# Generalisation of calendar arbitrage condition to options on futures

This question has discussed the condition on which calendar arbitrage opportunities arise for European call options on a stock. Do similar criteria exist for European options on futures?

The most important difference between the two types of underliers is the term structure. Futures all have a maturity, whereas stocks are perpetual. Therefore, options on futures with different maturities also have different underliers, i.e. futures contracts with different maturities.

I'll try to illustrate why different maturities can be a problem by the following example. Suppose we have two call options with maturities $$t_1 and two corresponding futures maturities $$T_1>t_1, T_2>t_2$$. Suppose the two options on futures have the same strike $$K$$, and at this point $$C(t=0,\text{option maturity}=t_1)>C(t=0,\text{option maturity}=t_2)$$. If we follow the ordinary calendar arbitrage strategy, we short the expensive and long the cheap, get proceedings $$P>0$$ and then at time $$t_1$$ we will be left with a payoff of $$Pe^{rt_1}-(F_{[t_1,T_1]}-K)_++C(t=t_1,\text{option maturity}=t_2)$$ where $$F_{[t_1,T_1]}$$ denotes the futures price at time $$t_1$$ whose maturity is $$T_1$$. It's not clear whether the payoff constitutes an arbitrage opportunity, because we cannot guarantee that the term $$C(t=t_1,\text{option maturity}=t_2)$$ dominates $$(F_{[t_1,T_1]}-K)_+$$. (The intrinsic value is $$(F_{[t_1,\color{red}{T_2}]}-K)_+$$ which is hard to compare against $$(F_{[t_1,T_1]}-K)_+$$.)

Nevertheless, does there exist other "more strict" variants of the original criteria $$C(t_1)>C(t_2)$$ which can definitely constitute a calendar arbitrage? For example can we find some a priori constant $$M$$ such that when $$C(t_1)>C(t_2)+M$$, we can be certain to find a calendar arbitrage?

• What are the futures on? For futures on a stock, the same conditions should be there, commodity futures will weaken the argument, the degree its weakened by depending on the difficulty of physically holding the commodity between the dates in the calendar spread. – will Jul 15 at 5:48
• @will commodities. In my specific case, soybean meals. – Vim Jul 15 at 5:50
• @will an additional difficulty with commodities seems to be the basis structure not being stable, e.g. frequent switches between backwardation & contago. – Vim Jul 15 at 5:56
• @will would you please offer a mathematical expression for the degree by which the argument is weakened in terms of costs of storage and convenience yield? Thanks! – Vim Jul 15 at 5:58
• I don't have a mathematical expression. But think about it, many commodities have a finite lifetime - oil products spoil, grains and softs too. Live cattle stop being live cattle, carbon emmisions can only be used to offset emmisions contracts in certain time frames, the list of issues goes on. The result is that you may not be able to take delivery in one month, and then use that same physical commodity to deliver another contract - they cease being the same underlying, the arbitrage is impossible to realise, so it doesn't exist. – will Jul 15 at 21:21