# Formula Sheet

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:
$str = \frac{1}{2} (\sigma_{25 \delta c} + \sigma_{25 \delta p}) - atm$
rr = σ25δc − σ25δp

Here str, rr, and atm denote, respectively, the strangle, risk reversal, and at-the-money volatility, and σ25δc and σ25δp 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 = a2 + 2ab + b2
• (ab)2 = a2 − 2ab + b2

#### Factoring formulae

• a2b2 = (a + b)(ab)
• a3b3 = (ab)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2ab + b2)

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:
a2 = b2 + c2 − 2bccos A

### Complex numbers

• De Moivre's Theorem:

Let z1 = r1(cos θ1 + isin θ1) and z2 = r2(cos θ2 + isin θ2).

1. z1z2 = r1r2(cos(θ1 + θ2) + isin(θ1 + θ2)), and
2. $\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))$.

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

1. zn = rn(cos nθ + isin nθ), and
2. zn = rn(cos nθ − isin nθ).

### Analysis

#### Series

• $\sum_{k = 0}^n k = \frac{1}{2} n (n + 1)$arithmetic progression, kth triangular number
• $\sum_{k = 0}^n r^k = \frac{1 - r^{n + 1}}{1 - r}$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
• $\sum_{k = 0}^n k^2 = \frac{n (n + 1) (2n + 1)}{6}$kth (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 2α + cos 2α = 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 | < 1infinite 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

### Ordinary differential equations (ODEs)

#### First order linear differential equation

Written in standard linear form:

$\frac{dy}{dx} + p(x) y = q(x)$

Here is the trick: by multiplying both sides by the integrating factor

$u(x) = \exp\left( \int p(x) dx \right)$

one can apply the chain rule to the left-hand side since $\frac{du}{dx} = p(x) u(x)$, so the equation becomes

(u(x)y)' = u(x)q(x)

Hence

$y = \frac{\int u(x) q(x) dx + C}{u(x)}$

If an initial condition is given, one can find the constant of integration C.

### Statistics

• Entropy of the distribution of random variable R whose ith occurrence in the distribution has probability pi:
$H(R) = - \sum_{i=1}^n p_i \log p_i$
• Relative entropy between an initial distribution P and a subsequent distribution Q:
$S(P, Q) = \mathbb{E}_Q [\log Q - \log P] = \sum_x Q(x) \log \left( \frac{Q(x)}{P(x)} \right)$

### Notation

A word on notation. The symbols we prefer to use for various quantities and operations. These conventions should hold for most of this site, but not necessarily all of it.

• C - value of an American call option to buy one unit of the asset
• c - value of a European call option to buy one unit of the asset; also denoted V
• F - forward/futures price of the asset
• F0 - forward/futures price of the asset today
• K - strike price of the option
• P - value of an American put option to buy one unit of the asset
• p - value of a European put option to buy one unit of the asset
• q - dividend yield (equity options); foreign risk-free rate (FX options)
• r - risk-free rate, continuously compounded, for an investment maturing at time T (equity options); domestic risk-free rate (FX options)
• S - spot price of the asset
• S0 - spot price of the asset today
• t - current time
• T,t *  - expiry time (expiration time, maturity time)
• V - value of a European call option to buy one unit of the asset; also denoted c
• μ - expected return on the asset per year
• σ - volatility of the spot price of the asset per year
• Φ - standard normal CDF
• ϕ - standard normal PDF
• τ - tenor of the option (time to expiry, time to maturity)