European Options
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 call/put option is a contract giving its holder the right but not the obligation to purchase/sell an underlying asset for some specified strike price \(K\) at some future maturity time \(T\).
For an underlying asset with price \(S\), the payoff \(h\) of a call and a put with strike \(K\) at maturity \(T\) is
\[ h_{\text{call}}(S_T) = (S_T - K)^+, \qquad h_{\text{put}}(S_T) = (K - S_T)^+. \]
Throughout, \(C(S_t,t)\) and \(P(S_t,t)\) denote the call and put prices at time \(t \le T\), and \(r\) is a constant risk-free interest rate. See Derivatives for payoff diagrams.
2 Pricing European Options
2.1 Risk-Neutral Approach
By risk-neutral pricing, the price of a European option at time \(t\) is the expected payoff under the risk-neutral measure \(\mathbb{Q}\), adjusted for the time value of money
\[ V(t) = \mathbb{E}_{\mathbb{Q}}\left[ D_{t,T}\, h(S_T) \mid \mathcal{F}_t \right], \]
where \(D_{t,T}\) is the discount factor from \(T\) to \(t\). For a constant rate, \(D_{t,T} = e^{-r(T-t)}\).
2.2 Black-Scholes Formulas
Let \(N\) denote the cumulative distribution function of the standard normal distribution,
\[ N(x) = \mathbb{P}(Z \le x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}}\,e^{-u^2/2}\,du, \qquad Z \sim \mathcal{N}(0,1). \]
It satisfies \(N(x) + N(-x) = 1\).
Under the Black-Scholes-Merton Model the prices of a European vanilla call and put option at time \(t\) with underlying stock \(S_t\), strike \(K\) and maturity \(T\) are
\[ \begin{aligned} C(S_t,t) & = N(d_{+})\,S_t - N(d_{-})\,Ke^{-r(T-t)}, \\ P(S_t,t) & = N(-d_{-})\,Ke^{-r(T-t)} - N(-d_{+})\,S_t, \end{aligned} \]
where
\[ d_{+} = \frac{1}{\sigma\sqrt{T-t}} \left( \log\frac{S_t}{K} + \left( r + \frac{\sigma^2}{2} \right)(T-t) \right), \qquad d_{-} = d_{+} - \sigma\sqrt{T-t}. \]
These follow by evaluating the risk-neutral expectation above, using that \(S_T\) is log-normal under \(\mathbb{Q}\). They also solve the Black-Scholes PDE. The sensitivities of these prices are the Option Greeks.
3 Put-Call Parity
For European options on a non-dividend-paying asset with the same strike \(K\) and maturity \(T\),
\[ C(S_t,t) - P(S_t,t) = S_t - Ke^{-r(T-t)}. \]
This relation does not depend on the model for \(S\). It follows from the payoff identity
\[ (S_T - K)^+ - (K - S_T)^+ = S_T - K, \]
since taking the discounted expectation under \(\mathbb{Q}\) and using that the discounted stock price \(e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale gives
\[ C(S_t,t) - P(S_t,t) = e^{-r(T-t)}\,\mathbb{E}_{\mathbb{Q}}[S_T - K \mid \mathcal{F}_t] = S_t - Ke^{-r(T-t)}. \]
The same relation comes from replication. A long call and short put replicate a forward contract with delivery price \(K\), which is worth \(S_t - Ke^{-r(T-t)}\) (buy the asset and borrow \(Ke^{-r(T-t)}\)). Any other price would allow an arbitrage.
As a check, the Black-Scholes formulas satisfy parity because \(N(x) + N(-x) = 1\),
\[ C - P = S_t\left[N(d_+) + N(-d_+)\right] - Ke^{-r(T-t)}\left[N(d_-) + N(-d_-)\right] = S_t - Ke^{-r(T-t)}. \]
If the asset pays dividends, \(S_t\) is replaced by \(S_t\) minus the present value of the dividends. See Black-Scholes-Merton Model. Parity also follows from the Carr-Madan Formula.
4 No-Arbitrage Bounds
Since option prices are non-negative, parity gives bounds that hold in any model
\[ \max\left(S_t - Ke^{-r(T-t)},\, 0\right) \le C(S_t,t) \le S_t, \]
\[ \max\left(Ke^{-r(T-t)} - S_t,\, 0\right) \le P(S_t,t) \le Ke^{-r(T-t)}. \]
The upper bounds hold because the payoffs satisfy \((S_T-K)^+ \le S_T\) and \((K-S_T)^+ \le K\).