# Option price constituents

Option’s pricing is no walk in the park. Different pricing models are implemented in every piece of options-trading software and used by traders every day. Ninety-nine percent of people take Black-Scholes and other models for granted, using them with no clue about what’s going on under the hood. Well, if they looked, they’d find a bunch of complicated math.

This article graphically shows how the option’s theoretical price is influenced by three main parameters:

- time to expiration
- underlying price volatility
- distance between the option’s strike and the current value of the underlying instrument price

Instead of tying ourselves to any particular options pricing model, let’s just use some common-sense reasoning.

Without losing generality, we’ll consider a call option. All reasoning applicable to put options.

Let’s simulate the underlying price random walks from some point in time up to the option’s expiration date and discuss how these walks affect the call’s price.

## Distance between option’s strike and underlying asset price

Let the initial conditions be as follows:

- underlying instrument price at Day 1 is 50 (some units, doesn’t matter)
- some volatility in the underlying instrument price
- option series time to expiration – 250 trading days (approximately 1 year)
- we observe three out-of-the-money call options with strikes 55, 60, and 65

Now, let’s simulate a reasonable number of underlying instrument price random walks. Each walk starts at Day 1, has a price of 50, and lasts for 250 trading days.

This video shows how generated price walks evolve over time and how some of them reach and cross strikes levels:

The 55^{th} strike was crossed 43 times, while the 60^{th} only 35 times and the 65^{th} 21. From the viewpoint of a call option seller, each cross means that the option goes in-the-money and the seller is in hot water.

Because the seller gets in trouble with the 55^{th} strike more often than with the 60^{th}, they’ll sell the 55^{th} call at a higher price than the 60^{th} and 65^{th} calls. Thus, the further the strike stands from the current underlying instrument price, the safer and cheaper an option at this strike is for the seller.

## Time to expiration

To visualize how time to expiration affects an option price, let’s start with the following:

- underlying instrument price at Day 1 is 50
- some fixed underlying instrument price volatility
- two expiration dates, two option series with 125 and 250 trading days till expiration
- a call option with the 60th strike in both series

In the video below, vertical lines denote two expiration dates. The horizontal line is a 60th strike level. Blue lines are the underlying instrument price walks.

Clearly, the strike line was crossed by underlying price walks twice as many times during the whole simulation time range (from the beginning to the second expiration date) than in the first half. From the option seller’s point of view, this means that an option with the later expiration date has more chances to get an in-the-money state than the same strike with sooner expiration. More time to expiration → more risk → greater price.

## Underlying instrument price volatility

The following video contains the same experiment as the first one. The only difference is that we simulate random walks of two underlying instruments, and the volatility of one is twice as big as another. Random walks lines have different colors. Blue lines are for the more volatile instrument.

Red marks denote crossings of the strike level by the more volatile instrument. Black marks are for the less volatile. We can see that red marks dominate black ones.

More volatile instrument → greater frequency in reaching the strikes → more expensive options.

Wednesday, 28 October 2020