Option Basics: Styles, Put-Call Parity and Greeks

Options are a common topic in mathematical finance. Beyond the Black Scholes Merton Equation (which I also discuss here), it is good to know some basics about options as a product.

A. Toponymy

Options are characterised by the rules around how optionality is exercised. The only difference between an option and a forward contract is that the execution is at the discretion of one party. They have a maturity, often noted \(T\) and a strike price called \(K\).

Two types of options are deemed "vanilla" 🍨, which means they are standardised.

• A European option can only be exercised at maturity
• An American option can be exercised at any time before maturity

There is a type of option where there are several (but discrete) points in time before maturity when it can be exercised, and it is called a Bermudan option. The name is derived from the fact that Bermuda 🌴 is between Europe and America.

Non-vanilla options are called exotics. The possibilities are endless, but here are the ones I have encountered the most often.

• An Asian option is one where the payoff is computed using an average over prices in a set period of time.
• The Lookback option can only be exercised at a maximum or minimum price recorded over a set period of time.
• A Barrier option can only be exercise is a pre-agreed limit price has been hit before maturity. A barrier option can also refer to an option which can't be exercised if such limit price was hit.
• A Binary or Digital options have a fixed payoff at maturity: it's either this amount or zero payoff.
• A Forward Start option has a start date in the future and a payoff computed in the future, but the contract is entered now, and the premium is paid now.
• A Cliquet or Ratchet option is when two consecutive forward start options are used.

B. Put-Call Parity

Each option is either a call \(C\), which gives the buyer the right to buy the underlying, or a put option \(P\), which provides the buyer with the right to sell an underlying.
Under no arbitrage assumption, combining a put and a call (short, long) with same strike price \(K\) should be the same as holding a forward \(F\) for the same maturity (and some adjustments) 💱.

\[C- P = D\times (F - K)\]with \(D\) the discount factor that is a function of maturity \(t\) and risk-free rate \(r_f\). There are other formulations, many are listed on Wikipedia (link), but this one is the most common.

The concept of put-call parity is often used for mathematical proofs in options theory 🎓.

C. Greeks

The sensitivity of the price of an option \(V\) against different variables is measured using standardised metrics. Because those metrics use letters of the greek alphabet, they are called option Greeks 🇬🇷.
There are several Greeks, but here are the most common ones.

C.1. Greeks against the price of the underlying

The most common Greeks are the ones against the price of the underlying asset \(S\). The first order is galled delta \(\Delta\). The second order is called gamma \(\Gamma\).

\[\begin{align} \Delta &= \frac{\partial V}{\partial S}\\ \Gamma &= \frac{\partial }{\partial S}\Delta = \frac{\partial^2 V}{\partial S^2} \end{align}\]

C.2. Greek against volatility

The option price against volatility 🎢 sigma \(\sigma\) is called vega \(\mathcal{V}\).

\[\mathcal{V} = \frac{\partial V}{\partial \sigma}\]

C.3. Greek against time

The option price will vary as it gets closer to maturity ⏲️. This is called time decay \(\tau\) and is measure with theta \(\Theta\). Note the negative sign for this Greek.

\[\Theta = -\frac{\partial V}{\partial \tau}\]

C.4. Greek against risk-free rate

Rho \(\rho\) quantifies how sensitive the price of an option is against the risk free rate \(r_f\).

\[\rho = \frac{\partial V}{\partial r_f}\]

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