LECTURE 8 (pas le temps de niaiser !) Flashcards
Why is the cst volatility hypothesis rejected by data ?
Volatility tend to cluser in time
Better to use conditional volatility since more relevant
What does volatility clustering suggest ?
Conditional return distribution is time-varying
Why is volatility a proxy for risk ?
- Forecasting return
- Pricing of derivative
- Asset allocation
- Risk mngmnt
What is volatility’s major problem ?
Not directly observable from returns
What is the unconditional volatility ?
sample std deviation
What are the several ways to measure volatility ?
- Squared & absolute returns
- Historical volatility
- EWMA
- RiskMetrics
- Square Root of time rule
What is the problem with squared returns ?
Noisy volatility estimator since underestimates true values
What are absolute returns ?
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
What is historical volatility ?
- 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
What is EWMA ?
• 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]
What is RiskMetrics ?
- 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
What is the square root of time rule ?
- Specific cases where possible to forecast variance for different horizon
- If not serially correlated, V[rt(k)] = tσ
- Not supported by empirical evidence
What does a volatility model describe ?
Evolution of σ(t)^2
What are the 2 types of volatility models ?
- Volatility = exact function → (G)ARCH models
* Volatility = stochastic function → stochastic volatility models
What is the basic idea of an ARCH model ?
Unexpected return is serially uncorrelated but dependent where the dependency is a quadratic funtion of lagged values
How is forecasting obtained on an ARCH model ?
Recursively
What is the characterisic of the excess kurtosis in ARCH models ?
Always positive
What are the ML estimation properties ?
- Consistency
- Asymptotic normality
- Asymptotic efficiency
- Invariance
What is the drawback of the Hessian form of Î(θ) ?
It relies on second-order derivatives of log-likelihood → measure very erratic.
What is the alternative of the hessian form ?
Use an alternative estimator of I(θ) based on first order derivatives called BHHH estimator or outer product of gradients estimator.
What are the various steps for estimation ?
- Estimate mean equation
- Select initial values for θ
- Compute conditional volatility
- Compute log-likelihood
- Change values of parameters
- Iterate steps 3-5 until convergence
What test is used for ARCH effects ?
The Lagrange Multiplier
What is the null of the LM test on ARCH ?
Ho : ϵt|It-1 ̴ N(0,σ^2) based on : α1 = … = αp = 0
What is the alternative of the LM test on ARCH ?
Ha : ϵt|It-1 ̴ ARCH(p) based on : αi ≥ 0 with at least on strict inequality.
What is the LM test statistic ?
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)
What is an alternative to the LM test ?
Ljung-Box for ϵ_t^2 with p lags → X^2(p)
What are the limits to the ARCH Models ?
- 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
