2007 FRM > 1st Movie

Quantitative Analysis(I.1 & I.2)

VolatilityVolatility

What is it?1

3

GARCH

4 EWMA

5

Key Takeaways6

Moving average

2 Forecasting approaches

Quantitative Analysis(I.1 & I.2)

What is it?

What is it?

What is it?

R~N( , ) Mean =1st momentStandard deviation = 2nd

Skewness = 3rd

Kurtosis = 4th

Mean =1st momentStandard deviation = 2nd

Skewness = 3rd

Kurtosis = 4th

What is it? Traditional risk measure

2 n

Contrast with downside measures

(negative semi-variance)

Contrast with downside measures

(negative semi-variance)

Approaches

Approaches

Approaches

Jorion calls

this

“Moving

average”

2 2

1

1 m

n n iiu

m

Un-weighted (MA)

Return =

Variance (Volatility2) =

Variance = Sample standard deviation2 of returns Volatility = Sample standard deviation of returns

ii

i

Su

S 1

ln

m

n n ii

u um

2 2

1

1( )

1

Also can use percentage (%) return instead of continuously compounded

(natural log) return –but the safe bet is continuous

Un-weighted (MA)

m

n n ii

u um

2 2

1

1( )

1

Variance (Volatility2) =

1. Assume ū = 0 2. Replace (m-1) with (m)

Variance (Volatility2) = m

n n iiu

m2 2

1

1

(m-1) is an “unbiased estimator.” (m) is “maximum likelihood”

Conditional

Unconditional

“Average volatility is 15%”

What isthe currentVolatility?

Implicit

Embedded in market prices

Conditional

A function ofYesterday’sVariance

(“conditional” on recent past)

EWMA

m

n n iiu

m2 2

1

1

EWMA

m

n i n ii

u2 2

1

WeightedScheme

Alphas are weights,

so they must sum to one

m

n n iiu

m2 2

1

1

EWMA

m

n i n ii

u2 2

1

WeightedScheme

Alphas are weights,

so they must sum to one

EWMA

m

n i n ii

u2 2

1

n n

n

n

u

u

u

2 0 21

1 22

2 23

(1 )

(1 )

(1 )

WeightedScheme

ExponentiallyWeighted

MovingAverage(EWMA)

Alphas are weights,

so they must sum to one

In EWMA weights

also sum to one,

howeverthey

decline in constant

ratio (lambda)

EMWA (cont)

2 2 2n n 1 n 1(1 )u

ExponentiallyWeighted

MovingAverage(EWMA)

2 2 2n n 1 n 1(0.94) (0.06)u RiskMetricsTM

(EWMA)

RiskMetricsTM is EWMA with a lambda (smoothing constant) of ~0.94

Lambda is the “persistence parameter” or “smoothing constant”

Lambda is the “persistence parameter” or “smoothing constant”

Weighting Schemes

EWMA

RiskMetricsTM

= Variancen-1 + (1-) Return 2 n-1

= (.94) Variancen-1 + (.06) Return 2 n-1

Weighting Schemes

GARCH(1,1)

= Variance (Average) + Variancen-1 + Return 2 n-1

EWMA

RiskMetricsTM

= Variancen-1 + (1-) Return 2 n-1

= (.94) Variancen-1 + (.06) Return 2 n-1

Weighting Schemes

2 2 2n L n 1 n 1V u

GARCH(1,1)

EWMA

RiskMetricsTM

2 2 2n n 1 n 1(1 )u

2 2 2n n 1 n 1(0.94) (0.06)u

A special case of GARCH(1,1) where gamma=0 and (alpha + beta = 1)

A special case of GARCH(1,1) where gamma=0 and (alpha + beta = 1)

EWMA with lambda = 0.94 EWMA with lambda = 0.94

GARCH(1,1)

2nσ = (weighted) Long-

run variance

+ lagged, squared return(1)

+ lagged variance (1)

2 2 2n L n 1 n 1V u If gamma < 0, then

GARCH(1,1) is unstable

If gamma < 0, then GARCH(1,1) is

unstable

GARCH(1,1)

GARCH(1,1) to forecast volatility

GARCH(1,1)• Using GARCH(1,1) to forecast volatility

2 2[ ] ( ) ( )tn t L n LE V V

GARCH(1,1) to forecast volatility

GARCH(1,1)• Long run variance = .001%• Weight to squared return = 0.15• Weight to variance = 0.75• Today’s variance estimate = 0.006%

• What is estimate volatility five days (5) forward?

