Formula Sheet

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1 Finance
- 1.1 Fundamentals
- 1.2 Options
2 Mathematics

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Finance

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Fundamentals

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

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

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

Continuous compounding factor:

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

Discrete discounting factor:

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

Continuous discounting factor:

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

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Options

Put-call parity:

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

Put-call delta parity:

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

Option pricing theory:

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

where $LaTex: Q$ is a risk-neutral measure.

Black-Scholes pricing formulae:

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

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

Greeks:

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

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

where

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

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

Volatility smile:

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

$LaTex: rr = \sigma_{25 \delta c} - \sigma_{25 \delta p}$

Here $LaTex: str$ , $LaTex: rr$ , and $LaTex: atm$ denote, respectively, the strangle, risk reversal, and at-the-money volatility, and $LaTex: \sigma_{25 \delta c}$ and $LaTex: \sigma_{25 \delta p}$ denote the implied volatilities of the 25 delta call and the 25 delta put.

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Mathematics

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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.

$LaTex: (a + b)^2 = a^2 + 2ab + b^2$
$LaTex: (a - b)^2 = a^2 - 2ab + b^2$

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Factoring formulae

$LaTex: a^2 - b^2 = (a + b)(a - b)$
$LaTex: a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
$LaTex: a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

These identities can be generalised as follows.

For integer $LaTex: n$ :

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

For odd $LaTex: n$ :

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

Pascal's triangle:

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

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Complex numbers

De Moivre's Theorem:

Let $LaTex: z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $LaTex: z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$ .

$LaTex: z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$ , and
$LaTex: \frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))$ .

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

$LaTex: z^n = r^n (\cos n \theta + i \sin n \theta)$ , and
$LaTex: z^{-n} = r^{-n} (\cos n \theta - i \sin n \theta)$ .

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Analysis

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Series

$LaTex: \sum_{k = 0}^n k = \frac{1}{2} n (n + 1)$ — arithmetic progression
$LaTex: \sum_{k = 0}^n r^k = \frac{1 - r^{n + 1}}{1 - r}$ — geometric progression
$LaTex: \sum_{k = 1}^n \frac{1}{k}$ , divergent — harmonic progression
$LaTex: \sum_{k = 0}^n k^2 = \frac{n (n + 1) (2n + 1)}{6}$ — pyramidal number

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Special functions

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Trigonometric functions

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Well-known values

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

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Trigonometric identities

The Pythagorean formula for sines and cosines:

$LaTex: \sin^2 \alpha + \cos^2 \alpha = 1$

Sum and difference formulae:

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

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Hyperbolic functions

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

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

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

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

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

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

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Inequalitites

Cauchy-Schwarz inequality:

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

Minkowski inequality:

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

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Power series

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

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

Taylor's series:

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

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Well-known power series

$LaTex: \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \ldots = \sum_{k=0}^{\infty} x^k$ for $LaTex: |x| < 1$ — infinite geometric series
$LaTex: \frac{1}{1 + x} = 1 - x + x^2 - x^3 + \ldots = \sum_{k=0}^{\infty} (-1)^k x^k$ for $LaTex: |x| < 1$
$LaTex: (1 + x)^{\alpha} = \sum_{k = 0}^{\infty} \left( \begin{array} \alpha \\ k \end{array} \right) x^k$ — binomial series
$LaTex: e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots = \sum_{k = 0}^{\infty} \frac{x^k}{k!}$
$LaTex: \ln(1 - x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \ldots = - \sum_{k=1}^{\infty} \frac{x^k}{k}$ for $LaTex: |x| < 1$
$LaTex: \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots = \sum_{k = 0}^{\infty} \frac{(-1)^k x^{2k + 1}}{(2k + 1)!}$ for all $LaTex: x$
$LaTex: \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots = \sum_{k = 0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!}$ for all $LaTex: x$