I'm reading Sinclair's Option Pricing and am confused by the proof for the maximum value of a call. It makes sense logically that a call can't be worth more than the underlying, and so:
c <= S
The proof the book uses however is as follows. Say there's a call trading for more than the underlying. Then, I will sell the call, and buy the underlying. At expiration (time T), our profit is:
I don't understand why we subtract the S_T at the end? Doesn't this imply a huge profit? Say the call was worth 110, S=100, and S_T = 105. Then the profit would be 110-(100-105) = 115. That doesn't make sense.
ALSO, does this just assume that the option that we sold expires OTM, so it's not exercised??