Proving Calendar Arbitrage in Calls with a Positive Interest Rate

Summary

The document examines three quoted European call prices with the same underlying and strike but different maturities. The longest-dated call is quoted below the nine-month call, prompting a request for an explicit arbitrage argument. The accepted response focuses on those two maturities and derives a price ordering rather than constructing a state-by-state trading portfolio.

It expresses each call price as a discounted risk-neutral expected payoff, conditions the longer-maturity payoff on information at the shorter date, and applies Jensen’s inequality to the convex call payoff. Under the assumed positive risk-free rate, the resulting inequalities imply that the longer-dated call cannot be cheaper than the shorter-dated one. The quoted prices therefore violate the derived ordering under the model assumptions. The argument depends on the risk-neutral stock growth and discounting setup used in the answer, including the absence of dividends or other carry adjustments; it does not address transaction costs, market frictions, or a detailed execution strategy.

Key ideas

• For calls with the same strike, the response derives a price ordering between shorter and longer maturities.
• Conditioning on information at the shorter maturity and applying Jensen’s inequality gives a lower bound for the longer call.
• With a positive risk-free rate under the stated assumptions, the longer-dated call must cost at least as much.
• A quoted violation of that ordering indicates an arbitrage opportunity in the frictionless model.
• Dividends, carry adjustments, and transaction costs are outside the proof as presented.

Tags

# Check for arbitrage - European calls with same strike price, different duration and price


# Check for arbitrage - European calls with same strike price, different duration and price












I tried a lot of different things to check for arbitrage on the following calls but didn't succeed.

Let's suppose we have a stock that is currently valued at 40. The interest rate is 0.05 and the strike price of the european calls is 35. Call 1: price 6.75, run time 6 months Call 2: price 7.93, run time 9 months Call 3: price 7.76, run time 12 months

I know that a lower price of Call 3 with longer duration in comparison to call 2 indicates arbitrage, but I don't know how to prove it. I hope someone can assist with an arbitrage strategy.

## Answer by Frido (score 3, accepted)

https://quant.stackexchange.com/a/75802

What I wrote in my comment is one way to show the arbitrage. For your own benefit it might be good if you try to understand also the following more mathematical reasoning.

Let $T = 12$ months, $t = 9$ months and $0$ is today. Then $$ C_3 = e^{-rT} E_0 (S_T - K)_+ $$ and $$ C_2 = e^{-rt} E_0 (S_t - K)_+ $$

By conditioning and using Jensen's inequality \begin{align} C_3 &= e^{-rT} E_0 (S_T - K)_+ \\ &= e^{-rT} E_0 E_t (S_T - K)_+ \\ &\geq e^{-rT} E_0 (E_t S_T - K)_+ \\ &= e^{-rT} E_0 (S_t e^{r(T-t)} - K)_+ \\ &= e^{-rt}e^{-r(T-t)}E_0 (S_t e^{r(T-t)} - K)_+ \\ &= e^{-rt}E_0 (S_t - Ke^{-r(T-t)})_+ \\ &\geq e^{-rt}E_0 (S_t - K)_+ \\ &= C_2 \end{align} So $C_3$ must always be more expensive than $C_2$ if risk-free rate is positive.

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