Chapter 23: Estimating Volatilities and Correlations

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Estimating Volatility

For an asset with price $S_i$ , define the continuously compounded daily return:

$ui=ln⁡(SiSi−1)u_i = \ln\left(\frac{S_i}{S_{i-1}}\right)$

The standard estimate of daily variance:

$σn2=1m−1∑i=1m(un−i−uˉ)2\sigma_n^2 = \frac{1}{m-1}\sum_{i=1}^{m}(u_{n-i} - \bar{u})^2$

where $uˉ\bar{u}$ is the mean of $u_i$ . Usually $uˉ\bar{u}$ is assumed to be zero for simplicity (daily mean return is negligible relative to volatility).

The annualized volatility:

$σannual=σdaily×252\sigma_{\text{annual}} = \sigma_{\text{daily}} \times \sqrt{252}$

Standard approach weighs all past observations equally — not ideal because recent data is more relevant.

EWMA Model

The Exponentially Weighted Moving Average model gives more weight to recent observations. The variance estimate is updated recursively:

$σn2=λσn−12+(1−λ)un−12\sigma_n^2 = \lambda \sigma_{n-1}^2 + (1 - \lambda) u_{n-1}^2$

where $λ\lambda$ is the decay factor ( $\lambda < 1$ ).

$λ=0.94\lambda = 0.94$ is commonly used (RiskMetrics)
As $λ→1\lambda \to 1$ , the model approaches equal weighting
As $λ→0\lambda \to 0$ , only the most recent observation matters

Advantages:

Less data needed to store
More responsive to volatility changes
Automatically captures volatility clustering

The GARCH(1,1) Model

Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) adds mean reversion:

$σn2=γVL+αun−12+βσn−12\sigma_n^2 = \gamma V_L + \alpha u_{n-1}^2 + \beta \sigma_{n-1}^2$

where:

$V_L$ = long-run average variance
$γ\gamma$ = weight on long-run variance
$α\alpha$ = weight on the most recent squared return
$β\beta$ = weight on the most recent variance estimate

And the constraint: $γ+α+β=1\gamma + \alpha + \beta = 1$

Setting $ω=γVL\omega = \gamma V_L$ :

$σn2=ω+αun−12+βσn−12\sigma_n^2 = \omega + \alpha u_{n-1}^2 + \beta \sigma_{n-1}^2$

$α+β\alpha + \beta$ measures persistence of volatility
If $α+β=1\alpha + \beta = 1$ , the model becomes the EWMA model (integrated GARCH or IGARCH)
If $α+β<1\alpha + \beta < 1$ , volatility reverts to $V_L$

GARCH captures two key features of financial returns:

Volatility clustering: Large/small returns tend to be followed by large/small returns
Mean reversion: Volatility tends toward a long-run average

Choosing Between Models

Criterion	EWMA	GARCH(1,1)
Responsiveness	Good	Good (tunable)
Mean reversion	No	Yes
Forecast horizon	Short-term only	Short and long term
Parameter estimation	Simple (choose $λ\lambda$ )	Requires MLE
Tracking structural changes	Better with small $λ\lambda$	Reverts to $V_L$

Maximum Likelihood Estimation (MLE) is used to estimate GARCH parameters.

Forecasting Future Volatility

Under GARCH(1,1), the expected variance $k$ days ahead:

$E[σn+k2]=VL+(α+β)k−1(σn+12−VL)E[\sigma_{n+k}^2] = V_L + (\alpha + \beta)^{k-1}(\sigma_{n+1}^2 - V_L)$

As $\to \infty$ , the forecast converges to the long-run variance $V_L$ . The speed of convergence depends on $α+β\alpha + \beta$ .

The volatility term structure: for an option maturing at $T$ , the average variance over the life:

$V_L + \frac{1 - e^{-aT}}{aT}[\sigma(0)^2 - V_L]$

where $\ln(1/(\alpha+\beta))$ .

Estimating Correlations

The covariance between two assets $X$ and $Y$ :

$covn=λ⋅covn−1+(1−λ)⋅xn−1⋅yn−1\text{cov}_n = \lambda \cdot \text{cov}_{n-1} + (1 - \lambda) \cdot x_{n-1} \cdot y_{n-1}$

(using EWMA), where $x_i$ and $y_i$ are the returns on day $i$ .

The correlation:

$ρn=covn(X,Y)σX(n)⋅σY(n)\rho_n = \frac{\text{cov}_n(X, Y)}{\sigma_X(n) \cdot \sigma_Y(n)}$

For multivariate GARCH, there are more sophisticated models like DCC (Dynamic Conditional Correlation) and BEKK that ensure the covariance matrix is positive definite.