Convexity

[colby2152]

I had trouble with this question on a TIA practice exam (Exam #6 - Problem 14).

An investor sees the following table of call options prices in the newspaper one morning for a non-dividend-paying stock. The investor exploits an arbitrage opportunity, using (either buying or selling short) a block of 100 options expiring in 6 months and a block of 100 options expiring in 2 years. What is the present value of the minimum guaranteed profit for the investor if the continuous risk-free rate of interest is 6%?

With there being an arbitrage opportunity, how do I know whether the first call should be 29, the second call should be 27.5, or the third call should be 24 (and the same situation for the 2 year calls)?

[jraven]

Hmm. I'm confused. Convexity seems to be fine for those. However there does seem to be a time-based arbitrage present.

For instance, we can sell 100 6-month calls with strike 100 and buy 100 2-year calls with strike 110. This results in an up-front profit of

After six months the worst case scenario is that the calls we sold are exercised, in which case we have to pay

On the other hand we still have the 100 2-year calls with strike 110 that we purchased, which according to put-call parity are each worth

So if we sold our calls to cut our losses then all told we would make at least

This would make the present value of all our cash flows at least

There are other arbitrages possible, but I think this one is probably the most profitable.

[colby2152]

Okay, so there are other arbitrage opportunities, but that is the best one.

I thought this was an example of convexity - the differing strike prices and premiums.

[jraven]

Well, there are other arbitrage opportunities, but they're all time-based. There are no convexity or slope-based arbitrages in the problem.