 1.0% and 1.2%, respectively. The prices of the assets at close of trading yesterday were $30.00 and $50.00; the prices of the two assets at close of trading today are $31.00 and $51.00. Therefore, the most recent daily price changes are Δ(X) = 31/30-1 = 3.33% and Δ(Y) = 51/50-1 = 2.00%. The estimate of the coefficient of correlation between the returns on the two assets made at this time was 0.50. Correlations and volatilities are updated using a GARCH( 1,1) model. The estimates of the model's parameters are α = 0.04 and β = 0.94. For the correlation ω = 0.000001 and for the volatilities ω = 0.000003. How is the correlation estimate updated? 44

Calculate covariance based using the EWMA and GARCH (1,1) models (continued) These are the assumptions given about the two variables: Using a GARCH(1,1) approach, we can update the previous correlation as follows: Covariance(X,Y) on day (n-1) = 0.50 * 1.00% * 1.20% = 0.000060 Updated covariance = 0.000001 + 0.040*3.33%*2.00% + 0.940*0.000060 = 0.0000841 Updated variance(x): σ(x)^2(n) = 0.000003 + 0.040*3.33%^2 + 0.940*1.00%^2 = 0.0001414; So the new volatility of (X) is 1.19% Updated variance(y): σ(y)^2(n) = 0.000003 + 0.040*2.00%^2 + 0.940*1.20%^2 = 0.0001544; So the new volatility of (Y) is 1.24% The updated correlation between X and Y, ρ(x,y) = 0.0000841 / (1.19% * 1.24%) = 0.5689 45

Calculate covariance based using the EWMA and GARCH (1,1) models Same calculation: Covariance(X,Y) on day (n-1) = 0.50*1.00%*1.20% = 0.000060 Updated covariance = 0.000001 + 0.040*3.33%*2.00% + 0.940*0.000060 = 0.0000841 Updated variance(x): σ(x)^2(n) = 0.000003 + 0.040*3.33%^2 + 0.940*1.00%^2 = 0.0001414; So new σ(x) is 1.19% Updated variance(y): σ(y)^2(n) = 0.000003 + 0.040*2.00%^2 + 0.940*1.20%^2 = 0.0001544; So new σ(y) is 1.24% The updated ρ(x,y) = 0.0000841 / (1.19% * 1.24%) = 0.5689 X[n-1] $30.00 X[n] $31.00 Y[n-1] $50.00 Y[n] $51.00 ΔX 3.33% ΔY 2.00% σ(x_n-1) 1.0% σ(y_n-1) 1.2% ρ(x_n-1,y_n-1) 0.50 cov(x_n-1,y_n-1) 0.000060 GARCH(1,1) updates alpha, α 0.04 beta, β 0.94 omega, ρ 0.0000010 omega, σ 0.0000030 cov(x_n,y_n) 0.0000841 σ^2(x_n) 0.0001414 σ(x_n) 1.19% σ^2(y_n) 0.0001544 σ(y_n) 1.24% ρ(x_n,y_n) 0.5689 46

Apply the consistency condition to covariance. A variance-covariance matrix that satisfies the condition Ω 0 is internally consistent, and is known as positive-semidefinite. Note: is simply the transpose of To ensure that a positive-semidefinite matrix is produced, covariances should be calculated using the exact same variables used for calculating variances. That is, weightages and lambda values used for calculation should be the same for variances as well as covariances. 47

Describe the procedure of generating samples from a bivariate normal distribution. The following three step procedure is used to generate samples for variables, let s say and (both having mean zero and standard deviation one), from a bivariate normal distribution: 1. Obtain independent samples Z1 and Z2 from a univariate standardized normal distribution 2. Set = 3. Set = + 1 48

Describe the procedure of generating samples from a bivariate normal distribution. First X(2) = 5.0 + 9.0* [ρ*ε(1) + ε(2)*sqrt(1-ρ 2 )] = 5.0 + 9.0*[ 0.80*-2.70 + -0.580*sqrt(1-0.80^2)] 49

Describe the properties of correlations between normally distributed variables when using a one-factor model. When defining correlations between normally distributed variables using a one factor model, each variable having a standard normal distribution has a component dependent on a common factor. For example, consider the CAPM (capital asset pricing model) for calculating expected return on stock i, with as the risk free rate: = + In this model, the expected return on a stock has a component dependent on the return from the market and an idiosyncratic component that is independent of the return on other stocks. An advantage of the one-factor model is that the resulting covariance matrix is always positive-semidefinite (internally consistent). 50

Describe the properties of correlations between normally distributed variables when using a one-factor model. The single-factor model where U(i) is dependent on the common factor, F, and both F and Z have standard normal distributions: = + The coefficient of correlation between U(i) and U(j) is given by: = 51

Define copula, describe the key properties of copula and copula correlation. Copula can be defined as a multivariate probability distribution for which the marginal probability distribution of each variable is uniform. Interdependence of returns of two or more assets is calculated using the correlation coefficient, which only works well with normal distributions, whereas in practice, distributions in financial markets are mostly skewed. Copula helps transform variables into new variables that have bivariate normal distributions for which it is easy to define a correlation structure. This is done by mapping the variables into new variables on a percentile-to-percentile basis. In this structure, the conditional mean of a dependent variable is linearly dependent on the independent variable, while the standard deviation of is constant. Note: The correlation between and is known as copula correlation and is not the same as correlation between and 52

Explain one tail dependence. Tail dependence is the tendency for extreme values for two variables to occur together. Tail dependence of a pair of random variables describes their comovements in the tails of the distributions. The choice of the copula affects tail dependence. Tail dependence is higher in a bivariate Student t-distribution than in a bivariate normal distribution. 53

Describe Gaussian copula, Student t-copula, multivariate copula and one factor copula. Gaussian copula: This is used to estimate the correlation structure between two variables to define their joint distribution. To use Gaussian copula, variables having marginal distributions are mapped into new variables that have standard normal distributions in order to arrive at a bivariate normal joint distribution. Student-t copula: This is very similar to Gaussian copula, with the exception that variables are assumed to have a bivariate Student-t distribution rather than a bivariate normal distribution. Multivariate copula: This is a form of Gaussian copula that is used to define a correlation structure between more than two variables hence the name multivariate. One-factor copula: This is a multivariate copula model that utilizes a factor model for estimating the correlation structure between mapped variables. 54

The End P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition 55