Options Arbitrage

European Put Convexity in Strike

Given P(30)=6, P(20)=4, and P(0)=0, do these prices violate convexity?

1. Plot the three quoted prices

The points are (0,0), (20,4), and (30,6).

2. Why this does not contradict convexity

A convex function does not need to be curved. A linear function is also convex. Therefore, the fact that the three prices lie on one straight line does not create a convexity violation.

For K₁ < K₂ < K₃:

[P(K₂)-P(K₁)] / [K₂-K₁] ≤ [P(K₃)-P(K₂)] / [K₃-K₂]

Here both slopes equal 0.20, so the inequality holds with equality.

3. Butterfly / convex-combination check

Because 20 is two-thirds of the way from 0 to 30,

20 = (1/3)·0 + (2/3)·30

Convexity therefore requires

P(20) ≤ (1/3)P(0) + (2/3)P(30)

Weighted price (1/3)·0 + (2/3)·6 = 4

Observed price P(20)=4

Difference 4 − 4 = 0

Conclusion: no convexity arbitrage from these three quotes.

When would there be an arbitrage?

Suppose instead that P(20) were greater than 4. Then the middle-strike put would lie above the chord joining the K=0 and K=30 puts:

P(20) > (1/3)P(0) + (2/3)P(30)

That would violate convexity and allow a butterfly-style arbitrage: sell the overpriced middle-strike put and buy the weighted combination of the surrounding strikes.

Put prices increase with strike + slope must not decrease ⇒ convex in strike