Glossary

Characteristic Function

The characteristic function of a random variable \(x\) is the Fourier transform of \({\mathbb P}_x\), where \({\mathbb P}_x\) is the distrubution measure of \(x\).

(83)\[\begin{equation} \Phi_{x,u} = {\mathbb E}\left[e^{i u x}\right] = \int e^{i u s} {\mathbb P}_x\left(d s\right) \end{equation}\]

If \(x\) is a continuous random variable, than the characteristic function is the Fourier transform of the PDF \(f_x\).

(84)\[\begin{equation} \Phi_{x,u} = {\mathbb E}\left[e^{i u x}\right] = \int e^{i u s} f_x\left(s\right) ds \end{equation}\]

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF), or just distribution function, of a real-valued random variable \(x\) is the function given by

(85)\[\begin{equation} F_x(s) = {\mathbb P}_x(x \leq s) \end{equation}\]

where \({\mathbb P}_x\) is the distrubution measure of \(x\).

Hurst Exponent

The Hurst exponent is a measure of the long-term memory of time series. The Hurst exponent is a measure of the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.

Check this study on the Hurst exponent with OHLC data.

Moneyness

Moneyness is used in the context of option pricing and it is defined as

(86)\[\begin{equation} \ln\frac{K}{F} \end{equation}\]

where \(K\) is the strike and \(F\) is the Forward price. A positive value implies strikes above the forward, which means put options are in the money and call options are out of the money.

Moneyness Time Adjusted

The time-adjusted moneyness is used in the context of option pricing in order to compare options with different maturities. It is defined as

(87)\[\begin{equation} \frac{1}{\sqrt{T}}\ln\frac{K}{F} \end{equation}\]

where \(K\) is the strike and \(F\) is the Forward price and \(T\) is the time to maturity.

The key reason for dividing by the square root of time-to-maturity is related to how volatility and price movement behave over time. The price of the underlying asset is subject to random fluctuations, if these fluctuations follow a Brownian motion than the standard deviation of the price movement will increase with the square root of time.

Probability Density Function (PDF)

The probability density function (PDF), or density, of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. It is related to the CDF by the formula

(88)\[\begin{equation} F_x(x) = \int_{-\infty}^x f_x(s) ds \end{equation}\]