The Rocket Science of Wall Street

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“In God we trust, All others must bring Data ”- W. Edwards Demming

The term “Data Science” is becoming increasingly common in the field of Finance. Data scientists like Robert Shiller [1], have demonstrated in the past that leveraging “Big Data” can be used to potentially predict stock market bubble and forecast early warnings about recessions and market meltdown. Ever since the latest financial crisis in 2011, the need for better albeit less tractable models are needed for highly challenging domains like stock price predictions, risk modelling, options pricing etc. Ongoing research tries to find tractable models which can capture interrelationships across verticals to help us make better predictions. In this blog we are primarily going to discuss some cutting edge techniques used in the financial services sector.[2]

Options Pricing

It is well known that stocks and bonds trading comes with intrinsic risk of market fluctuation. To protect their investment, huge financial investment institutions invest in what is known as “Options”. Options gives the investor a right (but not obligation) to buy or sell the underlying asset at a specified rate and a specified time. This works like an insurance on the stock. And since the rate or price is predetermined, the market fluctuations do not impact the investment. Pricing these options correctly to maximize their returns on investment is extremely crucial and requires complex mathematics.

In fact, the latest Options pricing methods are considered some of the most complex mathematical techniques. Huge Financial Institutions do options trading on a regular basis and these techniques are commonly used. Some of the most widely used models today are Black –Scholes Model and LIBOR model.

Black Scholes Model

This model is used for calculating a Black Scholes formula which is used to give a price estimate of Options. This model has a high margin of error but is widely accepted because the values predicted is very close to the actual prices of the options.

The model resembles a heat transfer equation and is as follows:

This model finds out the expected benefits gained from buying a stock outright and subtracts the present value of paying the price on the expiration day. This gives the fair market value of the option that is being considered. It has been found that this prediction is as close to the real market price as possible .There are various assumptions in this model which poses challenges and limits the model to some extent. However, this is still very widely used and relevant in Options pricing today.

Assumptions & Limitations:

1) This uses European Options trading terms i.e. the Options can be exercised only on the expiration day. This is not true in American markets, where options can be exercised anytime during the lifetime of the Options. However, this model can still be used because usually Options are not traded much before the expiration day because it will not give the expected benefits desired by the trader.

2) This model assumes that returns on stock are normally distributed.

3) This model uses risk free rate – discount rate on US government treasury bills with 30 days left until maturity. But this rate fluctuates a lot and these rates actually change and do not represent the risk free rate assumed in the model. This could lead to some uncertainty and errors. [3]

The LIBOR Market Model

The LIBOR market model also known as the Brace Gatarek Musiela Model is used to price interest rate derivatives and is used to predict market capital prices based on the volatility in the market [4] . The LIBOR rates calculated are based on variables called LIBOR dynamics which help in predicting forward rates of a given asset so that when a contract is signed in terms of selling or buying an asset in the future, its price is rightly estimated by both the buyer and the seller. The LIBOR Market model blends well with the Black Scholes Model, which is used to estimate a forward measure parameter, and is also consistent with the volatilities and correlations of various parameters being considered on the short term.

Challenges:

1) The LIBOR market model however cannot build a model on the basis of time as it considers volatility in the market to follow similar trends as compared to those observed over 5 years back.

2) Also the LIBOR Dynamics that are derived at the time of building the market model need to be chosen carefully and the input maturity values of the derivatives need to be a multiple of a very specific parameter called the tenor that is used in the model.

3) The Libor rate model also uses all features to predict the future price of an asset rather than taking into account just the important features. This kind of model at times could lead to over fitting issues as compared to an input features can be reduced by using a PCA based analysis.

Market Fluctuations

Market fluctuations and their predictions have been a matter of interest for a long time. This has enthralled mathematicians and financial analysts for a long time. In fact, some of the findings that are used today have been found as early as 1900s. One of those models is Bachelier Model. This model was then improvised by another mathematician Mandelbrot.

