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.
key points
- Computer errors:
- basic methods:
- Monte Carlo
- use randomness to solve problems that might be deterministic
- Generate inputs randomly from a probability distribution over the domain
- Compute on the inputs
- Aggregate the results
- Variance Reduction Techniques
- antithetic sampling
- stratified sampling
- common random numbers
- control variates
- importance sampling
- Quasi-Monte Carlo
- (Uniform distribution) use Weyl Sequence and Halton Sequence, but not random varibales
- (Current computer technology mostly uses pseudorandom number generators)
- display poor small sample properties compared with crude(original) Monte Carlo methods
- Numerical differentiation
- the difference of output divide by the difference of input, where the second difference is small enough (one-side or two-side)
- do it again for second order differential
- Automatic Differentiation (Autodiff)
- Autodiff is not symbolic differentiation
- save the numerical value but not a function for next step
- Autodiff and Back Propagation
- Back Propagation algorithm could be treated as a different verison of Autodiff, except the root node is the error term
- Autodiff (or other differentiation methods) is also an important part of Back Propagation algorithm
- Python实现自动微分(Automatic Differentiation)
- Automatic Differentiation Step by Step
- CSC321 Lecture 10: Automatic Differentiation
- forward mode
- start form one input varibale, compute with all nodes in order, and to the root node (the result)
- each partial differential need one cumputation flow
- reverse mode
- start from the root node (the result), and the differential of this node is just 1 (the differential of oneself is 1)
- compute with all nodes in reverse order, and to each input
- get all partial differentials only cumpute once
- Integration
- Gaussian Quadrature (for Normal distribution)
- Gauss-Hermite
- if $X\in N(\mu, \sigma)$
- $E[f(X)]\approx \cfrac{1}{\pi}\sum^n_{i=1}w_if(\sqrt{2}\sigma x_i+\mu)$
- $w_i$ and $x_i$ are the Gauss-Hermite quadrature weights and nodes
- Gauss-Laguerre
- Monte Carlo Integration
- N draws from $U(a, b): {x_1, ..., x_N}$
- $\int^b_af(x)dx\approx\cfrac{1}{N}\sum^N_{i=1}f(x_i)$
- Monte Carlo approximation is itself a random variable
- Dynamic Programming
the [QF] series finished here!