Estimating Volatilities and Correlations Following Options, Futures, and Other Derivatives, 5th edition by John C. Hull Chapter 17

Daniel HERLEMONT

Standard Approach to Estimating Volatility


Define σn as the volatility per day between day n-1 and day n, as estimated at end of day n-1
Define Si as the value of market variable at end of day i
Define ui= ln(Si/Si-1) σ n2 = u =

1 m ∑ (u − u ) 2 m − 1 i =1 n − i

1 m ∑u m i =1 n − i

Daniel HERLEMONT

Page 1 1

Simplifications Usually Made

Define ui as (Si-Si-1)/Si-1 Assume that the mean value of ui is zero Replace m-1 by m This gives

σ 2n =

1 m 2 ∑ u m i =1 n − i

Daniel HERLEMONT

Statistical properties 2  ^2  2 ERQM =E (σ n − σ )^ 2  = σ 4   n

If prices follow a lognormal process

Example: with 50 historical price, the relative error on variance estimate is still about 20% Daniel HERLEMONT

Page 2 2

Efficient estimators with Highs and Lows Parkinson Rogers Satchell Garman Klass

Exemple: with n=50, relative error is about 10%, to be compared with the 20% error

Daniel HERLEMONT

Volatility based on Open/Close/High/Low Source: Malik Magdon-Ismail, Amir F. Atiya Volatility Estimation Using High, Low, and Close Data

Daniel HERLEMONT

Page 3 3

Difficulty in evaluating and comparing volatility models is due to the fact that volatility is not directly observable. An approach to solve this particular problem is to compare the volatility forecasts, with squared returns

Daniel HERLEMONT

Weighting Scheme

Instead of assigning equal weights to the observations we can set

∑

σ 2n =

m i =1

α i u n2− i

where m

∑α

i

=1

i =1

Daniel HERLEMONT

Page 4 4

ARCH(m) Model

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

σ n2 = γ V L +

∑

m i =1

α i u n2 − i

where m

γ+

∑α

i

=1

i =1

Daniel HERLEMONT

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

σ n2 = λσ n2 −1 + (1 − λ ) u n2−1

Daniel HERLEMONT

Page 5 5

Attractions 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 RiskMetrics uses λ = 0.94 for daily volatility forecasting

Daniel HERLEMONT

GARCH (1,1) In GARCH (1,1) we assign some weight to the long-run average variance rate

σ n2 = γ V L + α u n2−1 + βσ 2n −1 Since weights must sum to 1 γ + α + β =1

Daniel HERLEMONT

Page 6 6

GARCH (1,1) continued

Setting ω = γV the GARCH (1,1) model is

σ 2n = ω + α u n2−1 + βσ 2n −1 and

VL =

ω 1− α − β

Daniel HERLEMONT

Example

Suppose

σ n2 = 0.000002 + 0.13un2−1 + 0.86σ n2−1 The long-run variance rate is 0.0002 so that the longrun volatility per day is 1.4%

Daniel HERLEMONT

Page 7 7

Example continued

Suppose that the current estimate of the volatility is 1.6% per day and the most recent percentage change in the market variable is 1%. The new variance rate is 0.000002 + 013 . × 0.0001 + 0.86 × 0.000256 = 0.00023336

The new volatility is 1.53% per day

Daniel HERLEMONT

GARCH (p,q)

p

σ

2 n

=ω +

∑α u i

i =1

q

2 n−i

+

∑β σ j

2 n− j

j =1

Daniel HERLEMONT

Page 8 8

Other Models

We can design GARCH models so that the weight given to ui2 depends on whether ui is positive or negative We do not have to assume that the conditional distribution is normal (e.g. Student residuals)

Daniel HERLEMONT

Variance Targeting

One way of implementing GARCH(1,1) that increases stability is by using variance targeting We set the long-run average volatility equal to the sample variance Only two other parameters then have to be estimated

Daniel HERLEMONT

Page 9 9

Maximum Likelihood Methods

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

Daniel HERLEMONT

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 outcome is

10p(1− p)9 We maximize this to obtain a maximum likelihood estimate: p=0.1

Daniel HERLEMONT

Page 10 10

Example 2

Estimate the variance of observations from a normal distribution with mean zero  1  − ui2   Maximize: ∏  exp    2v   i = 1  2 πv n  ui2  or: − ln( v ) − ∑  v  i =1  1 n This gives: v = ∑ ui2 n i =1 n

Daniel HERLEMONT

Application to GARCH

We choose parameters that maximize

 ui2  ∑ − ln(vi ) −  vi  i =1  n

Daniel HERLEMONT

Page 11 11

How Good is the Model?

The Ljung-Box statistic tests for autocorrelation We compare the autocorrelation of the ui2 with the autocorrelation of the ui2/σ σ i2 Is σi2 a predictor of ui+12 ? Perform a regression of σi2 versus ui+12

Daniel HERLEMONT

Forecasting Future Volatility A few lines of algebra shows that

E[σ 2n + k ] = VL + (α + β) k (σ 2n − VL ) The variance rate for an option expiring on day m is 1 m

m −1

∑ E [σ

2 n+k

]

k =0

Daniel HERLEMONT

Page 12 12

Correlations

Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1 Also σu,n: daily vol of U calculated on day n-1 σv,n: daily vol of V calculated on day n-1 covn: covariance calculated on day n-1 σu,n σv,n) The correlation is covn/(σ

Daniel HERLEMONT

Correlations continued

Under EWMA β covn-1 covn = (1− (1−β) un-1vn-1+β Under GARCH (1,1) covn = ω + α un-1vn-1+β β covn-1

Daniel HERLEMONT

Page 13 13