Estimating Volatilities and Correlations Chapter 21

Intro The methods here are useful for: The methods here are useful for: VaR calculations and VaR calculations and Derivative pricing Derivative pricing The methods just recognize that volatility changes over time The methods just recognize that volatility changes over time

Standard Approach to Estimating Volatility Define  n as the volatility per day between day n -1 and day n, as estimated at end of day n -1 Define  n as the volatility per day between day n -1 and day n, as estimated at end of day n -1 Define S i as the value of market variable at end of day i Define S i as the value of market variable at end of day i Define u i = ln( S i /S i-1 ) Define u i = ln( S i /S i-1 )

Simplifications Usually Made Define u i as (S i -S i-1 )/S i-1 Define u i as (S i -S i-1 )/S i-1 Assume that the mean value of u i is zero Assume that the mean value of u i is zero Replace m-1 by m Replace m-1 by m This gives

Weighting Scheme Instead of assigning equal weights to the observations we can set

ARCH(m) Model In an ARCH(m) model we also assign some weight to the long-run variance rate, V L :

EWMA Model In an exponentially weighted moving average model, the weights assigned to the u 2 decline exponentially as we move back through time In an exponentially weighted moving average model, the weights assigned to the u 2 decline exponentially as we move back through time This leads to This leads to

EWMA Model Example Suppose = 0.9,  n-1 = 1%, and u n-1 = 2%. Then, Suppose = 0.9,  n-1 = 1%, and u n-1 = 2%. Then, It means that  n = 1.14%. It means that  n = 1.14%. Also, Also,

Attractiveness of EWMA Relatively little data needs to be stored Relatively little data needs to be stored We need only remember the current estimate of the variance rate and the most recent observation on the market variable We need only remember the current estimate of the variance rate and the most recent observation on the market variable Tracks volatility changes with being the sensitivity to current changes in the market variable Tracks volatility changes with being the sensitivity to current changes in the market variable RiskMetrics uses = 0.94 for daily volatility forecasting RiskMetrics uses = 0.94 for daily volatility forecasting

GARCH (1,1) In GARCH (1,1) we assign some weight to the long-run average variance rate Since weights must sum to 1 

GARCH (1,1) continued Setting  V L the GARCH (1,1) model is and

Example Suppose Suppose 1-  -  =  -  = 0.01 The long-run variance rate is so that the long-run volatility per day is 1.4% The long-run variance rate is so that the long-run volatility per day is 1.4% 

Example continued Suppose that the current estimate of the volatility is 1.6% per day at t=n-1 and the most recent percentage change in the market variable is 1%. Suppose that the current estimate of the volatility is 1.6% per day at t=n-1 and the most recent percentage change in the market variable is 1%. The new variance rate is The new variance rate is The new volatility is 1.53% per day

Maximum Likelihood Methods In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring In maximum likelihood methods we choose parameters that maximize the likelihood of the observations occurring

Example 1 We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens? We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens? The probability of the event happening on one particular trial and not on the others is The probability of the event happening on one particular trial and not on the others is We maximize this to obtain a maximum likelihood estimate. Result: p =0.1 We maximize this to obtain a maximum likelihood estimate. Result: p =0.1

Example 2: Normal with constant variance Estimate the variance of observations from a normal distribution with mean zero

Application to GARCH (i.e., variance is time-dependent) We choose parameters that maximize

Excel Application (Table 21.1, page 485) Start with trial values of , , and  Start with trial values of , , and  Update variances Update variances Calculate Calculate Use solver to search for values of , , and  that maximize this objective function Use solver to search for values of , , and  that maximize this objective function Important note: set up spreadsheet so that you are searching for three numbers that are the same order of magnitude (See page 486) Important note: set up spreadsheet so that you are searching for three numbers that are the same order of magnitude (See page 486)

Forecasting Future Volatility A few lines of algebra shows that The variance rate for an option expiring on day m is

Volatility Term Structures The GARCH (1,1) model allows us to predict volatility term structures changes The GARCH (1,1) model allows us to predict volatility term structures changes It suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities It suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities

Impact of Volatility Changes Using the GARCH (1,1) model, estimate the volatility to be used for pricing 10-day options on the yen-dollar exchange rate, given Using the GARCH (1,1) model, estimate the volatility to be used for pricing 10-day options on the yen-dollar exchange rate, given the current estimate of the variance, (  n ) 2, is the current estimate of the variance, (  n ) 2, is  +  =  +  = V L = V L =

Forecasting Future Volatility continued (equation 21.14, page 489)

In our Japanese Yen example In our Japanese Yen example The volatility per year for a T-day option: The volatility per year for a T-day option:

Volatility Term Structures (Table 21.3) Yen/Dollar volatility term structure predicted from GARCH(1,1) Yen/Dollar volatility term structure predicted from GARCH(1,1) Option Life, T (days) Option Volatility (% per year)

Impact of Volatility Changes continued (equation 21.15, page 490) When  (0) changes by  (0),  (T) changes by When  (0) changes by  (0),  (T) changes by

Volatility Term Structures (Table 21.4) The GARCH (1,1) suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities The GARCH (1,1) suggests that, when calculating vega, we should shift the long maturity volatilities less than the short maturity volatilities Impact of 1% change in instantaneous volatility for Japanese yen example: Impact of 1% change in instantaneous volatility for Japanese yen example: Option Life (days) Volatility increase (%)