ANZ Finance blog | Option Pricing Model | McGraw Hill Australia New Zealand

A Cup of Coffee and an Option Pricing Model

Published 17 September 2020

By Dr Peter J Phillips, Associate Professor (Finance & Banking) University of Southern Queensland

Option Pricing Model -- McGraw Hill ANZ Finance Blog -- Australia and New Zealand

Where do finance ideas come from? A site that calls itself Ideas in Finance should probably answer this question! This is a theme that we shall return to in the future. For now, let us just say that many ideas in finance and economics trace their roots to physics. Ideas in finance and economics do not come from thin air. And they do not usually emerge from within finance and economics itself. They come from outside and begin life as analogies. When someone who is thinking about a finance problem happens to know about some other body of knowledge, a leap, a connection, is made. An analogy is created and a new finance theory is born.

I try to explain how this creative process works in one of my first ever articles. I called it Mathematics, Metaphor and Economic Visualisability. One of the most interesting things I learned while researching this article, which took about a year of fairly hard work, was that analogies and metaphors are the creative force in the social sciences, especially economics and finance. More interesting still is that analogies and metaphors are the creative force in almost all cutting-edge science. For example, in 1915, when Niels Bohr was trying to come up with a theory about what was happening deep inside the atom, his first model was an analogy between the atom and the solar system. He pictured the atom as a miniature solar system with electrons orbiting like little planets. This provided the starting point that the quantum mechanics needed.

Among the many analogies that were imported from physics into finance to give us the starting point needed to build finance theory is one that lies at the heart of options pricing. In the late 1960s, Fischer Black and Myron Scholes set out to see whether they could build a model to determine the price of an option. This is the famous Black-Scholes option pricing formula. By the way, Fischer Black has a PhD in applied mathematics (having switched from… physics!).

An option is a derivative security that derives its value from an underlying security. If you own an option on BHP shares, the price of the option depends on the ups and downs in the price of the BHP shares. The details are not important for now. We will provide a simple explanation of options in the future (or see Chapter 20 of the textbook). Finding a formula that could be used to price options was not easy. While price is always determined by supply and demand, like anything, people trading options consider a number of factors that determine how much they will pay (or accept). For example, because an option derives its value from an underlying security, the extent to which the price of that security moves up and down is an important factor to consider when deciding how much to pay for an option.

Black and Scholes teamed up at MIT and set to work. They eventually came up with what is now called the Black-Scholes equation. This Black-Scholes equation is the engine inside of the Black-Scholes formula (or model). So what has physics got to do with this? It turns out that the Black-Scholes equation, which is a partial differential equation (PDE), can be reduced to an equation from physics that describes how heat diffuses across a hotplate or how milk spreads through coffee. When you pour some milk into black coffee, it doesn’t all go to the bottom or to one side of the cup. It diffuses. It spreads out. Physicists have an equation to describe this diffusion and it’s essentially the same equation as the Black-Scholes equation! When their work led them to their PDE, Black and Scholes knew where to look for structure and possible solutions… The physics library at MIT.

Discussion Question

Finance theorists make widespread use of the idea of a random walk. That is, share prices shift from moment to moment as if they are following a random walk. There is no way to predict what will happen next. Look up ‘random walk model’. Does this have a physics analogue too? We will have more to say about this in future.

Further Reading

In Chapter 20 of the textbook we discuss options. We don’t get into higher level maths and the Black-Scholes model is discussed in general terms. This is still very important because the Black-Scholes model gives us a list of things that are important in determining the supply and demand (and therefore the price) of options.