From my book Financial Speculation

Francis Galton was the epitome of the wealthy upper class Englishman during the Victorian era, a polymath with a high degree of curiosity and a private income, he spent his entire life investigating and researching new ideas. His range of interests was diverse enough to encompass criminology, where he helped pioneer finger-printing techniques, weather patterns, where he devised the classification of cyclonic and anti-cyclonic weather systems; and he even conducted experiments to test the efficacy of prayer – though his results were not very encouraging on that front. And of more direct relevance to our interest in finance he spent a great deal of time looking at statistics and probability.

Whilst looking for ways to enliven his lectures on statistics Galton developed in the mid 1870’s a simple mechanical device he named a quincunx. The apparatus which he first demonstrated at the Royal Institution in London comprised a wooden box with a glass front and a funnel at the top. Metals balls of equal size and weight are dropped via the funnel to fall through a number of rows of pins spaced equally in the box. Each row was offset from the previous row so that the pins sat between the gaps of the row above. These pins then deflect each falling metal ball to the left or right with equal probability and at the bottom of the box each metal ball finished by falling into one of a number of compartments. After a number of metal balls are dropped through this device a pattern in the compartments below starts to emerge. The balls start to describe a binomial distribution which with a large number of rows approximates to our previous curve – the normal distribution.

With this device Galton sought to demonstrate that seemingly random events or facts do in fact tend to arrange themselves into a distribution. So it would appear that the distribution curve we examined earlier is a natural occurrence that appears even when seemingly random events take place. Galton went on to do a large number of experiments that looked to see if in fact distributions did appear in natural life. His researches conclusively proved that they do, and furthermore the outcomes were often quite close to the normal pattern described by the quincunx.

At this point we can depart from Francis Galton but use his ideas and clever box like device to look at financial derivatives. Derivatives have been around since finance began, some claim there is a reference in Aristotle to an option like instrument, and certainly by the Middle Ages very crude option like transactions were being executed. As we saw earlier with the story of Russell Sage, by the second half of the nineteenth century stock options were starting to emerge as a recognised, although specialist and niche, financial market. Of course options really took off with the publication of the Black-Scholes formula in 1973, which for the first time sought to accurately value options. The financial de-regulation of the late 1970’ and early 1980’s really boosted derivative trading and nowadays the global market has expanded massively. It is estimated by the latest (June 2008) Bank for International Settlements report to have an outstanding nominal value in excess of US$680 trillion. This is a truly eye watering number. To give you an idea of just how large – A trillion (being one million millions in modern usage) can be expressed in a number of ways – there are a trillion seconds in 31,710 years!

The Black-Scholes formula was just the start of a series of equations that sought to value and price options, and still remains one of the best known in the business, to calculate it, we need the following inputs:
1. The time to expiry of the instrument
2. The asset price; i.e. the stock, commodity or currency price
3. The strike price
4. The implied volatility of the instrument
5. The so called risk free interest rate – typically the yield on low risk short maturity government securities. E.g. 90 Day Treasury Notes

From these basic inputs we can get an option valuation, but it comes with a number of conditions and caveats, namely:

1. The asset price follows a log normal random walk
2. The risk free interest rate and volatility are known functions of time
3. No transaction costs in hedging portfolio
4. No dividends paid during the life of the option
5. No arbitrage possibilities
6. Continuous trading of underlying asset
7. Underlying asset can be sold short

A number of important problems strike one about these conditions; first and foremost an enormous assumption is being made that the underlying instrument is continuously traded, this of course ignores that most dangerous of foes – lack of liquidity. Secondly in the real world, brokerage, bid offer spreads, slippage (effectively the monetary cost of less than perfect liquidity) and taxes all loom large. In fact as we will see these charges can be quite punishing. So whilst Black-Scholes gives us the first serious approximation for pricing options risk it is hemmed in by a number of limitations. In fact it is probably true to say that it is more important to understand these limitations than to necessarily worry about the underlying maths. Risk assessment is not just about cold equations, the judgements we make about the softer more fuzzy elements of the decision process are often much more important. It is unlikely the maths alone will protect us – we have to know the context in which the result was calculated.

But let us go back to Mr. Galton’s box with its pins and metal balls, as it can produce a useful mental picture with which to consider and understand option pricing. Consider Chart Nine. The position of the funnel represents the current price of the underlying instrument; move it to the left to decrease the price; move it to the right to increase the price. Every row of pins is an increment in time, one day say, and the number of rows represents the time to maturity. The horizontal distance between neighbouring pins represents the volatility; moving the pins further apart increases volatility; moving them together decreases volatility. The figure shows the passage of one ball as it bumps down onto a pin and has to go either right or left, before falling to the next row. This represents the daily price movement of the underlying instrument and the movement of the option by one day towards maturity. At the bottom, the ball will drop into a box. The boxes are divided into two groups by the strike price. The boxes to the left hold winners for owners of put options; the boxes to the right hold winners for owners of call options. In each case, the boxes furthest from the strike price are the most valuable. Increase the strike price and there are more put winner boxes; decrease the strike price and there are more call winner boxes. When we drop a large number of balls, they finish up (expire) in the boxes distributed in the bell shaped curve that we met earlier. For the mathematically inclined, this requires us to deal in logarithms of prices, rather than the prices themselves, but ignoring this does not affect overall picture.

Many aspects of option behaviour can be understood from this model. In the example in the figure, the put option is ‘in the money’, while the call option is ‘out of the money’. You can see that adding more rows of pins (increasing the time to maturity) increases the number of winners that are far from the strike price, so generally increasing the value of the option. Increasing the distance between the pins (raising the volatility) has the same qualitative effect. The model also highlights the arbitrary nature of the underlying assumptions of the Black-Scholes formula. For example, why should all the pins be equi-distant?

 

Now in a way this is all just a parlour game it’s not meant to be a serious substitute for Black-Scholes or any of the myriad successor mathematical formulae for options and alike; but it does provide us once again with a quite vivid picture of option risk and a broad idea of how prices react in the three dimensional landscape of derivatives risk.