Lecture 9: Estimate Volatitily

Question 1

Volatility: ??

Answer 1

Volatility: Standard deviation

Question 2

Implied volatility: implied by an ?? model.

Issue: # options on same u.a. have same T but # K => # volatilities => volatility ‘smiles / smirks’

Answer 2

Implied volatility: implied by an option pricing model.

Issue: # options on same u.a. have same T but # K => # volatilities => volatility ‘smiles / smirks’

Question 3

Volatility clustering: due to heteroskedasticity

i.e. large changes tend to be followed by ? changes, of either sign, and small changes tend to be followed by ? changes.

Answer 3

Volatility clustering: due to heteroskedasticity
i.e. large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.

Question 4

Why are volatility & correlation important?
- model building approach to ?:
> We’re x% certain we won’t lose more than vega in next n days.
> require ? estimates of volatility and correlation
- valuation of derivatives requires ? over the whole life of derivative.

Answer 4

Why are volatility & correlation important?
- model building approach to VaR:
> We’re x% certain we won’t lose more than vega in next n days.
> require current estimates of volatility and correlation
- valuation of derivatives requires forecasts over the whole life of derivative.

Question 5

A key issue: volatility and correlation is not constant

• volatility that lasts a long time => ‘?’
• volatility that is short-lived => a ‘?’ component

Answer 5

A key issue: volatility and correlation is not constant

• volatility that lasts a long time => ‘persistent’
• volatility that is short-lived => a ‘news’ component

Question 6

Standard approach to volatility:
u_i = ln( S_i/ S_i-1)
Variance = (sigma_n)^2 = (1/ m-1) * SUM_i=1^m (u_n-i - average u)^2
Average daily return = average u = (1/m) * SUM_i=1^m (u_n-i)

Answer 6

Standard approach to volatility:
u_i = ln( S_i/ S_i-1)
Variance = (sigma_n)^2 = (1/ m-1) * SUM_i=1^m (u_n-i - average u)^2
Average daily return = average u = (1/m) * SUM_i=1^m (u_n-i)

Question 7

For VaR estimation:
u_i = (S_i - S_i-1) / S_i-1 (% return)
average u = 0
Variance = (sigma_n)^2 = (1/ m) * SUM_i=1^m (u_n-i)^2

Answer 7

For VaR estimation:
u_i = (S_i - S_i-1) / S_i-1 (% return)
average u = 0
Variance = (sigma_n)^2 = (1/ m) * SUM_i=1^m (u_n-i)^2

Question 8

Both of the standard approach to volatility and VaR estimation give ?? to all past observations => may be inappropriate where there’s ??

Answer 8

Both of the standard approach to volatility and VaR estimation give EQUAL WEIGHT to all past observations => may be inappropriate where there’s volatility clustering

Question 9

Weighted Average scheme:
(σ_n)^2 = ∑_i=1^m [α_i * (u_n-1)^2]
where ∑_i=1^m α_i = 1

Answer 9

Weighted Average scheme:
(σ_n)^2 = ∑_i=1^m [α_i * (u_n-1)^2]
where ∑_i=1^m α_i = 1

Question 10

ARCH(m) model:
(σ_n)^2 = γ* V_L +∑_i=1^m [α_i * (u_n-i)^2]
where γ +∑_i=1^m α_i = 1

Answer 10

ARCH(m) model:
(σ_n)^2 = γ* V_L +∑_i=1^m [α_i * (u_n-i)^2]
where γ +∑_i=1^m α_i = 1

Question 11

!! EWMA (Exponentially weighted moving average) model:
(σ_n)^2 = ??????

Intuitively, for eg., a trader wants to estimate volatility of stock for Tuesday on Tuesday morning (i.e. estimate σ_n). Volatility estimate of Monday morning (σ_n-1) and events in stock market during trading on Monday (u_n-1) both affect overall volatility of Tuesday. And weights are given to each component as λ and (1-λ).

Answer 11

!! EWMA (Exponentially weighted moving average) model:
(σ_n)^2 = λ(σ_n-1)^2 + (1-λ)(u_n-1)^2

Intuitively, for eg., a trader wants to estimate volatility of stock for Tuesday on Tuesday morning (i.e. estimate σ_n). Volatility estimate of Monday morning (σ_n-1) and events in stock market during trading on Monday (u_n-1) both affect overall volatility of Tuesday. And weights are given to each component as λ and (1-λ).

Question 12

GARCH (1,1) model:

(1,1) means 1 lag on most recent volatility estimate and 1 lag on daily squared return.

Answer 12

GARCH (1,1) model:

(1,1) means 1 lag on most recent volatility estimate and 1 lag on daily squared return.

Question 13

GARCH (1,1) model:
(σ_n)^2 = γ* V_L + α(u_n-1)^2 + β(σ_n-1)^2

V_L = long-run average variance rate = ω / (1 - α - β)

Answer 13

GARCH (1,1) model:
(σ_n)^2 = γ* V_L + α(u_n-1)^2 + β(σ_n-1)^2

V_L = long-run average variance rate = ω / (1 - α - β)

Question 14

!!! GARCH (1,1) model:

(σ_n)^2 = ????

Answer 14

!!! GARCH (1,1) model:

(σ_n)^2 = ω + α(u_n-1)^2 + β(σ_n-1)^2