Both payoffs increase linearly between $80 and $120.
So the collar should cost more than the spread since it's the same shaped payoff but shifted up by $80:
collar = spread + gap
Expanding the terms:
put80 + stock - call120 = call80 - call120 + gap
Cancelling the short calls and re-arranging:
gap = stock + put80 - call80
By put-call parity, that right side is just PV($80): if price at expiry is below $80 you exercise your put to sell your shares at $80; if it's above, your shares are called away for $80. No matter what happens, you end up with $80 at expiration.
So, finally,
collar = spread + PV($80)
We could have also just reasoned that "same payoff shape but shifted up by $80" by no-arbitrage gives us a gap of PV($80).
So the "varying amount" is just a bond price which moves with rates and expiry. The deferred-rate highs you flag as pushing forwards up are the same r shrinking this gap.
Using strikes of $80 and $120:
Collared stock: min payoff $80, max payoff $120
Call spread: min payoff $0, max payoff $40
Both payoffs increase linearly between $80 and $120.
So the collar should cost more than the spread since it's the same shaped payoff but shifted up by $80:
collar = spread + gap
Expanding the terms:
put80 + stock - call120 = call80 - call120 + gap
Cancelling the short calls and re-arranging:
gap = stock + put80 - call80
By put-call parity, that right side is just PV($80): if price at expiry is below $80 you exercise your put to sell your shares at $80; if it's above, your shares are called away for $80. No matter what happens, you end up with $80 at expiration.
So, finally,
collar = spread + PV($80)
We could have also just reasoned that "same payoff shape but shifted up by $80" by no-arbitrage gives us a gap of PV($80).
So the "varying amount" is just a bond price which moves with rates and expiry. The deferred-rate highs you flag as pushing forwards up are the same r shrinking this gap.
Fun puzzle.
yep...reduces to the cost of carry on the strike + intrinsic
call = put + (S-K) + cost of carry on the strike