List of topics for quantitative finance with concise key points and useful resources. For Quant interview preparation and quantitative finance introduction, or could be cheatsheets. Motivated by Ran Ding.
resources
key points
- Combinatorics
- Permutation
- $A^m_n=n(n-1)...(n-m+1)=\cfrac{n!}{(n-m)!}$
- Combinations
- $C^m_n=\frac{A^m_n}{m!}=\cfrac{n!}{m!(n-m)!}$
- Inclusion-Exclusion Principle
- $|A \cup B|=|A|+|B|-|A \cap B|$
- $\left|A_{1} \cup A_{2} \cup \ldots \cup A_{P}\right|=\sum_{1 \leq i \leq p}\left|A_{i}\right|-\sum_{1 \leq j_{1}<i_{2} \leq p}\left|A_{i_{1}} \cap A_{i_{2}}\right|+\sum_{1 \leq i_{1}<i_{2}<i_{3} \leq p}\left|A_{i_{1}} \cap A_{i_{2}} \cap A_{i_{3}}\right|-\ldots+(-1)^{p-1}\left|A_{1} \cap A_{2} \cap \cdots \cap A_{p}\right|$
- Bayes rule
- $P(A \mid B)=\cfrac{P(B \mid A) P(A)}{P(B)}$
- Randomness is a state of knowledge, not a state of nature.
- consider more than one model, Bayesian inference taking a weighted combination of all the models
- Variance and correlation
- Variance
- $\operatorname{var}(X)=\mathrm{E}\left[X^{2}\right]-\mathrm{E}[X]^{2}$
- Covariance
- "Pearson" correlation
- $\operatorname{cov}(X,Y)=\mathrm{E}[X Y]-\mathrm{E}[X] \mathrm{E}[Y]$
- Correlation
- $\rho(X, Y)=\cfrac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}}$
- Independence implies zero correlation
- The reverse is not true!
- If $\rho(X, Y)=0$ we say that $X$ and $Y$ are "uncorrelated."
- If $P(X=X, Y=y)=P(X=X) P(Y=y)$ we say "independent"
- Probability Distributions
- Discrete probability
- Binomial
- $\mathbb{P}(X=x)=C_{x}^{n}p^{x}(1-p)^{n-x}, k=0,1,2,\ldots$
- Poisson
- $\mathbb{P}(X=k)=\cfrac{\lambda^{k}}{k !} e^{-\lambda}, k=0,1,2, \ldots$
- Continuous probability
- Uniform
- $f(x)={\begin{array}{ll}\frac{1}{b-a} & \text { for } a<x<b \ 0 & \text { otherwise }\end{array}$
- Gaussian
- $f(x)=\cfrac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\cfrac{(x-\mu)^{2}}{2 \sigma^{2}}\right)$
- $\mathbb{E}\left(e^{X}\right)=e^{\mu+\frac{1}{2}\sigma^{2}}$
- Hash collision probability
- ref: Hash Collision Probabilities
- k randomly generated values
- each value is a non-negative integer less than N
- probability P that at least two of them are equal
- $P=1-\frac{N-1}{N} \times \frac{N-2}{N} \times \cdots \times \frac{N-(k-2)}{N} \times \frac{N-(k-1)}{N}\approx1-e^{\frac{-k(k-1)}{2 N}}\approx\frac{k^{2}}{2 N}$
- Universality of Uniform distribution (Probability integral transform)
- random variable X with c.d.f. be $F_X()$
- $Y=F_{X}(X)\sim Unif(0,1)$
- Order statistics: Expectation of Minimum
- Central Limit Theorem
- equally-weighted averages of samples from any distribution themselves are normally distributed