1 European Options
European options are the simplest kind of financial derivatives after forwards and futures. Options have become a standard tool in modern finance allowing investors to construct contracts to hedge portfolio risk, take advantage of perceived arbitrage and leverage risk.
A European put/call option is a contract giving its holder the right but not the obligation to sell/purchase an underlying asset for some specified strike price \(K\) at some future maturity time \(T\).
2 Pricing European Options
2.1 Risk-Free Measure Approach
Introducing typical mathematical notation, for underlying asset with price \(S\) the payoff function \(\Phi\) of a European call option with strike price \(K\) at maturity time \(T\) is given by \[ \Phi(T,S)=(S_{T}-K)_{+}. \] Therefore from risk-neutral pricing we have that the price of the European call option at present time \(t\) is given by the expected payoff of the option adjusted for the time-value of money \[ P(t, \Phi) = \mathbb{E}[D_{t}\Phi(T,S)], \] where \(D_{t}\) represents the discounting factor. The specific form of this discount depends on the assumed underlying model.