Expected Value of an Underlying Asset Versus a Call Option
Summary
The document explains why expected value differs for holding an underlying asset and holding a call option. An underlying position is exposed to the asset’s full range of possible prices, so its expected price weights every outcome by its probability. A call’s expiration payoff is zero at or below its strike and positive only above it. For a call with a strike of 100, the payoff in an above-strike outcome is the amount by which the underlying price exceeds 100, not the full underlying price.
The answer expresses these ideas as probability-weighted integrals: the underlying price is integrated across its possible outcomes, while the call payoff is integrated above the strike using the excess over the strike. This distinction concerns expected expiration payoff and does not by itself give the option’s current fair price. The explanation omits discounting, risk-neutral probabilities, early exercise, and other pricing assumptions, so the formulas should not be treated as a complete option valuation model.
Key ideas
- An underlying position’s expected price includes all possible terminal price outcomes.
- A call option has zero expiration payoff at or below its strike.
- Above the strike, a call’s payoff is the underlying price minus the strike.
- Expected payoff alone does not specify an option’s current price without further pricing assumptions.
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# How to calculate expected value for an underlying contract and expected value for an option?
# How to calculate expected value for an underlying contract and expected value for an option?
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 by Hans-Peter Schrei (score 2, accepted)
https://quant.stackexchange.com/a/74939
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.