How Option Prices Scale When the Underlying and Strike Scale Together
Summary
The document explains how a vanilla option’s price changes when the underlying price and strike are both rescaled by the same factor, with implied volatility, time to expiry, and other pricing inputs held constant. In its example, reducing both the underlying and strike to three quarters of their original amounts reduces the option price by the same proportion. This follows because the unit in which money is measured does not change the option’s relative economics.
The answer relates this scaling property to Black–Scholes pricing, while cautioning that relying on that formula entails accepting its assumptions. The discussion does not establish that options on different securities with equal implied volatility have identical returns: strike relative to spot and other contract or market details still matter. Its core lesson is proportional price scaling for matched contracts, not a general rule about covered-call returns.
Key ideas
- When spot and strike are scaled by the same factor, a vanilla option price scales by that factor if other inputs are unchanged.
- The ratio between strike and underlying price is preserved by this rescaling.
- The denomination of money does not affect the option’s proportional value.
- The Black–Scholes framework can illustrate the property, subject to its modeling assumptions.
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Full text
# Does an option's price "ratio" with the underlying security price? # Does an option's price "ratio" with the underlying security price? I'm trying to understand option pricing better. Let's say security ABC is \$40, and a 38 PUT option with 40% implied volatility (and 90 days till expiration) is priced at X. If security ABC then drops to \$30, should the price of a $28 PUT option with 40% implied volatility (and 90 days till expiration), now have a price of: ``` 30 ----- * price of $38 PUT? 40 ``` Similarly, do stocks at around \$100 / share, have options whose time value is twice as much as a stock trading at \$50 a share, everything else (e.g. volatility) being equal? P.S. Another similar area of this question is selling covered calls. For the static return, one of course divides the sale price of the option against the cost of the underlying security, and one is often seeking to maximize that return. So, I guess one could say/ask, will a stock trading at \$40 a share, generate the same return ratio as a stock trading at \$30 a share, if those options have the same implied volatility? ## Answer by Jitse Niesen (score 4, accepted) https://quant.stackexchange.com/a/2443 Almost. If you compare the price of a \$38 put option on a security worth \$40 and the price of a \$28.50 put option on a security worth \$30, then the price of the second option is indeed 3/4 of the price of the first option (assuming the other parameters are all the same). The reason is that the unit of money does not matter. If the price of a put option with a strike of 38 dollars on a security worth 40 dollars is X dollars, then the price of a put option with a strike of 38 cents on a security worth 40 cents is X cents, and the price of a put option with a strike of 38 units of 0.75 dollar on a security worth 40 units of 0.75 dollar is X units of 0.75 dollar. You can also see this from the Black-Scholes formula, but then you have to believe all the assumptions that are made in deriving the formula.
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