How to calculate expected value for an underlying contract and expected value for an option?

Question

In Sheldon Natenberg's Options Volatility & Pricing, he writes:

There is an important distinction between an option position and an underlying position. The expected value for an underlying contract depends on all possible price outcomes. The expected value for an option depends only on the outcomes that result in the option finishing in the money. Everything else is zero.

What does he mean by option position and underlying position? How should I understand this paragraph?

Is the expected value of an underlying calculated by $E[\text{price}] = \int_{0}^{\infty} xf(x)dx$?

Is the expected value of a call option with 100 strike calculated by $E[\text{price}] = \int_{100}^{\infty}xg(x)dx$?

Answer 1

In options trading, an option position refers to the ownership of an option contract or a combination of option contracts, whereas an underlying position refers to the ownership of the asset or security underlying the option contract.

The paragraph you cited is discussing the difference between the expected value of an underlying position and the expected value of an option position. The expected value for an underlying contract is the average value of all possible price outcomes, while the expected value for an option contract only considers the outcomes that result in the option finishing in the money.

For example, if you own a call option with a strike price of \$100, the option will only have value if the underlying asset's price is above \$100 at expiration. Therefore, the expected value of the call option only takes into account the potential price outcomes above $100. In contrast, the expected value of the underlying asset would take into account all possible price outcomes.

To answer your other questions, yes, the expected value of an underlying asset is calculated using the formula $E[\text{price}] = \int_{0}^{\infty} xf(x)dx$, where $f(x)$ is the probability density function of the underlying asset's price.

However, the formula for the expected value of a call option with a strike price of \$100 would be $E[value]=\int_{100}^{\infty} (S-100)g(S)dS$, where $g(S)$ is the probability density function of the underlying asset's price, and $(S-100)$ represents the value of the option if the underlying asset's price is above the strike price.