To develop a deep understanding of valuation it is useful to be familiar with several probability and expected value concepts. Let's explore following five questions, all from the perspective of a risk-neutral world that both forms expectations and discounts using the risk-free rate:
Question 1: What is the probability that the underlying asset price will be greater than a given strike price at expiration in a risk-neutral world?
Question 2: What is the probability that the underlying asset price will be less than a given strike price at expiration in a risk-neutral world?
Question 3: What is the expected underlying asset price at expiration in a risk-neutral
world?
Question 4: If the underlying asset price will be greater than the strike price when the option expires, what is the expected underlying asset price in a risk-neutral world?
Question 5: If the underlying asset price will be less than the strike price when the option expires, what is the expected underlying asset price in a risk-neutral world?
Table 6.4 provides the answer to each of these questions. Let's explore each question and answer in-depth.
Table 6.4 Probability and expected value concepts
Question Answer
Q1: What is the probability that the underlying asset price will be greater than a given strike price at expiration in a risk-neutral world?
Q2: What is the probability that the underlying asset price will be less than a given strike price at expiration in a risk-neutral world?
Q3: What is the expected underlying asset price at expiration in a risk- neutral world?
Q4: If the underlying asset price will be greater than the strike price when the option expires, what is the expected underlying asset price in a risk- neutral world?
Q5: If the underlying asset price will be less than the strike price when the option expires, what is the expected underlying asset price in a risk-
neutral world?
Question 1: What is the probability that the underlying asset price will be greater than a given strike price at expiration in a risk-neutral world?
The probability that the underlying asset price will be greater than a given strike price in a risk-neutral world is:
Consider the following example. The underlying asset price is $24 and the strike price is
$20. Further, the risk-free interest rate is 3%, and the underlying asset volatility is 25%.
The probability that the underlying asset price will be greater than the strike price in three years is:
Question 2: What is the probability that the underlying asset price will be less than a given strike price at expiration in a risk-neutral world?
The probability that the underlying asset price will be less than a given strike price at expiration in a risk-neutral world is:
This is unsurprising. After all, is equal to . Since probabilities always sum to one it makes sense that:
For example, consider a situation where the underlying asset price is $31 and the strike price is $25. Further, the risk-free interest rate is equal to 3%, and the underlying asset volatility is 25%. The probability that the underlying asset price will be less than the strike price in three years is:
Question 3: What is the expected underlying asset price at expiration in a risk-neutral world?
The expected underlying asset price at expiration in a risk-neutral world is:
For example, consider a situation where the underlying asset price is $1.6, and the risk- free interest rate is 2%. The expected underlying asset price in two years is:
Question 4: If the underlying asset price will be greater than the strike price when the option expires, what is the expected underlying asset price in a risk-neutral world?
Let's explore this question to make sure we understand it. This question is not asking
“what is the expected underlying asset price when the option expires?” Instead, this question is asking: What is the expected underlying asset price in a risk-neutral world, conditional on it being greater than the strike price? This value is notated as . The vertical line (i.e., “|”) is “conditional on.” The answer is:
For example, consider a situation where the underlying asset price is $31 and the strike price is $25. Further, the risk-free interest rate is 3% and the underlying asset volatility is 25%. The expected underlying asset price in three years conditional on the market price being greater than the strike price is:
Question 5: If the underlying asset price will be less than the strike price when the option expires, what is the expected underlying asset price in a risk-neutral world?
Let's explore this question to make sure it is understandable. This question is asking:
What is the expected underlying asset price in a risk-neutral world, conditional on it being less than the strike price? This value is notated as . The answer is:
For example, consider a situation where the underlying asset price is $75 and the strike price is $93. Further, the risk-free interest rate is 3% and the underlying asset volatility is 25%. The expected underlying asset price in three years conditional on the market price being less than the strike price is:
Knowledge check
Q 6.19: What is the probability that the underlying asset price will be greater than a given strike price at expiration in a risk-neutral world?
Q 6.20: What is the probability that the underlying asset price will be less than a given strike price at expiration in a risk-neutral world?
Q 6.21: What is the expected underlying asset price at expiration in a risk- neutral world?
Q 6.22: If the underlying asset price will be greater than the strike price when the option expires, what is the expected underlying asset price in a risk-neutral world?
Q 6.23: If the underlying asset price will be less than the strike price when the option expires, what is the expected underlying asset price?