Why Equivalent Log-Return Percentiles Give Different Option Premiums

Summary

The document poses a pricing puzzle under Black–Scholes assumptions: with spot at 100, zero interest, and volatility chosen so the one-year one-standard-deviation price range is 50 to 200, the call at 200 costs more than the put at 50. It asks why strikes described as equally distant in log-return terms do not produce equal premiums.

The text provides the inputs and quoted option values but no answer or derivation. It therefore serves as a question about how option prices depend on the full risk-neutral distribution and each contract’s payoff, rather than on percentile distance alone. The stated percentile range does not by itself establish equal expected payoffs or equal option values. The document assumes Black–Scholes is accurate and log returns are normal; it does not discuss dividends, alternative payoff comparisons, or extensions beyond that setup.

Key ideas

• The document compares a call and put whose strikes are symmetric in log-return terms around spot.
• It reports different premiums despite equal assumed interest rates and volatility.
• Equal percentile distance does not by itself imply equal option payoffs or prices.
• The document raises the question but does not provide a resolution.

Tags

# Black-Scholes premium mismatch for equivalent percentile strikes?


# Black-Scholes premium mismatch for equivalent percentile strikes?












For this question, assume that Black-Scholes is an accurate model and that log returns are actually normally distributed.

First, take a stock with spot price $S_0 = 100$ and an implied volatility $σ = 0.69314.$

Since $σ$ is equivalent to 1 standard deviation of log returns, and since $exp(-0.69314) = 0.5$ and $exp(0.69314) = 2.0$, then the expected range for the stock price in one year $S_{365}$ within 1 standard deviation of the mean should be $50 <= S_{365} <= 200$.

However, given time to expiration $T = 1$ and risk free rate $r = 0$ (to make the problem as simple as possible), why is the Black Scholes premium for the $X = 200$ call $C_0 = 7.86$ substantially higher than the premium for the $X = 50$ put $P_0 = 3.93$?

I had assumed these should be equivalent, since there is equal density under the curve between those strikes and the current spot price, and there is no risk free rate to justify a higher call 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.