Volatility Modelling¶

Historical/Sample Volatility Measures¶

Annualized Vol $$ \text{Annualized Vol} = \begin{cases} s \sqrt{365.25} & \text{Daily data} \ s \sqrt{52} & \text{Weekly data} \ s \sqrt{12} & \text{Monthly data} \end{cases} $$

• Simple Moving average of volatility
• Exponential Moving average of volatility
• Simple regression of volatility

Geometric Brownian Motion Model¶

Assumes that volatility over time is stationary $$ \begin{aligned} dy_t &= \mu y_t dt + \sigma y_t dW_t \ \implies \dfrac{dy_t}{y_t} &= \mu dt + \sigma dW_t \end{aligned} $$

• \(dy_t\) is infinitesimal increment in series
• \(\mu\) is mean difference (per unit team) [drift term]
• \(\sigma\) is volatility of series
• \(dW_t\) is infinitesimal increment of standard Brownian motion/Wiener process
• \(dW_t = N(0, t'-t)\)
• \(dW_i\) and \(dW_j\) are independent

This is general case, to incorporate the possibility that the observations are at equal or unequal time intervals

• If \(\Delta j = 1\)
• \(\hat \mu = \bar R\)
• \(\hat \sigma^2 = E[(R_t - \bar R)^2]\)
• else

Garman-Klass Estimator¶

More information than just closing

Sample information of

• Period-close (last)
• Period-high (max)
• Period-low (min)
• Period-open (first)

Assume that

• \(\mu=0, \Delta_j = 1\)
• \(f \in (0, 1)\) denote the fraction of the day prior to the market open
• \(f \times T\) between the previous day’s closing and the current day’s opening, and an interval of length
• \((1-f) \times T\) between the current opening and the current closing, during which the market is open for trading
• \(T\) is the time frame between two consecutive closing prices

These expectations & variances of volatility estimate are obtained as standard deviation follows \(\chi^2\) distribution

Poisson Jump Diffusion Model¶

Assumes that volatility over time is stationary

Over time, a Brownian motion process is fully-continuous. It experiences shocks according to a Poisson process.

• Model is a Poisson mixture of Gaussian distributions
• Moment-Generating function derived as that of random sum of independent random values
• EM Algorithm expressible in closed form
• Shocks treated as latent variables which simplify computations
• Algo provides posterior estimates of number of shocks per time period

• \(dy_t\) = infinitesimal increment in series
• \(\mu =\) mean return (per unit time)
• \(\sigma =\) diffusion volatility of process
• \(dW_t =\) increment in standard Wiener process
• \(d \Pi_t =\) increment of a Poisson process with rate \(\lambda\) modelling the shock process
• \((\gamma \sigma) \times Z_t\) is the magnitude of a return shock
• \(Z_t = N(0, 1)\)
• \(\gamma =\) scale (\(\sigma\) units) of shock magnitudes

ARCH¶

AutoRegressive Conditional Heteroskedacity models

• Handles time-dependent volatility
• Specifies relative to discrete-time process for time series
• Implies that \((u_t)^2\) is an AR process

Basically AR model of variance

Type of Volatility Clustering

flowchart LR
yt1["y<sub>t-1</sub>"] --> sigmat["&sigma;<sub>t</sub>"]

sigmat & ut["u<sub>t</sub>"] --> yt["y<sub>t</sub>"]

where

• \(\mu_t\) is the mean return conditional on \(F_{t-1}\)
• \(F_{t-1}\) is the information available till time \(t-1\)

Adding \((u_t)^2 - \sigma^2_t\) to both sides of ARCH model $$ (u_t)^2 = \alpha_0 + \sum_{i=1}^p \alpha_i (u_{t-i})^2 + (u_t)^2 - \sigma^2_t $$ where

• \(E[(u_t)^2 - \sigma^2_t \vert F_t] = 0\)
• \(\text{Var}[(u_t)^2 - \sigma^2_t \vert F_t] = \text{Var}[{(u_t)^2}] = 2 \sigma^4_t\)

Lagrange Multiplier Test¶

1. Fit \(AR(p)\) on \((u_t)^2\)

The \(AR(p)\) estimates of parameters are not MLEs under gaussian assumptions; they correspond to quasi-MLE

1. LM test statistic = \(nR^2\), where \(R^2=\) R-Squared of fitted \(AR(p)\)

2. Under \(H_0\), \(n R^2 \approx {\chi^2}_p\)

Limitation¶

It can be “bursty”: Sudden jumps in volatility rather than smooth ones. Hence, it cannot effectively model persistent volatility

GARCH¶

Generalized AutoRegressive Conditional Heteroskedacity models

Basically ARMA model of variance

Type of Volatility Clustering

flowchart LR
yt1["y<sub>t-1</sub>"] & sigmat1["&sigma;<sub>t-1</sub>"] --> sigmat["&sigma;<sub>t</sub>"]
sigmat & ut["u<sub>t</sub>"] --> yt["y<sub>t</sub>"]

GARCH(1, 1)¶

• Fits most financial time-series
• Implies a ARMA(1, 1) with \((u_t)^2 - \sigma^2_t \sim WN(0, 2 \sigma^4)\)

Diagonal Vectorization¶

\(\Sigma_t\) is conditional covariance

Advantage

• Simple element-wise GARCH model

Disadvantage

• \(\Sigma_t\) not guaranteed to be positive-definite
• Prone to overfitting, so dimensionality reduction required

Stochastic Volatility Models¶

Implied Volatility¶

From options/derivatives

Volatility Clustering¶

• Large \(u_t^2\) follow large \(u_{t-1}^2\)
• Small \(u_t^2\) follow large \(u_{t-1}^2\)

GARCH Models can prescribe

• Large \(\sigma_t^2\) follow large \(\sigma_{t-1}^2\)
• Small \(\sigma_t^2\) follow small \(\sigma_{t-1}^2\)

Extended GARCH¶

• EGARCH
• TGARCH
• PGARCH
• GARCH-In-Mean

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