what is a call option?

it’s an option which give right (=option) to buyer to buy a stock at fixed price called strike

example of call option

company A buys a european call option on Microsoft stock with strike 10 and maturity 1 jan 2015

it means that if on 1/jan/2015 MSFT costs 12$ A would buy from B MSFT stocks at 10$ each . making profit of 2$/option

if MSFT costs less than 10$ ,then A would not buy the stock and let the option expire

Payoff

Black-Scholes Valuation

european option (exercicable only on maturity date) can be valued with Black-Scholes formula

Black-Scholes formula explanation

payoff could be written as $$ (S-K)^+ $$ ‘+’ means that if expression inside is negative we take it as 0
formula de Black-Scholes tells us that call’s price is an approximation to payoff formula as equals
$$ Call= S Proba1 – DiscountFactor K Proba2 $$
where Proba2 is a probability that S> K at time T and Proba1 is the same , but “in another probability measure” (whoch makes a small difference between Proba1 y Proba2)