Returns
Log Returns
\begin{aligned}
r_t &= log\left(\frac{P_t}{P_{t-1}}\right)\\
P_t &= P_{t-1}e^{r_t}\\
\\
P_T &= P_0e^{r_1}e^{r_2}e^{r_3}...e^{r_T}\\
&= P_0e^{r_1 + r_2 + r_3+ r_T}
\end{aligned}Simple Returns
\begin{aligned}
R &= e^r - 1\\
E[R] &= e^\mu - 1\\
Var(R) &= e^{2\mu}(e^{\sigma^2} - 1)\\
\text{where }\mu &= E[r],\;\; \sigma^2 = Var(r)
\end{aligned}Standardized Random IID
\begin{aligned}
\text{If }z_t &\text{\textasciitilde} iid(0, 1)\\
\\
E[z_t] &= 0\\
E[z_t^2] &= 1\\
E[z_tz_t'] &= 0
\end{aligned}Linear Models
Generalized Random Walk Model
\begin{aligned}
\text{If } X_T &= r_1 + r_2 + …. r_T \text{ where }r_t = σ*z_t + μ \text{ and }z_t \text{\textasciitilde} iid(0,1)\\
\text{Mean }&= T\mu\\
\text{Variance }&= T\sigma^2
\end{aligned}AR(1) Model
R_t - μ = -λ(R_{t-1} - μ) + σz_tPartial Differentiation Rules
Notation
\begin{aligned}
f_x &= \frac{\partial f}{\partial x}\\
f_{xx} &= \frac{\partial^2f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)\\
f_{yy} &= \frac{\partial^2f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)\\
f_{xy} &= \frac{\partial^2f}{\partial x \partial y} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)\\
f_{yx} &= \frac{\partial^2f}{\partial y \partial x} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)\\
\end{aligned}Chain Rule I
\begin{aligned}
F(s) &= f(x(s), y(s))\\\\
\frac{\partial F}{\partial s} &= \frac{\partial f}{\partial x}\frac{dx}{ds} + \frac{\partial f}{\partial y}\frac{dy}{ds}
\end{aligned}Note: Both x and y are in terms of s (i.e. the variable we are differentiating with respect to).
Chain Rule II
\begin{aligned}
F(u, v) &= f(x(u, v), y(u, v))\\\\
\frac{\partial F}{\partial u} &= \frac{\partial x}{\partial u}\frac{\partial f}{\partial x} + \frac{\partial y}{\partial u}\frac{\partial f}{\partial y}\\\\
\frac{\partial F}{\partial v} &= \frac{\partial x}{\partial v}\frac{\partial f}{\partial x} + \frac{\partial y}{\partial v}\frac{\partial f}{\partial y}
\end{aligned}Note: Both x and y are in terms of u and v (i.e. the variables we are differentiating with respect to).
2nd Order Derivative
\begin{aligned}
\frac{\partial }{\partial z}\left(\frac{\partial y}{\partial x}\right) &= \frac{\partial }{\partial x}\left(\frac{\partial y}{\partial x}\right).\frac{\partial x}{\partial z}\\
&= \frac{\partial^2 y}{\partial x^2}.\frac{\partial x}{\partial z}
\end{aligned}Taylor Expansion
\begin{aligned}
f(x+\delta x, t+\delta t) = &f(x, t)\\
&+\frac{\partial f}{\partial x}\delta x + \frac{\partial f}{\partial t}\delta t\\
&+ \frac{1}{2}\left[\frac{\partial^2f}{\partial x^2}\delta x^2 + \frac{\partial^2f}{\partial t^2}\delta t^2 + 2\frac{\partial^2f}{\partial x \partial t}\delta t \delta x\right] + ...
\end{aligned}Normal Distribution
\begin{aligned}
x &\text{\textasciitilde} N(\mu, \sigma^2)\\
PDF, f(x) &= \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2}\\
\int_{-\infty} ^{\infty} e^{-x^2}dx &= \sqrt{\pi}
\end{aligned}