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 \(x\).

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

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

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

Characteristic Exponent

The characteristic exponent \(\phi_{x,u}\) is defined from the characteristic function \(\Phi_{x,u}\) by

\[\begin{equation} \Phi_{x,u} = e^{-\phi_{x,u}} \end{equation}\]

The library implements the characteristic_exponent for several stochastic processes, including Brownian motion, Poisson and compound Poisson processes, the CIR square-root diffusion, Ornstein-Uhlenbeck processes, Heston and Double Heston stochastic volatility models, jump-diffusion models, and the Barndorff-Nielsen-Shephard (BNS) model. Having an analytic form of the characteristic exponent for these processes enables efficient option pricing via Fourier inversion methods such as the Lewis (2001) and Carr-Madan (1999) approaches.

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

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

Feller Condition

The Feller condition is a parameter constraint on a square-root diffusion process (such as CIR) that ensures the process remains strictly positive. For a process of the form

\[\begin{equation} dx_t = \kappa(\theta - x_t)\,dt + \sigma\sqrt{x_t}\,dw_t \end{equation}\]

the condition is

\[\begin{equation} 2\kappa\theta \geq \sigma^2 \end{equation}\]

where \(\kappa\) is the mean reversion speed, \(\theta\) is the long-run mean, and \(\sigma\) is the diffusion coefficient. When the condition holds, the origin is an inaccessible boundary, so \(x_t > 0\) for all \(t > 0\) almost surely.

In the Heston model the variance process \(v_t\) is a CIR process, so the same condition applies with \(\sigma\) being the vol of vol. The CIR.is_positive property checks whether the condition holds. The HestonCalibration class provides a feller_enforce flag (default True) that imposes this as a hard inequality constraint during optimisation.

Forward Space

Forward space is the unit-free convention in which option prices are normalised by the forward price.

For a call \(C\) and put \(P\) with strike \(K\), maturity \(T\), and forward \(F\), the forward-space prices are

\[\begin{equation} c = \frac{C}{F}, \qquad p = \frac{P}{F} \end{equation}\]

Forward-space prices are dimensionless and depend only on the log-strike \(k = \log(K/F)\), the implied volatility, and the time to maturity. They are the natural output of Fourier-based pricers and of Black pricing.

The conversion to quote-currency prices is a single multiplication by \(F\):

\[\begin{equation} C = c\, F, \qquad P = p\, F \end{equation}\]

Quantflow uses forward space everywhere downstream of the input layer. The inverse flag on OptionPrice only controls how the input price field is stored: for inverse options (option premium paid in the underlying) it already is in forward space; for non-inverse options (premium paid in the quote currency) it is the absolute quote-currency price and must be divided by \(F\) to enter forward space. The price_in_forward_space property handles both cases uniformly.

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.

Log-Strike

Log-strike, or log strike/forward ratio, is used in the context of option pricing and it is defined as

\[\begin{equation} k = \ln\frac{K}{F} \end{equation}\]

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 (ITM) and call options are out of the money (OTM). The log-strike is used as input for all Black-Scholes type formulas.

Moneyness

Moneyness is used in the context of option pricing in order to compare options with different maturities. It is defined as

\[\begin{equation} m = \frac{1}{\sqrt{\tau}}\ln{\frac{K}{F}} = \frac{k}{\sqrt{\tau}} \end{equation}\]

where \(K\) is the strike, \(F\) is the Forward price, and \(\tau\) is the time to maturity. It is used to compare options with different maturities by scaling the log-strike by the square root of time to maturity. This is because 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.

Moneyness Vol Adjusted

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

\[\begin{equation} m_\sigma = \frac{1}{\sigma\sqrt{\tau}}\ln\frac{K}{F} \end{equation}\]

where \(K\) is the strike, \(F\) is the Forward price, \(\tau\) is the time to maturity and \(\sigma\) is the implied Black volatility.

Parseval's Theorem

Parseval's theorem states that for two square-integrable functions \(f\) and \(g\) with Fourier transforms \(\hat{f}\) and \(\hat{g}\)

\[\begin{equation} \int_{-\infty}^\infty f(s)\, g(s)\, ds = \frac{1}{2\pi} \int_{-\infty}^\infty \hat{f}(u)\, \overline{\hat{g}(u)}\, du \end{equation}\]

where \(\overline{\hat{g}(u)}\) denotes the complex conjugate of \(\hat{g}(u)\).

If \(g\) is real-valued, then

\[\begin{equation} \overline{\hat{g}(u)} = \hat{g}(-u) \end{equation}\]

It is used in the derivation of the Lewis option pricing formula.

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

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

Put-Call Parity

Put-call parity is a no-arbitrage relationship between the prices of European call and put options with the same strike \(K\) and maturity. Denoting forward-space prices \(c = C/F\) and \(p = P/F\) (see Black Pricing), the relationship reads:

\[\begin{equation} c - p = 1 - \frac{K}{F} = 1 - e^k \end{equation}\]

where \(k\) is the log-strike. In quoting currency terms, multiplying through by \(F\):

\[\begin{equation} C - P = F - K \end{equation}\]

Time To Maturity (TTM)

Time to maturity is the time remaining until an option or forward contract expires, expressed in years. It is calculated using a day count convention applied to the interval between the reference date and the expiry date. For a reference date \(t_0\) and expiry date \(T\):

\[\tau = \text{dcf}(t_0, T)\]

where \(\text{dcf}\) is the day count fraction function (Act/Act by default in quantflow). TTM is denoted \(\tau\) throughout the pricing and calibration formulas.