Calendar Arbitrage with Options on Futures
Summary
The document examines whether the usual calendar spread arbitrage condition for European calls on a stock carries over to options on futures. The key complication is that options with different expirations can reference futures contracts with different delivery dates, so their underlying prices need not match. A higher price for the shorter-dated option therefore does not alone establish an arbitrage.
The example considers shorting the more expensive option and buying the cheaper one, then compares the remaining longer-dated option value at the first expiry with the payoff on the earlier option. Because those values depend on different futures maturities, their relative size is not assured. The document asks whether a stricter price gap could guarantee arbitrage, but supplies no criterion, proof, or answer. It frames the issue and its limitation rather than providing a usable trading rule.
Key ideas
- Options on futures with different expirations may have different futures contracts as underlyings.
- A stock option calendar spread argument does not directly apply when the underlyings differ.
- At the first option expiry, the later option and the exercised futures payoff may reference distinct futures maturities.
- The document raises, but does not resolve, whether a stricter price inequality can guarantee arbitrage.
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# Generalisation of calendar arbitrage condition to options on futures
# 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 $t_1<t_2$ 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.