 # Option Limits- reading 73 losK

was reading Schweser and the reading says there is no reason for early exercise of a American call because the value (St-X) is less than the minumim value, which is given as St-X/(1+RFR)T-t. This doesn’t seem to make sense since the denomiator in the second equation has to be > 1. Can anyone explain why there is no reason for the early exercise?

So St - X < St - X/100000

Why does that not make sense? Ex: St = 60 X = 50 RFR = .05 T = 15 t = 5 St - X = 10 St -X/(1.05)^10 = 29.30 10 < 29.30

OK, maybe I am being dense, but for some reason this is the first thing I have come across that has really stumped me. but 60-50=10,and 10 divided by any number >1 has to be less than 10. (1.05)^10= 1.6289, and 10/1.6289=6.1391. And 6.1391 is not > 10. So can anyone explain this to me. If I exercise now the call is worth 10. but the PV is only \$6.14. So why is there no value in the early exercise of the call?

I think you need to look at the parentheses there. How about this explanation though: Portfolio A: Call option with strike X on S(t) expiring at time T Cash = X*Exp(-r*T) Portfolio B: Share of stock valued at S(t) Suppose that you hold portfolio A and decide to exercise the option early at time q. Then portfolio A is worth S(q) - X + X*Exp(-r*(T-q)) which is certainly less than S(q). So if you exercise early, portfolio A is certainly worth less than portfolio B. But if you hold to expiration, Portfolio A is worth Max(S(T), X) which is >= S(T) so at expiration portfolio A is worth at least as much as portfolio B. So (this is the part you need to get comfortable with) Portfolio A is worth at least as much as Portfolio B at time T for sure and for certain and that’s known at all times prior to T. That means that A must be worth at least as much as B at all times prior to T otherwise there is an arbitrage. So if you have a straegy like early exercise that makes A worth less than B, it must be a crappy strategy because you have blown away value that must be there. If you have an American option that you think needs to be exercised early, you have an American option that needs to be sold to someone else.