LECTURE 8 (pas le temps de niaiser !)

Question 1

Why is the cst volatility hypothesis rejected by data ?

Answer 1

Volatility tend to cluser in time

Better to use conditional volatility since more relevant

Question 2

What does volatility clustering suggest ?

Answer 2

Conditional return distribution is time-varying

Question 3

Why is volatility a proxy for risk ?

Answer 3

• Forecasting return
• Pricing of derivative
• Asset allocation
• Risk mngmnt

Question 4

What is volatility’s major problem ?

Answer 4

Not directly observable from returns

Question 5

What is the unconditional volatility ?

Answer 5

sample std deviation

Question 6

What are the several ways to measure volatility ?

Answer 6

• Squared & absolute returns
• Historical volatility
• EWMA
• RiskMetrics
• Square Root of time rule

Question 7

What is the problem with squared returns ?

Answer 7

Noisy volatility estimator since underestimates true values

Question 8

What are absolute returns ?

Answer 8

εt ̴ N(0,σt^2) then E[|εt|] = σt 2√π

where σ’t = |εt|/2√π is a proxy of σt

Goes back to 0 = mkt closure

Question 9

What is historical volatility ?

Answer 9

• Very simple measure = moving avg
• Problem : same weight to old & new information

• Drawback : generates pronounced ghost features
o After extreme event → huge increase in historical volatility stays as long as avging period

• Useful to measure LT volatility but not successful in ST

Question 10

What is EWMA ?

Answer 10

• Overweights new information → weight drops over time : φ ϵ [0;1]
o σ^2(t) = φσ^2(t-1) + (1 – φ)(r(t-1) – μ’)^2

• (1 – φ)(r(t-1) – μ’)^2 = measure of intensity of volatility‘s reaction to mkt event
• φσ^2(t-1) = measure of persistence in volatility
• Recommended value φ ϵ [0.75;0.98]

Question 11

What is RiskMetrics ?

Answer 11

• Special case of EWMA
• Log-returns = conditionally normal
• Daily volatility constructed assuming μ’=0 and φ=0.94
• Same approach to produce forecasts of covariance
• Allows estimation of large-dimensional covariance matrices
• Use same φ so covariance matrix = semi-positive definite

Question 12

What is the square root of time rule ?

Answer 12

• Specific cases where possible to forecast variance for different horizon
• If not serially correlated, V[rt(k)] = tσ
• Not supported by empirical evidence

Question 13

What does a volatility model describe ?

Answer 13

Evolution of σ(t)^2

Question 14

What are the 2 types of volatility models ?

Answer 14

• Volatility = exact function → (G)ARCH models

* Volatility = stochastic function → stochastic volatility models

Question 15

What is the basic idea of an ARCH model ?

Answer 15

Unexpected return is serially uncorrelated but dependent where the dependency is a quadratic funtion of lagged values

Question 16

How is forecasting obtained on an ARCH model ?

Answer 16

Recursively

Question 17

What is the characterisic of the excess kurtosis in ARCH models ?

Answer 17

Always positive

Question 18

What are the ML estimation properties ?

Answer 18

• Consistency
• Asymptotic normality
• Asymptotic efficiency
• Invariance

Question 19

What is the drawback of the Hessian form of Î(θ) ?

Answer 19

It relies on second-order derivatives of log-likelihood → measure very erratic.

Question 20

What is the alternative of the hessian form ?

Answer 20

Use an alternative estimator of I(θ) based on first order derivatives called BHHH estimator or outer product of gradients estimator.

Question 21

What are the various steps for estimation ?

Answer 21

1. Estimate mean equation
2. Select initial values for θ
3. Compute conditional volatility
4. Compute log-likelihood
5. Change values of parameters
6. Iterate steps 3-5 until convergence

Question 22

What test is used for ARCH effects ?

Answer 22

The Lagrange Multiplier

Question 23

What is the null of the LM test on ARCH ?

Answer 23

Ho : ϵt|It-1 ̴ N(0,σ^2) based on : α1 = … = αp = 0

Question 24

What is the alternative of the LM test on ARCH ?

Answer 24

Ha : ϵt|It-1 ̴ ARCH(p) based on : αi ≥ 0 with at least on strict inequality.

Question 25

What is the LM test statistic ?

Answer 25

TR^2 where T is the sample size & R^2 is computed from ϵ_t^2=α_0+α_1 ϵ_(t-1)^2+⋯+ α_p ϵ_(t-p)^2+v_t
It is distributed as a X^2(p)

Question 26

What is an alternative to the LM test ?

Answer 26

Ljung-Box for ϵ_t^2 with p lags → X^2(p)

Question 27

What are the limits to the ARCH Models ?

Answer 27

• Need many lags to capture dynamics of volatility
• Complicated constraints on parameters for series to be well behaved

• Positive & Negative shocks assumed to have same effect on volatility
→ In general negative shocks stronger effect on volatility

• Likely to overpredict volatility because respond slowly to large isolated shocks to return series