# Optimal exercise of American BOND option

I know that early exercise for American options has been discussed extensively, but I have not seen a lot in relation to BOND options with American optionality and there are few things I cannot get it right.

Any help or thought would be much appreciated.

I tend to consider bond option by nature different from stock options as the bond has a maturity and its price will inevitably converge to par (let’s take the most common situation for a bond paying its notional at maturity, so 100%).

Then, if I have an option with maturity (of the option) very close to the maturity of the bond, my underlying dynamics will be strongly influenced by the border constraint … and at some point, it will even become somewhat deterministic.

Now, allow me to take a specific example: I have a bond American call option with maturity T_option = 10 May 2022, for an underlying ZERO COUPON bond with maturity T_bond = 10 June 2022. The strike of the call is par and then I can formulate the payoff as

Max(PV_bond – 100%,0)

What is the optimal exercise here?

If I see the bond at a certain point for any fundamental reason to quote at, say 120%, and I know from my market analysis that this is somewhat the highest I can reasonably get, why it is not optimal (or at least sub-optimal) to exercise, pay 100%, sell the bond in the market and have a profit of 20%?

Yes, I could

1-short the bond to earn 120%,

2-exercise the call option at maturity and obtain the bond paying no more than 100%

3-sell the stock in the market (in the shallow window of 1 month in my example) or wait for its maturity

My profit will be at least of 20%, so the first idea to exercise early and sell directly is sort of sub-optimal (but rather good anyway).

In any case “do nothing” and simply wait for the call exercise at expiration will certainly not be optimal, right (and this should be true also for American call option on non-paying dividend stock)?

Am I missing something?

Thank you very much in advance