Put Option Convexity and Payoff-Based Price Bounds

Summary

The document asks whether the price of a put struck at 100 is bounded by the average prices of puts struck at 90 and 110, and whether proving the inequality requires the Black–Scholes formula. The response points to a payoff comparison: the two outer-strike puts together produce at least as much at expiry as the middle-strike put, with a larger payoff in some underlying-price scenarios.

This illustrates how convexity of option payoffs can establish price relationships without relying on a particular pricing formula. If one portfolio's payoff is never lower, its value should be no lower under standard no-arbitrage pricing assumptions. The exchange is brief and refers readers elsewhere for a fuller explanation; it does not spell out the payoff algebra, specify contract conventions, or discuss market frictions. The displayed inequality's exact scaling also depends on how the put portfolios are defined, so the payoff comparison should be checked against those quantities.

Key ideas

• Option price inequalities can be examined by comparing expiration payoffs.
• A portfolio that pays at least as much in every scenario should not be cheaper under no-arbitrage pricing.
• Convexity of vanilla option payoffs underlies relationships among prices at different strikes.
• A payoff argument can establish a price bound without using Black–Scholes.

Tags

# Option proofing: Analytical solution for option math


# Option proofing: Analytical solution for option math












How do I prove the following equation:

P(X=100)≤(P(X=110)-P(X=90))/2

I am not sure how to start and whether it involves using the Black-Sholes formula or not (something like this: https://www.youtube.com/watch?v=LM6iMfHbQDs&t=939s). Also, please note that this is an option price of a put, not a probability.

Thank you!

## Answer by StackG (score 1)

https://quant.stackexchange.com/a/57433

The pair of puts has to be more valuable. Consider the payoff at expiry - it looks like this:

The pair of puts pay off the same or more in every scenario (usually the same, but more when you're between 90 or 110) so they MUST be more valuable.

This is actually related to the convexity of the option payoff profile... it's duscussed further in this question and answers

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.