Formula Sheet

From Thalesians

Here is a list of important formulae. At least those important in our work. This list covers both basic maths, statistics, and financial mathematics.

Finance

Fundamentals

Discrete compounding factor with $n$ calculations per year in $t$ years from today, the (constant) annual interest rate being $r$ :

$\text{CPDF}(n, t) = \left( 1 + \frac{r}{n} \right)^{tn}$

Values of

n

: 1 = annual, 2 = semi-annual, 4 = quarterly, 12 = monthly, 52 = weekly, 365 = daily, 8760 = hourly, minute-by-minute compounding frequency.

Continuous compounding factor:

$\text{CPDF}(\infty, t) = \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{tn} = \left\{ \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n/r} \right\}^{rt} = e^{rt}$

Discrete discounting factor:

$\text{DDFT}(n, t) = 1/\text{CPDF}(n, t) = \left(1 + \frac{r}{n}\right) = \left(1 + \frac{r}{n}\right)^{-tn}$

Continuous discounting factor:

$\text{CDCF}(t) = 1/\text{CPDF}(\infty, t) = e^{-rt}$

Options

Put-call parity:

$c - p = S_0 - \frac{K}{(1 + rT)}$

Put-call delta parity:

$\Delta_c - \Delta_p = e^{-r_f \tau}$

Option pricing theory:

$V(S, t) = e^{-r(T - t)} \mathbb{E}_Q [ \text{Option payoff at } T | S, t ]$

where $Q$ is a risk-neutral measure.

Black-Scholes pricing formulae:

$c = S_0 e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2)$

$p = K e^{-r_d T} N(-d_2) - S_0 e^{-r_f T} N(-d_1)$

Greeks:

$\Delta_c = e^{-r_f T} N(d_1)$

$\Delta_p = e^{-r_f T} [N(d_1) - 1]$

where

$d_1 = \frac{\ln(S_0 / K) + (r_d - r_f + \sigma^2 / 2) T}{\sigma \sqrt{T}}$

$d_2 = \frac{\ln(S_0 / K) + (r_d - r_f - \sigma^2 / 2) T}{\sigma \sqrt{T}} = d_1 - \sigma \sqrt{T}$

Volatility smile:

$str = \frac{1}{2} (\sigma_{25 \delta c} + \sigma_{25 \delta p}) - atm$

r r = σ 25δ c - σ 25δ p

Here $s t r$ , $r r$ , and $a t m$ denote, respectively, the strangle, risk reversal, and at-the-money volatility, and $σ 25δ c$ and $σ 25δ p$ denote the implied volatilities of the 25 delta call and the 25 delta put.

Mathematics

Algebraic identities

An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under particular conditions.

$(a + b) 2 = a 2 + 2 a b + b 2$
$(a - b) 2 = a 2 - 2 a b + b 2$

Factoring formulae

$a 2 - b 2 = (a + b)(a - b)$
$a 3 - b 3 = (a - b)(a 2 + a b + b 2)$
$a 3 + b 3 = (a + b)(a 2 - a b + b 2)$

These identities can be generalised as follows.

For integer $n$ :

$a^n - b^n = (a - b) (a^{n-1} + a^{n-2} b + \ldots + a^{n-1-k} b^k + \ldots + a b^{n-2} + b^{n-1})$

For odd $n$ :

$a^n + b^n = (a + b) (a^{n-1} - a^{n-2} b + \ldots + (-1)^k a^{n-1-k} b^k + \ldots + (-1)^{n-2} a b^{n-2} + (-1)^{n-1} b^{n-1})$

Pascal's triangle:

$\left( \begin{array}{c} n \\ r \\ \end{array} \right) + \left( \begin{array}{c} n \\ r-1 \\ \end{array} \right) = \left( \begin{array}{c} n+1 \\ r \\ \end{array} \right)$

Trigonometry

Sine Rule:

$\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}$

Cosine Rule:

a 2 = b 2 + c 2 - 2 b c cos A

Complex numbers

De Moivre's Theorem:

Let $z 1 = r 1 (cos θ 1 + i sin θ 1)$ and $z 2 = r 2 (cos θ 2 + i sin θ 2)$ .

$z 1 z 2 = r 1 r 2 (cos(θ 1 + θ 2) + i sin(θ 1 + θ 2))$ , and
$\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))$ .

As a corollary, if $n$ is a positive integer, then

$z n = r n (cos n θ + i sin n θ)$ , and
$z - n = r - n (cos n θ - i sin n θ)$ .

Analysis

Series

$\sum_{k = 0}^n k = \frac{1}{2} n (n + 1)$ — arithmetic progression, $k$ th triangular number
— geometric progression
- For example: $\sum_{k = 0}^{\infty} \frac{1}{2^k} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots =2$
$\sum_{k = 1}^n \frac{1}{k}$ , divergent — harmonic progression
$\sum_{k = 0}^n k^2 = \frac{n (n + 1) (2n + 1)}{6}$ — $k$ th (square) pyramidal number
$\sum_{k = 0}^n k^3 = \left( \sum_{k = 0}^n k \right)^2$ — squared triangular number, Nicomachus theorem

Special functions

Trigonometric functions

Well-known values

$α$	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$sin α$	0	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	1
$cos α$	1	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	0
$tan α$	0	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$\infty$

Trigonometric identities

The Pythagorean formula for sines and cosines:

$sin 2 α + cos 2 α = 1$

Sum and difference formulae:

$\sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
$\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

Hyperbolic functions

$\cosh x = \frac{e^x + e^{-x}}{2}$

$\sinh x = \frac{e^x - e^{-x}}{2}$

$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

$\coth x = \frac{1}{\tanh x} = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$

$\text{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$

$\text{cosech} x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$

Inequalitites

Cauchy-Schwarz inequality:

$\%7c\langle\mathbf{x},\mathbf{y}\rangle%7c\leq%7c%7c\mathbf{x}%7c%7c.%7c%7c\mathbf{y}%7c%7c$

Minkowski inequality:

$||\mathbf{x}%2b\mathbf{y}||\leq||\mathbf{x}||%2b||\mathbf{y}||$

Power series

Maclaurin's series expresses the function $f (x)$ in terms of its successive derivatives at $x = 0$ :

$f(x) = f(0) + x f'(0) + \frac{x^2}{2!} f''(0) + \ldots + \frac{x^n}{n!} f^{(n)}(0) + \ldots$

Taylor's series:

$f(a + h) = f(a) + h f'(a) + \frac{h^2}{2!} f''(a) + \ldots + \frac{h^n}{n!} f^{(n)}(a) + \ldots$

Well-known power series

$\frac{1}{1 - x} = 1 + x + x^2 + x^3 + \ldots = \sum_{k=0}^{\infty} x^k$ for $| x | < 1$ — infinite geometric series
$\frac{1}{1 + x} = 1 - x + x^2 - x^3 + \ldots = \sum_{k=0}^{\infty} (-1)^k x^k$ for $| x | < 1$
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (1 + x)^{\alpha} = \sum_{k = 0}^{\infty} \left( \begin{array} \alpha \\ k \end{array} \right) x^k

— binomial series

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots = \sum_{k = 0}^{\infty} \frac{x^k}{k!}$
$\ln(1 - x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \ldots = - \sum_{k=1}^{\infty} \frac{x^k}{k}$ for $| x | < 1$
$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots = \sum_{k = 0}^{\infty} \frac{(-1)^k x^{2k + 1}}{(2k + 1)!}$ for all $x$
$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots = \sum_{k = 0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!}$ for all $x$