Chapter 23: Estimating Volatilities and Correlations

Estimating Volatility #

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

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

The standard estimate of daily variance:

${\sigma }_n^2=\frac{1}{m-1}{\sum }_{i=1}^m(u_{n-i}-\bar{u}{)}^2$

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

The annualized volatility:

${\sigma }_{\text{annual}}={\sigma }_{\text{daily}}\times \sqrt{252}$

Standard approach weighs all past observations equally, which is 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:

${\sigma }_n^2=\lambda {\sigma }_{n-1}^2+(1-\lambda )u_{n-1}^2$

where $\lambda$ is the decay factor ($0<\lambda <1$).

• $\lambda =0.94$ is commonly used (RiskMetrics)
• As $\lambda \to 1$, the model approaches equal weighting
• As $\lambda \to 0$, 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:

${\sigma }_n^2=\gamma V_L+\alpha u_{n-1}^2+\beta {\sigma }_{n-1}^2$

where:

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

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

Setting $\omega =\gamma V_L$:

${\sigma }_n^2=\omega +\alpha u_{n-1}^2+\beta {\sigma }_{n-1}^2$

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

GARCH captures two key features of financial returns:

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

Choosing Between Models #

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

Forecasting Future Volatility #

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

$E[{\sigma }_{n+k}^2]=V_L+(\alpha +\beta {)}^{k-1}({\sigma }_{n+1}^2-V_L)$

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

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

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

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

Estimating Correlations #

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

${\text{cov}}_n=\lambda \cdot {\text{cov}}_{n-1}+(1-\lambda )\cdot x_{n-1}\cdot y_{n-1}$

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

The correlation:

${\rho }_n=\frac{{\text{cov}}_n(X,Y)}{{\sigma }_X(n)\cdot {\sigma }_Y(n)}$

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