# Chapter 5The Black-Scholes Framework

Derivatives products have been traded in one form or another for centuries, but the development of the Black-Scholes model in the 1970s enabled financial derivatives markets to flourish by enabling volatility to be consistently priced.

Financial mathematics books generally give the derivation of the Black-Scholes formula and list the reasons why the assumptions underpinning it aren't correct in practice. Traders don't need to know how to derive the Black-Scholes formula from scratch. However, it is vital that they understand the *features* of the Black-Scholes framework since it is the foundation for all derivatives valuation.

## Black-Scholes Stochastic Differential Equation (SDE)

The Black-Scholes framework assumes that the price of the underlying (i.e., the FX spot rate) follows a geometric Brownian motion. The Black-Scholes stochastic differential equation (SDE) is:

where is the price of the underlying (spot) at time , is the change in underlying at time , and are continuously compounded (see Chapter 10) CCY1 and CCY2 interest rates respectively, ...