A $(K_1, t_1, K_2, t_2)$ double call option is one that can be exercised either at time $t_1$ with strike price $K_1$ or at time $t_2$ ($t_2 > t_1$) with strike price $K_2$. Argue that you would never exercise at time $t_1$ if $K_1 > e^{-r(t_2-t_1)}K_2$.
This was exercise 5.26 on p. 90 of Sheldon M. Ross's "An Elementary Introduction to Mathematical Finance", 3rd edition, Cambridge University Press, 2011.
I'd appreciate help in solving this problem. My foremost problem is formalizing the phrase "you would never exercise at time $t_1$". Does it mean "You would never exercise at time $t_1$ and then just hold the underlying asset until time $t_2$"? What if I exercise at time $t_1$, but, rather than holding the underlying asset until time $t_2$, I partly invest it in something and partly deposit it in a savings account? Is there no such investment/savings strategy that would make it more advantageous for me to exercise at time $t_1$?
My attempted solution
I will show that, under a certain extra assumption (namely $(*)$ below), if the inequality $K_1 > e^{-r(t_2-t_1)}K_2$ holds, then the expected profit when exercising at $t_1$ is no greater than the expected profit when exercising at $t_2$ (i.e. $(**)$ below).
But will this solve the problem? Well, it depends on two factors:
(a) Is this inequality of expected profits (i.e. $(**)$ below) what is meant by the phrase "you would never exercise at time $t_1$"? If not, what mathematical formula does this phrase translate into?
(b) Is the extra assumption that I am going to make (i.e. $(*)$ below) reasonable? If not, how can the argument be modified to avoid relying on this assumption?
We shall start by introducing a couple symbols to denote the double call option as well as the underlying asset that the double call option derives from. Denote the underlying asset by $S$ and denote the option by $X$. $S$ and $X$ are stochastic processes $S = (S_t)_{t \in [0, \infty)}, X = (X_t)_{t \in [0, \infty)}$, where, for every $t \in [0, \infty)$, $S_t$ is the underlying's payoff at time $t$ and $X_t$ is the double call option's payoff at time $t$. As always the payoffs $S_0, X_0$ at the present time $t = 0$ are taken to be constant.
Furthermore, denote the market's fixed interest rate by $r$ (we assume $r \in [0, \infty)$), and denote the profit process associated with $X$ by $P$, in other words, for every $t \in [0, \infty)$, $$ P_t = e^{-rt}X_t - X_0 $$
We shall now make the extra assumption, to which we alluded above, that $$ e^{-rt_1}S_{t_1} \sim e^{-rt_2}S_{t_2} \tag{*} $$ i.e. the distribution of the underlying's present value at time $t_1$ is the same as its distribution at time $t_2$.
We may now state our main result: $$ E(P_{t_1}) \leq E(P_{t_2}) \tag{**} $$ where the expectation is calculated w.r.t. the actual market probability measure (as opposed to the EMM probability measure, if such exists, though note that the argument to follow does not make any assumption regarding the existence or absence of arbitrage in the market. In fact, as far as mathematical correctness is concerned, the following chain of equalities and inequalities holds no matter what probability measure the expectations are calculated with respect to, as long as it's the same one throughout).
$$ \begin{align} E\left(P_{t_1}\right) &= E\left(e^{-rt}X_{t_1} - X_0\right) \\ &= E\left(e^{-rt_1}(S_{t_1} - K_1)_+ - X_0\right) \\ &= E\left((e^{-rt_1}S_{t_1} - e^{-rt_1}K_1)_+ - X_0\right)\\ &= E\left((e^{-rt_2}S_{t_2} - e^{-rt_1}K_1)_+ - X_0\right)\\ &\leq E\left((e^{-rt_2}S_{t_2} - e^{-rt_2}K_2)_+ - X_0\right)\\ &= E\left(P_{t_2}\right) \end{align} $$ where the inequality is due to the fact that, by assumption, $K_1 > e^{-r(t_2-t_1)}K_2$, which is equivalent, after rearrangement of terms, to $$ K_1e^{-rt_1} > K_2e^{-rt_2} $$
There are a few missing (but standard) assumptions underlying this. As a technical point, you have to assume that the market is liquid and arbitrage free. in particular, you have to assume that there is a rational market available into which you could sell the unexercised option at any time. Or, similarly, a market wherein you can short the asset liquidly without incurring special financing costs. Also, and more critically, you have to assume that the asset "carries at the riskless rate". The claim may be false otherwise (if, say, the asset is like oil, which incurs non-trivial storage costs, or if the asset finances at a rate other than the riskless rate, or if the asset releases cash, like a bond or a dividend paying stock).
Note: to tie these assumptions into the approach begun in the question, the assumption that the asset carries at the riskless rate is equivalent to the desired claim that the expected value of the stock at a future date is just the forward value of the spot price. Again, this is an assumption. If, say, your stock pays dividends then it is untrue (without appropriate corrections).
Assuming all of that, then imagine the two scenarios sketched above:
Scenario A: use exercise at $t_1$ and just hold the asset until $t_2$.
Scenario B: you do not exercise early, instead you invest the $K_1$ at the riskless rate and wait until $t_2$.
Now, let's look at $t_2$. The asset is trading at X_2, say. Of course, your cash has grown to something more than $K_2$ (by your inequality on the strikes). If $X_2 ≥ K_2$ then of course you exercise the call, if you are in B. Thus, in both cases you own the asset, but in B you are slightly better off (because you made a little cash on your investment).
If, on the other hand, $X_2 < K_2$ then in scenario B you are even better off. YOu have a loss from scenario A which you are spared since you needn't exercise.
thus in call cases Scenario B dominates scenario A.
One might worry a bit about what happens between times $t_1$ and $t_2$. For example, if the asset shoots up the day after $t_1$ and crashes to $0$ some while after that, you can imagine that scenario A looks better (since you could sell the asset to exploit the bubble). But in that case the technical assumptions I stressed at the beginning of my answer kick in. You can sell the option just as easily as you could sell the asset and, as the call price is always at least the value of the forward purchase you can still exploit the bubble. Or you could short the stock against the forward call exercise.
As an exercise: find where, in the above argument, the assumption on the "riskless carry" was used.