2.2d.1. Options

Options

An option is a financial instrument (traded both OTC and on exchanges) that gives the holder the right, but not the obligation, to buy or sell an underlying asset at a fixed predetermined price called the strike price (or exercise price) on or before a future maturity date, after which the option is invalidated:

• A call option gives the holder the right to buy the underlier, whereas a put option gives the holder the right to sell the underlier.

  • The short side must exercise the opposite action once the option is exercised.

• American options can be executed before or on the maturity date, whereas European options (common form for non-US options) can only be executed on the maturity date.

The price (or premium) of an option at time $t\ge 0$ can be decomposed into two components–its intrinsic value (payoff), and its time value–such that
$P_t={\text{IV}}_t+{\text{TV}}_t.$
Here the intrinsic value (payoff) is the value the option would have if the TTM was zero:

\max(S_{t}-F_{0}, 0) & \text{call option}, \
\max(F_{0}-S_{t}, 0) & \text{put option},
\end{cases}$$

where $S_t$ is the spot at time $t$, and $F_0$ is the strike.

Thus, if $P_t>{\text{IV}}_t$ (or equivalently, ${\text{TV}}_t>0$), then one should either sell the option or hold it rather than exercise it. Note that as the option approaches maturity (i.e., $t\to T$), ${\text{TV}}_t\to 0$ and ${\text{IV}}_t\to P_t$.

A basic no-arbitrage restriction linking calls, puts, the stock, and the risk-free asset is put–call parity. For European options with the same strike $K$ and maturity $T$, the call–put spread must equal the stock price minus the present value of the strike:

$C_t-P_t=S_t-K{e}^{-r(T-t)}.$

This identity matters because it states that two portfolios with identical payoffs must have the same price. In particular, $C_t-P_t+K{e}^{-r(T-t)}$ replicates holding one share of stock, so any violation implies a law-of-one-price inconsistency (ignoring frictions).

The more volatile that the underlying asset is, the higher the probability that it will be in-the-money at expiration. Given the uncertainty of options pricing, models can be used to estimate option values–one of which is the Black-Scholes equation.

Examples

Example