Chapter 3: Volatility and comovement

Question 1

What is volatility?

Answer 1

• The most traditional measure of risk.
• The standard deviation of logreturns per unit of time.
  
  • Unit of time in risk management is a day
  • For IID normal logreturns: standard deviation of a day is square 250*standard deviation day

Question 2

How do we obtain an estimate of volatility?

Answer 2

• Different approaches existto obtain an estimate of volatility:

1. Historical based volatility = gives a backward looking estimate:
   
   • Most common and most simple approach
   • It can easily be modified to allow for time-variation
2. Implied volatility:
   
   • Gives a forward looking estimate as implicit in observed market prices of options

Question 3

What is the operational version used for risk management purposes for volatility?

Answer 3

• Mean logreturn is zero
• Simple returns are a proxy of logreturns
• ML estimate instead of unbiased estimate

Question 4

What are the main issues when computing volatility based on an historical sample?

Answer 4

1. How to define the sample size?
   
   • Large samples yield more accurate estimates
   • long-time ago observations might be less relevant
   • There might be stationarity issues: this would imply a violation of the ‘identical’ assumption of IID returns.
     
     • Stationarity = statistical properties are constant over time
2. Are equal weights reasonable?
   
   • More recent observations might be more predictive for the near future

Question 5

What are the methods to account for time-variation in volatility?

Answer 5

1. Different weighting scheme: extra alpha introduced which is the weight given to the observation of i days ago.
   
   • Special case: exponentially weighted moving average model
2. Different weighting scheme and reversion to the mean:
   
   • Special case: autoregressive conditional heteroskedasticity model
   • Vl is the long-run variance rate with weight ypsilon.

Question 6

What is the EWMA model?

Answer 6

• Exponentially weighted moving average (EWMA) volatility: introduces exponentially decreasing weights, with a decay parameter between 0 and 1. This weighting scheme gives less weight to older observation.
• Responsiveness by the decay parameter to recent daily changes:
  
  • If decay paramter is small: quick response of volatility to new information.
    
    • Eg. daily data: typical 0.9
  • If decay paramter is large: slow response of volatility to new information.
    
    • Eg. typical: 0.97.
• Nice characteristic: simple updating rule for volatilities where today’s volatility is a weighted average of previous forecast and latest % change.

Question 7

How can we evaluate the EWMA volatility?

Answer 7

• Major advantage is the simple updating rule:
  
  • quickly obtain a volatility estimate
  • Few data to store/input
• Major disadvantage: no build-in mean reversion
  
  • EWMA variance could drift away in principle
  • However: it is an empirical fact that the variance rate pulls back to the long run mean
• A realistic volatility model should include a long-run variance component.

Question 8

What is the GARCH model?

Answer 8

• Generalized autoregressive conditional heteroskedasticity volatility
• Has a long-run average variance rate, with weight.
• Has a decay parameter.
• The GARCH(1,1) model is a weighted average of the previous forecast, the latest %-change and the long run variance rate
• The EWMA volatility is a special case of the GARCH(1,1) volatility.

Question 9

What are implied volatilities?

Answer 9

• Implied volatilities are volatilities as implied in observed market prices of options.
  
  • Since market prices are forward looking: implied volatilities are also forward looking.
  • Intuitively this is more appropriate to estimate future volatility.
• To obtain an estimate of the implied volatility, we need a particular option pricing model.
  
  • Extract volatility: the volatility, when plugged into the option priving model, gives the observed market price is the implied volatility.
  • Most common option pricing: Black & Scholes.

Question 10

What is comovement?

Answer 10

• When evaluating risk in a portfolio, a standalone volatility analysis needs to be complemented with an analysis of the comovement between the different portfolio components.
  
  • The risk profile of the portfolio can be very different depending upon the degree of comovement of the portfolio constituents.
  • On the level of the portfolio, comovement reflects benefits of diversification = which leads to lower portfolio volatility.

Question 11

What are the different approaches to model comovement?

Answer 11

• Most traditional measure: Pearson correlation
  
  • Also possible to introduce variation over time, so similar to the approach in volatilities
  • Main drawback: makes strong distributional assumptions
• More general measures of comoveemnt can be applied to more general distributions
  
  • Rank correlations: more general measures of concordance
  • Copulas: tool of defining a correlation structure and joint distribution between variables - regardless of shape of their marginal probability distributions.

Question 12

What is the Pearson correlation?

Answer 12

• Correlation = standardized covariance of the logreturns of asset x and y per unit of time
• Possible to monitor time-variation in correlations:
  
  • Exponentially weighted moving average correlations
  • Generalized autoregressive conditional heteroskedasticity correlations
    
    • The variance-covariance matric needs to be estimated and updated consistently. Simple in the EWMA model, but trickier in the GARCH setting, and reuires a multivariate GARCH model.

Question 13

How can we evaluate the Pearson correlation?

Answer 13

• Most common measure as it is easy to compute.
• Shortcomings:
  
  1. Only appropriate for elliptical distributions. In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution.
  2. It only captures linear dependence

Question 14

How can we capture association more generally?

Answer 14

• To measure the assocation between non-normally distributed variables: we can use:
  
  1. Rank correlation: statistic of association between the two variables
     
     1. Spearman correlation
     2. Kendaly tau corrlation
  2. Copules: allows us to link marginal distribution of variables into a joint distribution

Question 15

What is the Spearman correlation?

Answer 15

• Non-parametric measure of correlation based on data ranks.
• Spearman calculates differences between ranks.

Question 16

What is Kendall’s Tau?

Answer 16

• Kendall’s tau is a non-parametric measure of correlation based on the number of concordances and discordances in paired observations:
  
  • Any pair of observations is concordant is the ranks for both elements agree and disconcordant otherwise.
• Kendall’s tau measures the proportion of concordant pairs vs disconcordant pairs.

Question 17

How do we choose the quantitative factors, the VaR parameters when we want to use it as a potential loss measure?

Answer 17

• The choice of c is arbitrary as the VaR is never the worst loss, it is only the loss corresponding to a particular significance level.
• The choice of hoirzon T is determined by the period over which the portfolio is assumed static.
  
  • Determined by its liquidity.

Question 18

How do we choose the quantitative factors, the VaR parameters when we want to use it as a capital cushion?

Answer 18

• c is set high to capture tail risk (99%-99.9%)
• The choice of T is determined by the period over which the portfolio is assumed static:
  
  • Up to 10 days for market risk
  • 1 year for credit risk

Question 19

How do we choose the quantitative factors, the VaR parameters when we want to use it as a backtesting?

Answer 19

• Here we allow to observe frequent breaches.
• c is set rather low as to allow VaR to be breached frequently.
• The horizon T is chosen rather short as to observe independent VaR breaches.
• Using a 2 week VaR horizon implies 26 independent observations per year. A 1-day VaR horizon has 250 observations over the same year.
  
  • Using a daily VaR, you can compare your P&L number 250 times to VaR and determine how many times P&L exceeds VaR.