Bachelier and Mandelbrot Model

In 1900, a young Doctoral student Louis Bachelier proposed that bond prices followed a random pattern and they changed in way that is impossible to predict. However unknowingly, this student shed light on random walk movement or the “Brownian motion” which was discovered 5 years later by Einstein and earned him a Nobel Prize. Bachelier’s model, which says that stock prices move under the influence of random market shocks just like how particles behave in nature. [5]But Bachelier, made one huge assumption which became the core of the financial industry ever since, that the stock and bond price changes follow well behaved randomness portrayed by the Gaussian curve.[6]

After many decades, Benoit Mandelbrot came up with changes to the initial model given by Bachelier. He suggested that the model would work fine except for the basic assumption which goes wrong. Stocks and bond price fluctuations are not normally distributed, i.e. normal distribution of stock and bond price distributions do not represent the real situation. If stock markets were Gaussian then stock market crashed would have happen once in a Billion years. Mandelbrot’s model insists that financial market show extreme events more often than what is presented through the bell curve, that is to say that the tails are fatter in the distribution than the typical Gaussian curve. He recommends that instead of using normally distributed bell curve, a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance should be used. He observed that the independence assumption in his suggested model does not fully reflect reality in that on closer inspection, “large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.” [7]

This assumption of Gaussian distribution even today has led to many failures. A lot of other models also have this assumption as the basis of their model. The long-term Capital Management mishap in 1998 and the subprime crisis of 2007-2009 could have been avoided by the use of Mandelbrot’s model. Ever since then, Mandelbrot’s fractal model has been widely used and appreciated as it takes into account the less well behaved nature of the market fluctuations.[8]

Monte Carlo Simulation

The most famous model in the domain of financial analysis is the Monte Carlo Simulation Model. The model is known to perform effective risk analysis by repeatedly sampling a range of uncertain values (probability distribution) and calculating the outcome for every sample [9]. The simulation then produces a distribution of the outcomes and these results are further analyzed for sensitivity and scenario analysis. The effect of Monte Carlo Simulation has a major impact on Scenario analysis mainly because it considers various scenarios being considered and gives probabilistic and graphical results of possible outcomes along with its likelihood. Besides the range of values predicted, the simulation also determines the correlation between various inputs considered in areas of stock market prediction and help financial managers make better decisions.

Challenges:

1) The Monte Carlo model though considers various possibilities; the different types of uncertainties to be considered by the model are eventually determined by the investor/user. [10]

2) This method is heavily dependent on the data being provided and the model fails when enough empirical information has not been collected. This requires analysts to make several assumptions on information that is not present and could lead to human errors.[11]

3) The Monte Carlo model also fails to continuously consider partial ignorance that is usually considered by the model being used (frequentist approach).[12]

The potential these techniques have is tremendous. Optimal utilization of these approaches could bring a lot of benefits to the table. However, it is also critical to understand that some of these methods have several intricacies associated with them and are computationally expensive. Hence, in order to extract maximum value from these techniques we have to build a thorough understanding of their applications.

References

[1] http://www.mastersindatascience.org/industry/finance/

[2] statweb.stanford.edu/~ckirby/lai/pubs/2013_DataScience.pdf

[3] http://bradley.bradley.edu/~arr/bsm/pg04.html

[4] http://www.yetanotherquant.de/libor/tutorial.pdf

[5] http://www.nybooks.com/articles/archives/2013/oct/24/random-walk-louis-bachelier/

[6] http://papers.ssrn.com/sol3/papers.cfm?abstract_id=78588

[7] http://rocketscienceofwallstreet.blogspot.com/2013/05/benoit-mandelbrots-contribution-to.html

[8] http://www.nybooks.com/articles/archives/2013/oct/24/random-walk-louis-bachelier/

[9] http://www.palisade.com/risk/monte_carlo_simulation.asp

[10] http://www.spe.org/twa/print/archives/2006/2006v2n2/twa2006_v2n2_pillars.pdf

[11] http://www.tandfonline.com/doi/abs/10.1080/10807039609383659#.U0imiGRdW1Q