#### Louis Bachelier, "A theory of speculation"

- Bachelier's dissertation ("A theory of speculation") proposed that random walks could be used to model stock prices.
- Valid if the trade is a fair bet. Intuitively a trade means that buyer believes information is positive, seller believes information is negative, therefore the trade price is the price at probability of going up == probability of going down. Equivalent logic to the Efficient Market Hypothesis (Fama, Chicago School)
- If stock prices follow a random walk, probability of distance from starting point is normal, variance increases with time. Distribution of future price at a time is normal, mean is at the starting price, variance depends on the time. As time increases, variance increases, normal curve becomes flatter
- Bachelier extended the model to options/derivatives. Fair price of an option is price that would make it a fair bet. Used random walk to calculate probability of a future price, and derived a fair price estimate.
- Basis for model is somewhat flawed, e.g if efficient market hypothesis was true, bubbles could not happen. Also, model was not fully validated with real data.

#### Maury Osborne, "Brownian motion in the stock market".

- If Bacheliers' hypothesis were right, stock prices would be normally distributed, which was not supported by real data. Osborne showed that returns were normally distributed, so stock prices follow a log normal distribution. Rate of returns follow a random walk (prices change by a fixed %age, not by a fixed amount) i.e. prices are log normally distributed
- Has an intuitive basis: Investors do not care about absolute price, they care about rate of return. Also, from the Weber-Fechner psychological principle: logarithms model human response to stimuli

- Hypothesis that markets are random seems to indicate that in the long term, investments will yield no gain, However, estimating future values of options can be used to develop instruments that yield a profit.
- Later Osborne, rejected the memoryless efficient market hypothesis in favour of the memory based models: after prices go up they are likely to go down and vice versa. Fundamental change in assumption from random walk

#### Benoit Mandelbrot, "The fractal geometry of nature"

- Mandelbroit's work showed that real market returns are governed by Levy stable distributions with 1 < alpha <2 i.e long tailed distributions

Extreme events occur much more often than predicted. Makes random walk based models obsolete. - Mandelbrots theories emerged around the same time that random walks were gaining traction in financial modeling, but were unable to gain much traction because of complexity/tractability. Random walk gives good results "most" of the time. Long tailed models are not tractable.
- Notes on long tailed distributions

- Levy-stable distribution: Alpha characterizes the tail. Normal: 2, Cauchy 1, (<1 => distribution has no average). Self similar features have no average
- Zipf's law: Frequency of occurrence of events related to ranking.
- Pareto principle: 80:20 rule
- Cauchy distribution: Long tailed distributions

#### Edward Thorp, "Delta hedging"

- Bachelier, Osborne, Mandelbrot did not apply their theories to real investments. Ed Thorp was the first to apply their theories to the market
- Card counting strategy to achieve a favourable strategy for 21. If you have a strategy (edge) that is probabilistically profitable in the long run, how can you estimate betting amount to avoid "Gamblers ruin". Thorp used Kelly'c criteria (information theory) Probability of likelihood of correctness when a message is distorted by noise. Calculated the optimal amount to gamble when betting in a favoured bet. Kelly criteria specifies fraction to bet given the advantage and payout. Shows that rate of return equals information rate
- Applied the strategy to options (warrants):
- Used Osborne/Bachelier equations to estimate how much a warrant should be worth. Thorp found most options were overpriced using pricing theory. This provided an edge in the warrant market (not the stock market).
- Used short selling of options to exploit the edge. Short selling:allows investors to bet against a stock, without owning it.
- Thorp hedged short sale of warrants against underlying stock. - The first hedge fund. Underlying stock protect against increase in option value. Protects against all but large changes in stock value. Controls risk, but does not eliminate it.
- Procedure: Fair price of option is price at which it is a fair bet.Assume stock prices are log normal, calculate option price, calculate proportion of stocks and options to execute delta hedging
- Even though hedging guarantees profit, long and short term profits are taxed differently, reducing profits

#### Fischer Black, "Black-Scholes-Merton options pricing model", "Dynamic hedging"

- CAPM: Capital Aset Pricing Model: Proposed model that assigned a price to risk. Linked risk and return via a cost benefit analysis of risk premiums
- Dynamic hedging: It is always possible to construct a portfolio consisting of an asset and its option that is always risk free
- Procedure:
- Assume there exists a mix of stock/options to construct a risk free portfolio
- Use CAPM to calculate risk free rate
- Calculate price of options in order to realize the risk free return
- Allowed banks to construct options to sell them. Banks could sell options, and reduce risk, by buying corresponding asset
- 1987 crash:Portfolio insurance based hedge fund. O'connor used a moidified BlackScholes model that accounted for long tail events- was not impacted by the 1987 crash

#### The Prediction Company, Lorenz, "Chaos theory"

- Lorenz developed Chaos theory - Sensitive dependence of state on initial conditions
- The Prediction Company was started by a group of physicists with expertise in chaos theory, and prediction algorithms. Their objective was to find the signal in the noise, applying understanding of chaos, genetic algorithms Developed statistical arbitrage on correlated assets e.g. Pairs trading, algorithms around voting for trades
- Most significant contribution was Black box modelling - building balck boxes that predicted based on accuracy on past real data (training sets, etc.)
- One premise was that markets are inherently unpredictable, obey efficient market hypothesis, which implies they should be impossible to predict. However if anamolies (e.g a stock price is away from its normal, expected value) are detected they can be exploited before market returns to equilibrium. Need computation power and speed to detect, take action

#### Didier Sornette: "Self organisation"

- Ruptures in physical systems result from a self organization of components. Self organisation:uncorrelated entities begin to join together in correlated behaviour

Log periodic patterns predict ruptures. Used to predict breaking of water tanks, earthquakes. - Specific crashes (Dragon Kings) may be caused by state of the market rather than a particular event. More extreme than long tailed events, may be predictable through log periodic observations. Self organisation is difficult to predict, has fractal properties, but log periodic behaviour in properties may indicate system is in 1a dragon king state
- Predicted the 1997 Asian currency crash, 2000 dotcom crash

#### New Manhattan Project

- Gauge theory and its application in calculating a new CPI

#### Notes:

- Renaissance Technologies:
- Medallion Hedge Fund, approx 2500% return (compared to 1700% Soros)
- 40% over lifetime, compared to 20% (Berkshire Hathaway)
- 80% return in 2008 during the crisis
- One asset is usually a derivative

- Derivative: Contract based on some kind of security: stock, bond, commodity
- Objective: Reduce risk (historically with commodity futures), now with stock futures

- Hedge fund: Counterbalanced protofclio comprising asset and its derivate. Calculate relationship between derivative prices and underlying asset price, quantify risk of a fund based on derivatives, keep portfolio in balance.

- 1971: Chicago Securities Board allowed the first options market
- Breton Woods 1944 agreement. Fixed exchange rate, all currently tied to dollar, dollar tied to gold. Abandoned by Nixon on recommendation from Milton Friedman (Chicago school). Currency futures became widely traded after this.
- 1987 crash: Portofolio insurance: Hedge: Buying a stock, short sell futures. Volatility smile: Abnormality in options pricing graphs becuase of short comings of the Black Scholes model
- 2007 crash: Banks needed an asset hat was like a treasury bond (low risk), that they could provide as collateral on deposits from corporations/other banks (shadow banking system). They used consumer debt (mortgage, credit card, student loans) - Collateral Debt Obligations (CDO). Shadow banking system collapsed when underlying assets became toxic. Mathematical models made a flawed assumption of the independence of failure of individual assets (mortgages). Failure was followed by run on the banks.