Introduction

In this post, we will be covering what a vanilla option is, the types of options, features of options, calculating the payoff of options, and some examples.

What is an Option

An option is a contract between two parties, that gives the holder the right, but not obligation, to buy/sell an underlying asset at a predetermined price within a certain timeframe.

A plain vanilla option is a type of option that has no special features or special terms.

Call vs Put Options

There are two types of vanilla options: call options and put options.

The holder(or buyer) of a call option has the right, but not obligation, to buy from the seller one unit of the asset for a predetermined pirce K, called the strike. For this right, the buyer pays C(t) at time t < T to the seller of the call.

The holder of a put option has the right, but not obligation, to sell from the seller one unit of the asset for a predetermined pirce K, called the strike. For this right, the buyer pays P(t) at time t < T to the seller of the put.

Both calls and puts have a predetermined expiration date T in the future, called the maturity, which puts a limit on the time the holder has to exercise the option.

European vs American Option

• European Option: the buyer can only exercise the right to buy/sell the underlying asset at maturity or time T.

• American Option: the buyer can exercise the right to buy/sell the underying asset at any time before maturity or time t < T.

In, Out, and At the money

Let S(t) be the price of an underlying asset at time t and K be the predetermined price:

A call option is:

• in the money (ITM) if S(t) > K
• out the money (OTM) if S(t) < K
• at the money (ATM) if S(t) = K

Similarly, a call option is said to expire ITM, OTM, or ATM at time T if S(T) > K, S(T) < K, or S(T) = K respectively.

A put option is:

• in the money (ITM) if S(t) < K
• out the money (OTM) if S(t) > K
• at the money (ATM) if S(t) = K

Similarly, a put option is said to expire ITM, OTM, or ATM at time T if S(T) < K, S(T) > K, or S(T) = K respectively.

Calculating Payoff of a Vanilla Option

The payoff of a call option at maturity is:

\[C(T) = max(S(T) - K, 0) = \begin{cases} S(T) - K & \text{if } S(T) > K \\ 0 & \text{if } S(T) \le K \\ \end{cases}\]

The payoff of a put option at maturity is:

\[C(T) = max(S(T) - K, 0) = \begin{cases} 0 & \text{if } S(T) \ge K \\ S(T) - K & \text{if } S(T) < K \\ \end{cases}\]

Example