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Preventing Call Spread Arbitrage with Convex Strike Prices

Article Quant Q&A · Author: Jojo
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Summary

Call prices with a common maturity must decrease as strike rises and must be convex across strikes. These conditions prevent arbitrage through portfolios of calls at different strikes: a middle-strike call cannot be priced above the linear interpolation of calls at surrounding strikes.

The example fixes prices for calls struck at 110 and 140, then gives upper bounds for the prices at strikes 120 and 130. Setting the intermediate prices at their respective interpolation bounds satisfies the stated convexity constraints. The example illustrates one valid construction, not a uniquely determined market price schedule; the document does not address other pricing inputs or constraints.

Key ideas

  • For calls with the same maturity, a higher strike must have a lower or equal price.
  • Call prices must be convex as a function of strike to prevent spread arbitrage.
  • A middle-strike price cannot exceed the linear interpolation of prices at surrounding strikes.
  • The stated endpoint prices impose upper bounds on the intermediate call prices.

Tags

OptionsArbitrageDerivatives pricing
Full text
# Pricing Calls with different Strikes to prevent arbitrage


# Pricing Calls with different Strikes to prevent arbitrage












Provide option prices for call options stuck at 110,120,130, and 140 for an asset at 100 so that there is no arbitrage. I was not sure how to construct the market quickly.

This is an interview question. I understand that we want to prevent the person from buying and selling a call spread and that the prices would be decreasing. But how do I calculate prices numerically for this question?

Thanks

## Answer by Gordon (score 4)

https://quant.stackexchange.com/a/36220

You basically need to maintain the convexity with respect to the strike. For example, let's assume that the price of the option struck at 110 is 5 and for the option struck at 140 is 1. Then the price $x$ for the option struck at 120 and the price $y$ for the option struck at 130 satisfy the following inequalities \begin{align*} x &\le \frac{5+y}{2},\\ y &\le \frac{x+1}{2},\\ x &\le \frac{2}{3} \times 5 + \frac{1}{3} \times 1 = \frac{11}{3},\\ y &\le \frac{1}{3} \times 5 + \frac{2}{3} \times 1 = \frac{7}{3}. \end{align*} We can set $x=\frac{11}{3}$ and $y=\frac{7}{3}$.

## Answer by nbbo2 (score 3)

https://quant.stackexchange.com/a/36221

You can (also) solve the problem by the repeated application of two rules:

Rule 1. For any two calls (with same maturity) the higher strike call is cheaper than the lower strike call.

Rule 2. For three calls, the middle strike one is cheaper than the linear interpolation of the prices of the other two calls

(Of course, it is the same thing as the convexity property, just a different way to remember it).

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.