Getting to the Formula and the Correct Probability Distribution

Demetri Papacostas and Francesco Tonin

This chapter will finally bring us to the derivation of the Black-Scholes formula. To get there, we employ a less intuitive and more formal approach to understanding how option probabilities and returns on FX can be calculated. This chapter will explore the heart of option pricing, volatility, and tie together a few disparate concepts about the premium.

We will try to resolve the issue of how to come up with the correct probability distribution for the future, and use that distribution to arrive at the right premium. Specifically, we discover the central limit theorem and get a glimpse into the future. We will begin to understand that assumptions about how a currency moves have a significant impact on the ultimate price of the option. Then, we look at historical performance to see if that can help predict the future. Unfortunately, we find history has a limited capacity to divine the future, but if we start with history the market can then add its subjective assessment to arrive at a probability density distribution and a price. Realising how key the standard deviation is to the price of the option, we delve deeper into understanding volatility, as well as taking a look at the