Ideas from quantitative finance

I have created this site as a notebook for people interested in {mathematical | computational | quantitative} finance. Note that it has neither the goal to be accurate nor complete.

Mathematical finance can be split into:

1. Pricing models (i.e. Black-Scholes)

2. Risk management (i.e. portfolio optimisation)

3. Trading strategies

Getting Started

If you have an endless supply of time, read John C. Hull, Options, Futures, and Other Derivatives, Global Edition. Otherwise, the below “glossary” should get you acquainted with some of the basic concepts of the quant trade. The list is a non-exhaustive work in progress and is biased in the author's interest.

Financial markets are supposed to be chaotic. In the seventies, academia was quite confident that financial markets process new information immediately (such as news about a company that may drive a stock higher). It concluded that it would therefore be impossible to predict the market with publicly available information, because the news’ consequences are immediately absorbed by the market. They called such markets efficient markets. In this view, the market can be represented by a random walk. To consistently make profit in an efficient market requires to exploit non-public information (aka insider knowledge) that has not yet been absorbed by the market. Many hedge funds do exactly this (that's my claim).

Not much later, MIT mathematician James Simons et al. had the idea to model financial markets mathematically, with tremendous success to this date. By doing so, they directly proved that markets weren't fully efficient, because they operated on information every market participant also had (prices, trade volumina, …) but were highly profitable. It was the first quantitative hedge fund, today often opposed to the qualitative or macro hedge fund, such as Peter Thiel's Clarion Fund that uses “human reasoning” to find investment strategies.

Today, there are many quantitative hedge funds, such as D.E. Shaw, Two Sigma, Jane Street (ironically, D. E. Shaw's step father was strong adherent of the efficient-market hypothesis). Investment banks, although not so enigmatic as hedge funds (and playing with unsuspecting people's money) have recognised the potential of quantitative methods. For example, Goldman Sachs Securities DB analysed the risk of securities and helped the bank surviving the 2007 financial crisis (while also being one of the culprits of it).

The Black Scholes Merton model (BSM) is one of the best known tools in mathematics to find the price of a financial derivative (they are called derivatives because they derive their price and properties from a stock, and are tradable themselves). The formula can be solved for another variable and given a certain price, the volatility of the derivative can be determined. This way, one can find out what “the market” has deemed to be the (implied) volatility. Although the BSM is standard lecture and textbook material, it has severe limitations and is not thought to accurately represent reality. Trivia note: Long-Term Capital Management was a fund co-founded by Myron Scholes and Robert Merton (the ones receiving a Nobel prize for the BSM). They made use of their own models, and the fund capsized four years after the funds inception in the 1998 Russian financial crisis.

The formula assumes that the “risky asset” drifts in a Brownian Motion, which is again a random walk (we already talked abut this!). Because the Black Scholes model is typically not analytically solvable, numerical methods are utilised. One of them is Monte Carlo (and back we are in gambling territory 🎲). However, Brownian Random Motion follows a standard distribution, but financial markets usually don't.

A more primitive way to model derivatives are binomial trees, that represent the two outcomes of options (look at Sal Khan's excellent series about options or read Hull's Book P. 34ff).

Epilogue

When speaking about random walks, the notion of Hidden Markov Models may come to mind. Their parameters are found using Baum-Welch's Algorithm, similar to Gaussian Mixture Models. Bayesian Networks also sound interesting, as they seem to approximate probabilities instead of computing them exactly.

Recently, quantum computers have been studied to do exactly this Monte Carlo part faster than classical computers (which is highly interesting, and referred below). Also, I wonder whether the Monte Carlo Tree Search (highly effective in Chess programs) or other Learning Algorithms could be used to evaluate binomial trees.

Further Reading and ideas

• Nicholas Nassim Taleb's, The Black Swan is a sceptical view on the “standard” mathematical approaches and epistemological problems when modelling reality.

• Patrick Rebentrost et al., Quantum computational finance: Monte Carlo pricing of financial derivatives demonstrates how to do the Monte-Carlo part (faster) with a quantum computer. Something I have yet to dive into.

• finmath seems to be an interesting Java implementation of various financial algorithms and data structures. They have interesting examples on their website.

• JQuantLib is another one.

• The MIT Topics in Mathematics with Applications in Finance course is a nice companion to this

• Wired article about algorithmic trading and the use of new methods

• As the financial market is a highly interconnected system, Gaussian models often do not hold (as they arise with independence) ideas from biology and ecology may apply. One paradigm to examine may be Chaos Theory.

YOLO 🚀🌕 Meme Corner

• Barclays Robin Hood Explotation Strategy