FRM 2007 1. Volatility

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The Bionic Turtle presents the first tutorial in the schedule for the 2007 FRM exam

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  • FRM 2007 1. Volatility

    1. 1. Quantitative Analysis (I.1 & I.2) 2007 FRM > 1st Movie
    2. 2. Volatility What is it? 1 3 GARCH 4 EWMA 5 Key Takeaways 6 Moving average 2 Forecasting approaches Quantitative Analysis (I.1 & I.2)
    3. 3. What is it?
    4. 4. What is it?
    5. 5. What is it? Mean =1 st moment Standard deviation = 2 nd Skewness = 3 rd Kurtosis = 4 th
    6. 6. What is it? Traditional risk measure Contrast with downside measures (negative semi-variance)
    7. 7. Approaches
    8. 8. Approaches
    9. 9. Approaches Jorion calls this “ Moving average”
    10. 10. Un-weighted (MA) <ul><li>Return = </li></ul><ul><li>Variance (Volatility 2 ) = </li></ul><ul><li>Variance = Sample standard deviation 2 of returns </li></ul><ul><li>Volatility = Sample standard deviation of returns </li></ul>Also can use percentage (%) return instead of continuously compounded (natural log) return – but the safe bet is continuous
    11. 11. Un-weighted (MA) <ul><li>Variance (Volatility 2 ) = </li></ul><ul><li>1. Assume ū = 0 2. Replace (m-1) with (m) </li></ul><ul><li>Variance (Volatility 2 ) = </li></ul>(m-1) is an “unbiased estimator.” (m) is “maximum likelihood”
    12. 12. Conditional Unconditional “ Average volatility is 15%” What is the current Volatility? Implicit Embedded in market prices Conditional A function of Yesterday’s Variance (“conditional” on recent past)
    13. 13. EWMA
    14. 14. EWMA Weighted Scheme Alphas are weights, so they must sum to one
    15. 15. EWMA Weighted Scheme Alphas are weights, so they must sum to one
    16. 16. EWMA Weighted Scheme Exponentially Weighted Moving Average (EWMA) Alphas are weights, so they must sum to one In EWMA weights also sum to one, however they decline in constant ratio (lambda)
    17. 17. EMWA (cont) Exponentially Weighted Moving Average (EWMA) RiskMetrics TM (EWMA) RiskMetrics TM is EWMA with a lambda (smoothing constant) of ~0.94 Lambda is the “persistence parameter” or “smoothing constant”
    18. 18. Weighting Schemes EWMA RiskMetrics TM =  Variance n-1 + (1-  ) Return 2 n-1 = (.94) Variance n-1 + (.06) Return 2 n-1
    19. 19. Weighting Schemes GARCH(1,1) =  Variance (Average) +  Variance n-1 +  Return 2 n-1 EWMA RiskMetrics TM =  Variance n-1 + (1-  ) Return 2 n-1 = (.94) Variance n-1 + (.06) Return 2 n-1
    20. 20. Weighting Schemes GARCH(1,1) EWMA RiskMetrics TM A special case of GARCH(1,1) where gamma=0 and (alpha + beta = 1) EWMA with lambda = 0.94
    21. 21. GARCH(1,1) If gamma < 0, then GARCH(1,1) is unstable GARCH(1,1) 
    22. 22. GARCH(1,1) to forecast volatility GARCH(1,1) <ul><li>Using GARCH(1,1) to forecast volatility </li></ul>
    23. 23. GARCH(1,1) to forecast volatility GARCH(1,1) <ul><li>Long run variance = .001% </li></ul><ul><li>Weight to squared return  = 0.15 </li></ul><ul><li>Weight to variance  = 0.75 </li></ul><ul><li>Today’s variance estimate = 0.006% </li></ul><ul><li>What is estimate volatility five days (5) forward? </li></ul>
