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Convexity of European Call Prices Across Strike Prices

Article Quant Q&A · Author: Cindy Philip
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Summary

The document asks how to interpret an inequality relating prices of three European calls with the same expiration and increasing strikes. The accepted explanation invokes convexity of the call price as a function of strike: the price at an intermediate strike lies below the straight-line interpolation between prices at the two surrounding strikes. The coefficients in the inequality weight the outer option prices according to the intermediate strike's relative location.

This is a no-arbitrage shape restriction on call prices and provides a way to reason about consistency across strikes. The response gives the general convexity principle but does not derive it from option payoffs, discounting, or an arbitrage portfolio. It also states the inequality strictly, while convexity alone generally gives a non-strict bound; strictness requires additional conditions.

Key ideas

  • For a fixed expiration, European call prices are convex as a function of strike under standard no-arbitrage assumptions.
  • An intermediate-strike call price is bounded above by a weighted average of prices at lower and higher strikes.
  • The interpolation weights reflect the intermediate strike's location between the outer strikes.
  • Convexity gives a non-strict inequality unless additional conditions justify strictness.

Tags

OptionsDerivatives pricingArbitrage
Full text
# Relationship among three European call options


# Relationship among three European call options












Consider three European call options with strikes $K_1<K_2<K_3$ all at the same expiration time T. If we assume the absence of arbitrage at all earlier times t, there is a derived equation from the properties of option $$C(K_2)<\frac{K_3-K_2}{K_3-K_1}C(K_1)+\frac{K_2-K_1}{K_3-K_1}C(K_3)$$ Could someone explain how to think about this equation?

## Answer by Bob Jansen (score 1, accepted)

https://quant.stackexchange.com/a/61959

$C(K)$ is a convex function of strike, therefore it holds that:

$$C\left( t K_1 + (1-t) K_2 \right) \leq t C\left( K_1 \right) + (1-t) C\left( K_2 \right) $$ with $t = \frac{K_3-K_2}{K_3-K_1}$.

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.