Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

14Options and the Black–Scholes formula

14.1 Introduction

A financial option is a contract that gives its holder the right, but not the obligation, to perform some financial transaction. The holder will or will not exercise that right according to what is more favourable. Let us look at a couple of examples:

• A company in the European Union has imported some goods at the cost of one million US dollars, to be paid in 60 days. Suppose that the current exchange rate for buying dollars is €0.85 but the importing company needs or wishes to buy the required dollars for payment at the payment expiration date. The company is, however, afraid that, at that date, the exchange rate might be at an unacceptable level, higher than €0.90. To solve this problem, the company can acquire a call option of one million dollars with expiration date in 60 days at the exercise price of €0.90 per dollar. That option will give the company the right to buy, at the expiration date, one million dollars at the price of €0.90 per dollar. Should the market exchange rate at the expiration date be higher than €0.90 per dollar, the company will exercise the option and will buy the dollars at the agreed price of €0.90 per dollar, since this is more advantageous. If, however, the market exchange rate at the expiration date is lower than €0.90 per dollar, the company is not obliged and will not exercise the option, but rather will buy the one million dollars at the market at the more favourable market exchange rate. ...