GARCH(1,1) to forecast volatility

GARCH(1,1)• Long run variance = .01%• Weight to squared return = 0.15• Weight to variance = 0.75• Today’s variance estimate = 0.06%

• What is estimate volatility five days (5) forward?

2 2[ ] ( ) ( )tn t L n LE V V

GARCH(1,1) to forecast volatility

GARCH(1,1)• Long run variance = .01%• Weight to squared return = 0.15• Weight to variance = 0.75• Today’s variance estimate = 0.06%

2 2

5

[ ] ( ) ( )

0.01% (0.15 0.75) (0.06% 0.01%)

0.01% (59.05%)(.05%)

.035245%

tn t L n LE V V

EWMA

GARCH(1,1)Problem #1

• Yesterday’s (daily) volatility was 1%• Yesterday’s daily return was +2%• Lambda () = 0.97

What is the EWMA estimate for today’s volatility?

EWMA

GARCH(1,1)Problem #1

• Yesterday’s (daily) volatility was 1%• Yesterday’s daily return was +2%• Lambda () = 0.97

What is the estimate for today’s volatility?

2 2 2n n 1 n 1(1 )u

EWMA

GARCH(1,1)Problem #1

• Yesterday’s (daily) volatility was 1%• Yesterday’s daily return was +2%• Lambda () = 0.97

2 2 2(0.97)(1%) (1 0.97)(2%)

0.000097 0.000012 0.000109

0.010440 1.044%

n

GARCH(1,1)

2 2 2n L n 1 n 1V (0.3)u (0.6)

GARCH(1,1)Problem #2 (tough):

• Yesterday’s (daily) volatility was 2%• Yesterday’s daily return was +10%• The long-run average daily variance is 0.0003• The applicable GARCH(1,1) model parameters are: 0.3

weight to alpha and a 0.6 weight to beta:

What is the estimate for today’s volatility?

GARCH(1,1)

2 2 2n L n 1 n 1V (0.3)u (0.6)

• What is gamma, weight assigned to long-run variance?Since ++ = 1, gamma = 1 - 0.3 - 0.6 = 0.1

• Since long-run variance/day is 0.0003, the first term = (0.1 0.0003 = 0.00003)

• The second term is yesterday’s weighted, squared return: (0.3) (10%)2 = 0.003

• The third term is yesterday’s weighted variance:(0.6)(2%)2 = 0.00024

• Today’s variance = 0.0003 + 0.003 + 0.00024 = 0.00327

Remember: square volatility Remember: square volatility

Remember: square the return Remember: square the return

GARCH(1,1)

2 2 2n L n 1 n 1V (0.3)u (0.6)

• What is gamma, weight assigned to long-run variance?Since ++ = 1, gamma = 1 - 0.3 - 0.6 = 0.1

• Since long-run variance/day is 0.00003, the first term = (0.1 0.0003 = 0.00003)

• The second term is yesterday’s weighted, squared return: (0.3) (10%)2 = 0.003

• The third term is yesterday’s weighted variance:(0.6)(2%)2 = 0.00024

• Today’s variance = 0.00003 + 0.003 + 0.00024 = 0.00327

Remember: square volatility Remember: square volatility

Remember: square the return Remember: square the return

Volatility 0.057Volatility 0.057

GARCH(1,1)

2 2 21 1

2 2 21 10.00003 (0.04) (0.9)

n n n

n n n

u

u

GARCH(1,1)Problem #3:• Yesterday’s (daily) volatility was 2% and yesterday’s daily

return was +10%• The applicable GARCH(1,1) model parameters:

omega = 0.00003, 0.04 weight to alpha, 0.9 weight to beta

What is the long-run variance (LV)?

GARCH(1,1)

GARCH(1,1)Problem #3:• Yesterday’s (daily) volatility was 2%

0.00003.0005 .05%

1 1 0.04 0.9

VL

1 1

1

V VL L

2 2 21 1

2 2 21 10.00003 (0.04) (0.9)

n n n

n n n

u

u

EWMA vs. GARCH(1,1)