    24. 24. GARCH(1,1) to forecast volatility GARCH(1,1) <ul><li>Long run variance = .01% </li></ul><ul><li>Weight to squared return  = 0.15 </li></ul><ul><li>Weight to variance  = 0.75 </li></ul><ul><li>Today’s variance estimate = 0.06% </li></ul><ul><li>What is estimate volatility five days (5) forward? </li></ul>
    25. 25. GARCH(1,1) to forecast volatility GARCH(1,1) <ul><li>Long run variance = .01% </li></ul><ul><li>Weight to squared return  = 0.15 </li></ul><ul><li>Weight to variance  = 0.75 </li></ul><ul><li>Today’s variance estimate = 0.06% </li></ul>
    26. 26. EWMA GARCH(1,1) <ul><li>Problem #1 </li></ul><ul><li>Yesterday’s (daily) volatility was 1% </li></ul><ul><li>Yesterday’s daily return was +2% </li></ul><ul><li>Lambda (  ) = 0.97 </li></ul>What is the EWMA estimate for today’s volatility?
    27. 27. EWMA GARCH(1,1) <ul><li>Problem #1 </li></ul><ul><li>Yesterday’s (daily) volatility was 1% </li></ul><ul><li>Yesterday’s daily return was +2% </li></ul><ul><li>Lambda (  ) = 0.97 </li></ul>What is the estimate for today’s volatility?
    28. 28. EWMA GARCH(1,1) <ul><li>Problem #1 </li></ul><ul><li>Yesterday’s (daily) volatility was 1% </li></ul><ul><li>Yesterday’s daily return was +2% </li></ul><ul><li>Lambda (  ) = 0.97 </li></ul>
    29. 29. GARCH(1,1) GARCH(1,1) <ul><li>Problem #2 (tough): </li></ul><ul><li>Yesterday’s (daily) volatility was 2% </li></ul><ul><li>Yesterday’s daily return was +10% </li></ul><ul><li>The long-run average daily variance is 0.0003 </li></ul><ul><li>The applicable GARCH(1,1) model parameters are: 0.3 weight to alpha and a 0.6 weight to beta : </li></ul>What is the estimate for today’s volatility?
    30. 30. GARCH(1,1) <ul><li>What is gamma, weight assigned to long-run variance? Since  +  +  = 1, gamma = 1 - 0.3 - 0.6 = 0.1 </li></ul><ul><li>Since long-run variance/day is 0.0003, the first term = (0.1  0.0003 = 0.00003) </li></ul><ul><li>The second term is yesterday’s weighted, squared return: (0.3) (10%) 2 = 0.003 </li></ul><ul><li>The third term is yesterday’s weighted variance: (0.6)(2%) 2 = 0.00024 </li></ul><ul><li>Today’s variance = 0.0003 + 0.003 + 0.00024 = 0.00327 </li></ul>Remember: square volatility Remember: square the return
    31. 31. GARCH(1,1) <ul><li>What is gamma, weight assigned to long-run variance? Since  +  +  = 1, gamma = 1 - 0.3 - 0.6 = 0.1 </li></ul><ul><li>Since long-run variance/day is 0.00003, the first term = (0.1  0.0003 = 0.00003) </li></ul><ul><li>The second term is yesterday’s weighted, squared return: (0.3) (10%) 2 = 0.003 </li></ul><ul><li>The third term is yesterday’s weighted variance: (0.6)(2%) 2 = 0.00024 </li></ul><ul><li>Today’s variance = 0.00003 + 0.003 + 0.00024 = 0.00327 </li></ul>Remember: square volatility Remember: square the return Volatility  0.057
    32. 32. GARCH(1,1) GARCH(1,1) <ul><li>Problem #3: </li></ul><ul><li>Yesterday’s (daily) volatility was 2% and yesterday’s daily return was +10% </li></ul><ul><li>The applicable GARCH(1,1) model parameters: </li></ul><ul><ul><li>omega = 0.00003, </li></ul></ul><ul><ul><li>0.04 weight to alpha, </li></ul></ul><ul><ul><li>0.9 weight to beta </li></ul></ul>What is the long-run variance (L V )?
    33. 33. GARCH(1,1) GARCH(1,1) <ul><li>Problem #3: </li></ul><ul><li>Yesterday’s (daily) volatility was 2% </li></ul>
    34. 34. EWMA vs. GARCH(1,1)

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