Calls on the same expiry are quoted at 7.00 for the 95 strike, 4.80 for the 100 and 2.40 for the 105. Is the surface arbitrageable, what do you trade, and what does the violation say about the implied distribution?

Calls on the same expiry are quoted at 7.00 for the 95 strike, 4.80 for the 100 and 2.40 for the 105. Is the surface arbitrageable, what do you trade, and what does the violation say about the implied distribution?

Approach: Test convexity of the call price in strike using the equally spaced butterfly, then connect the second derivative in strike to the risk-neutral density.

Yes, buy the 95/100/105 butterfly for a credit of 0.20. The butterfly costs C(95) - 2*C(100) + C(105) = 7.00 - 9.60 + 2.40 = -0.20, so you are paid 0.20 to hold a payoff that is zero at 95 and below, zero at 105 and above, and worth up to 5.00 at 100. A non-negative payoff bought for a negative price is a static arbitrage. Call prices must show convexity in strike because the Breeden-Litzenberger result makes the second derivative in strike equal to the discounted risk-neutral density, so a negative butterfly is the market pricing a negative probability of finishing near 100. The practical check on a live surface is to run every equally spaced butterfly and every call spread for sign before quoting off a fitted smile, since a smile fitted with an unconstrained polynomial produces exactly this in the wings.

Follow-up: Your fitted smile is arbitrage free in price but the density has two modes. Is that a violation, and would you quote off it?

Key concepts: butterfly arbitrage, convexity in strike, implied density, Breeden-Litzenberger.