In Chapter 8, pg 92 of the CMP, there is a line that says "Futures have no time value and hence no time decay", since the Theta for futures is 0. How do we reconcile this with pg 10 (which is calculating the delta for a future) of the same Chapter? For example, if we take \( \frac{\partial F_t}{\partial t} = \frac{\partial S_t e^{r(T-t)}}{\partial t} = -r S_t e^{r(T-t)}\) which is not equal to 0? Thanks a lot in advance!
Your maths is right, but the way you're using the terminology doesn't line up with the way people speak about this stuff in practice. Theta is used for talking about time decay on options, but the concept of time decay for options doesn't quite extend to futures and forwards (so called 'delta one' products, to distinguish them from products involving options). I've tried to explain why time decay is different for options to delta one below. For futures and forwards, depending on the asset class of the underlying, you might hear people call dF/dt something like the 'carry'. Calling it a theta would likely cause confusion, because theta refers mainly refers to the fact that an option has smaller chance of expiring in the money when time to expiry decreases. This concept doesn't exist for futures and forwards. Theta is also a significant factor driving PnL for an options book - especially if you are have a lot of options close to the money you can lose a lot of money every day due to theta / time decay. This isn't true for futures and forwards, where day to day changes in value are driven mainly by changes in spot price. What makes time decay for options different to delta one The value of an option is higher when there is more chance for it to expire in the money. There is a better chance of this if the volatility is larger and if there is more time left to expiry, because the variance of the underlying until expiry which is priced into the option is proportional to sigma ^ 2 * (T - t). Therefore reducing the time to expiry decreases the value of the option, hence theta tends to be negative for long call and put positions. This concept only exists for options (and products involving options), as futures and forwards don't have the same concept of expiring in the money. However, not *all* of an option's theta is due to this 'chance of expiring in the money' effect. Theta still exists for an option even as the volatility approaches zero (see the below example of a call option). This 'zero volatility' part of the theta is similar to what you calculated as 'theta' for a futures contract. But it's not the important part of an option's theta and tends to be pretty small in most cases compared to the part of theta related to the smaller chance of expiring in the money when time to expiry decreases. Theta for a call option as volatility approaches 0 As volatility approaches zero the value of a call option becomes for S > K * exp(-r(T-t)): S - K * exp(-r(T-t)) for S < K * exp(-r(T-t)): 0 (not too tricky to check this yourself if you want to). Hence theta becomes: for S > K * exp(-r(T-t)): - K * exp(-r(T-t)) for S < K * exp(-r(T-t)): 0 The case for S > K * exp(-r(T-t)) is very similar to your formula for dF/dt. In other words, roughly speaking, if there wasn't any volatility, the option theta would line up with your dF/dt formula. But in reality there will be volatility, and this component of the call's theta won't be very important compared with the component related to the smaller chance of expiring in the money when the time to expiry decreases. Final note In practice theta tends to be calculated as 'value in 1 day's time, assuming inputs remain unchanged' minus 'current value'. I work in the market risk team for a money manager and if I looked on my screen at work many of the instruments would have a non-zero 'Theta' value even if they aren't options. Despite this, people don't really refer to this as the 'theta' for these instruments, because of the discussion above. I think depending on the asset class it might not make much sense, might not be important, or might be referred to using different word(s).
What I said here was poorly typed and confusing. I should have said: So, in conclusion, when the CMP said futures have no time value and hence no time decay, that's true if you're talking about the part of the value relating to the chance of expiring in the money. I would also say that in practice people don't call the partial derivative with respect to time of delta one instruments 'theta'. But your formula is right.
Thank you so much for the detailed answer CapitalActuary. What you said makes a lot of sense. My main confusion was mainly arising from the differences in theory (and hence, what should be written in exam), and what is being used in practice. Thanks a lot for clarifying
