Week 12 (Chapter 9)

Volatility and Volatility Models

• Predicting the "volatility" of a financial variable is often of interest.

• Conditional variance (or standard deviation) of a financial variable is commonly referred to as “volatility”.

AR(1) Model and Volatility

Conditional variance of a financial variable, commonly referred to ‘volatility’.

• Recall a stationary AR(1) model: xt = ϕ0 + ϕ1x{t-1} + u_t.

• Unconditional mean: E(xt) = \frac{ϕ0}{1 - ϕ_1}. (1)

  • The fraction arises from solving for long-run average of the series.

• Conditional mean: E(xt|Ω{t-1}) = ϕ0 + ϕ1x_{t-1}. (2)

• Unconditional variance: Var(xt) = \frac{σ^2}{1 - ϕ1^2}. (3)

• Conditional variance: Var(xt|Ω{t-1}) = σ^2, (4) where σ^2 = Var(ut|Ω{t-1}) = Var(u_t).

Unconditional variance is bigger in this model because it doesn't account for available information, unlike conditional variance (think of restricted and unrestricted model).

Forecasting with AR(1)

• Forecast x{T+1} given ΩT

  • 

    This is the conditional mean forecast.

  • ϕ0 and ϕ1 need to be estimated.

• Conditional variance:

  

Assuming normality of u_t:

Forecasting example:

• Blue: population observations

• Black: conditional mean of the distribution

• Pink: Forecast interval (95% CI)

Time series models are good at modeling time varying conditional mean, but if we are to assume TS4 for testing, then it contradicts to forecasting.

Conditional Distribution and Interval Forecasts

• Conditional distribution of x{T+1} given ΩT depends on the distribution of u_t.

• Assuming normality of ut: x{T+1}|ΩT \sim N(ϕ0 + ϕ1xT , σ^2). (7)

• 95% confidence intervals: ϕ0 + ϕ1x_T ± 2σ.

• In practice, computed as: \hat{ϕ}0 + \hat{ϕ}1x_T ± 2\hat{σ}.

Issues with Constant Conditional Volatility

• Constant conditional volatility assumption may not always hold.

• In some periods, the assumption of conditional homoskedasticity performs poorly. For example, 20.8% of returns fall outside the 95% CI, while we'd expect only 5%.

• Conditional homoskedasticity may not be a good assumption.

Time-Varying Conditional Mean vs. Variance

• Time series models (e.g., AR(1)) are good at modeling time-varying conditional mean.

  • xt = ϕ1x{t-1} + ut

  • E[xt|Ω{t-1}] = ϕ1x{t-1} if E[ut|Ω{t-1}] = 0.

• However, under conditional homoskedasticity, Var(ut|Ω{t-1}) = σ^2, which is constant over time, are not fit to explain variable of interest.

• It is often necessary to allow “time-varying conditional variance” or “time-varying volatility”.

ARCH Models: Modeling Time-Varying Variance

• Empirical evidence of time-varying conditional variance is seen in volatility clustering.

  

• The time series is modeled as follow, xt = ϕ0 + ϕ1x{t-1} + u_t under conditional homoskedasticity.

  • Then estimate the model via OLS and obtain residuals \hat{u}t = xt − \hat{ϕ}0 − \hat{ϕ}1x_{t-1}.

  • Check if the conditional variance u_t is time-varying after obtain the time series residual.

Auto Regressive Conditional Heteroskedasticity (ARCH) Model

• If the variance of ut doesn't look stable over time, the assumption Var(ut|Ω_{t-1}) = σ^2 is problematic.

• ARCH(1) model:

  

  • c represents a constant term, and γγ (gamma) represents the coefficient that measures the sensitivity of the conditional variance to the previous period's squared error term (u_t−1²)

• This is an AR(1) framework for the time series of u_t^2.

• Conditional variance is time-varying.

• Conditional standard deviation is commonly referred to as “volatility”.

• ARCH considers time-varying volatility of a financial variable (returns).

Details on ARCH Models

• ARCH(1) model: E(ut^2 |Ω{t-1}) = c + γu_{t-1}^2 .

• Volatility: \sqrt{c + γu_{t-1}^2}

• The model is often written with a martingale difference sequence \etat w.r.t. Ωt.

• \etat is a martingale difference sequence w.r.t. Ωt if E[\etat|Ω{t-1}] = 0.

• ARCH(1): ut^2 = c + γu{t-1}^2 + η_t.

• Generalizations:

  • ARCH(2): ut^2 = c + γ1u{t-1}^2 + γ2u{t-2}^2 + ηt

Value-at-risk

When portfolio returns are normally distributed, VaR tells us the investment loss at the first percentile of the return distribution (FINC2011).

• Ignores the magnitudes of potential further losses.

• In other words, 1% of the return portfolio may be viewed as worst-case scenario, so we test how much loss is from it.

VaR can be determined by mean and SD of the distribution:

→ In conclusion: there is a 1% chance that portfolio value will fall by more than 2.33 (based on normal distribution).