# 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.

• Short sell call 2 and buy call 3 from the proceeds and put the (positive) remainder in your bank account. In 9 months time, since call 3 will always be worth more than its extrinsic value (which is the value of call 2 at expiration) you will either still have money left if call 2 is exercised at expiration, and if call 2 is not exercised at the very least you'll have a non-negative amount at expiration of call 3. Plus you still have the positive amount in your bank account from the original short sale of 2 and purchase of 3. Arbitrage. Jun 8 at 7:51

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.