Estimating the Probability of an Options Contract Expiring Out of the Money: A Step-by-Step Guide

Understanding the likelihood of an options contract expiring out of the money (OTM) is crucial for investors seeking to manage risk. While precise predictions cannot be guaranteed due to market volatility, the Black-Scholes model offers a robust framework to estimate such probabilities. This guide will walk you through the steps to calculate the probability of an option expiring OTM.

Key Concepts

In the world of options trading, an option is said to have expired out of the money (OTM) if, at expiration, the price of the underlying asset does not reach the strike price. This phenomenon can occur for both calls (when the underlying asset price is below strike for a call) and puts (when the underlying asset price is above strike for a put). The probability of this happening can be estimated using the Black-Scholes model for European options, which is widely accepted and implemented in financial markets.

Understanding the Black-Scholes Model

The Black-Scholes model is a mathematical model used to price European options. It provides a framework for estimating the theoretical value of European call and put options, based on several key parameters:

S: Current price of the underlying asset. K: Strike price of the option. r: Risk-free interest rate, which is annualized. σ: Volatility of the underlying asset, also annualized. T: Time to expiration, in years.

Steps to Calculate the Probability of Expiring OTM

1. Identifying the Parameters

First, identify the necessary parameters that will be used in the Black-Scholes model:

S: Current stock price K: Strike price of the option r: Risk-free interest rate (annualized) σ: Volatility of the underlying asset (annualized) T: Time to expiration (in years)

2. Calculating the d2 Parameter

The Black-Scholes model utilizes two parameters, d1 and d2, which are calculated as follows:

d1 (ln(S/K) (r σ^2/2) * T) / (σ * sqrt(T))

d2 d1 - σ * sqrt(T)

3. Calculating the Probability

The probability of a call option expiring OTM (underlying asset price is less than the strike price) can be calculated using the cumulative distribution function of the standard normal distribution (N(-d2)). Similarly, the probability of a put option expiring OTM (underlying asset price is greater than the strike price) is calculated as N(d2).

PCall OTM N(-d2)

PPut OTM N(d2)

Example Calculation

Let's walk through an example to see how these calculations work in practice.

S 100 K 105 r 0.05 (or 5%) σ 0.2 (or 20%) T 0.5 (6 months)

First, calculate d1 and d2:

d1 (ln(100/105) (0.05 0.2^2/2) * 0.5) / (0.2 * sqrt(0.5))

d1 ≈ -0.15

d2 -0.15 - 0.2 * sqrt(0.5) ≈ -0.15 - 0.1414 ≈ -0.2914

Now, find PCall OTM and PPut OTM using standard normal distribution tables or calculators:

PCall OTM N(-0.2914) ≈ 0.613

PPut OTM N(0.2914) ≈ 0.387

Conclusion

By following the steps outlined above, you can determine the probability that an options contract will expire out of the money. While this method is primarily applicable to European options, it assumes that the underlying asset follows a log-normal distribution. Understanding these probabilities is crucial for managing risk and making informed trading decisions in the options market.