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Estimating Volatilities and Correlations Chapter 21

Learn different methods and models for estimating volatilities and correlations in order to calculate VaR and price derivatives effectively. Understand the standard approach, ARCH(m) model, EWMA model, GARCH(1,1) model, and maximum likelihood methods. Discover how to forecast future volatilities, predict volatility term structures, and assess the impact of volatility changes in pricing options and risk management.

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Estimating Volatilities and Correlations Chapter 21

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  1. Estimating Volatilities and Correlations Chapter 21

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

  3. Standard Approach to Estimating Volatility • Define sn as the volatility per day between day n-1 and day n, as estimated at end of dayn-1 • Define Si as the value of market variable at end of day i • Define ui= ln(Si/Si-1)

  4. Simplifications Usually Made • Defineui as (Si-Si-1)/Si-1 • Assume that the mean value of ui is zero • Replace m-1 by m This gives

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

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

  7. EWMA Model • In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time • This leads to

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

  9. Attractiveness of EWMA • 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 • Tracks volatility changes with  being the sensitivity to current changes in the market variable • RiskMetrics uses l = 0.94 for daily volatility forecasting

  10. GARCH (1,1) In GARCH (1,1) we assign some weight to the long-run average variance rate Since weights must sum to 1 g + a + b =1

  11. GARCH (1,1) continued Setting w = gVL the GARCH (1,1) model is and

  12. Example  • Suppose • 1-  -  = 0.01 • The long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%

  13. 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%. • The new variance rate is The new volatility is 1.53% per day

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

  15. 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? 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

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

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

  18. Excel Application (Table 21.1, page 485) Start with trial values of w, a, and b Update variances Calculate Use solver to search for values of w, a, and b 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)

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

  20. Volatility Term Structures • 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

  21. 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 • the current estimate of the variance, (n)2, is 0.00006 •  +  =0.9602 • VL = 0.00004422

  22. Forecasting Future Volatility continued (equation 21.14, page 489)

  23. Forecasting Future Volatility continued (equation 21.14, page 489) • In our Japanese Yen example • The volatility per year for a T-day option:

  24. Volatility Term Structures (Table 21.3) Yen/Dollar volatility term structure predicted from GARCH(1,1)

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

  26. 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 Impact of 1% change in instantaneous volatility for Japanese yen example:

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