Lesson 1 of 20 · 6 min
Why an option has a price before expiry at all
Before expiry the outcome is unknown, and an unknown outcome with a favourable floor is worth money.
At expiry an option's value is trivial: it is worth its intrinsic value, which is arithmetic on two numbers. Everything difficult about option pricing is the question of what it is worth before then, and the answer starts with one observation.
An option's outcome is asymmetric. If the underlying moves favourably, the value grows without limit in one direction. If it moves unfavourably, the value stops at zero and does not go negative. So the holder faces a distribution of outcomes that has been truncated on one side, and a truncated distribution has a positive average even when the current intrinsic value is nothing.
That is the whole intuition. A contract that pays something in some futures and nothing in the rest, and never requires you to pay more, has a positive expected value. Pricing is the work of putting a number on it.
You can see the effect without any model. Take a $100 strike call with the stock at exactly $100. Its intrinsic value is zero. But consider two possible outcomes at expiry, $90 and $110, equally likely: the contract pays nothing in the first and ten dollars in the second, averaging five dollars. Change the outcomes to $80 and $120 and the average becomes ten dollars. Nothing changed except the width of the distribution, and the value doubled.
This is why volatility is an input to price and not a footnote. Wider possible outcomes make a truncated payoff worth more, regardless of direction.
Key terms
- Expected value
- The probability-weighted average of all outcomes. For an option it is positive whenever any favourable outcome is possible.
- Asymmetric payoff
- A payoff that is truncated on one side. An option holder's loss stops at the premium while the gain does not stop.
- Discounting
- Converting a future value into a present one, because money available later is worth less than money available now.
Worked example
Value from width alone
A $100 strike call. The stock is at $100. Consider two-outcome futures, each outcome equally likely.
- Outcomes $95 / $105
- Average payoff $2.50
- Outcomes $90 / $110
- Average payoff $5.00
- Outcomes $80 / $120
- Average payoff $10.00
- Intrinsic value in all three cases
- $0.00
- What changed
- Only the width of the distribution
$0 and $5, halved.
The option's value quadrupled while the current price, the strike and the time all stayed the same. Width alone did it.
Check yourself
A $100 call on a $100 stock has zero intrinsic value. Why does anyone pay for it?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.