equating call bull spread to put bull spread

This took me a couple of hours, wasted, I am sure many of you already can do this, but I just figured I post it in-case it helps someone with their understanding.

I knew that it is possible to do the bull spread with both puts and calls, I knew they had to workout exactly the same despite different cash flow timeing, however I had to do the math to be convinced. So I took this example.

All options are priced in BS model*

xC1=50 xC2=60 xP1=50 xP2=60 c1=0.2235 c2=0.0015 p1=5.41 p2=14.29 S0=40 rf=10 volatility=10

First method to build the bull spread is using the 50 and 60 calls

Buy the 50 call for 0.2235 and sell the 60 call for 0.0015, total cost is 0.222

assume you borrow the 0.222, you will pay it back in one year as 0.2442

so if ST at expiary is 60, get 10-0.2442=9.7558 and if ST is 50, you pay 0.2442


Second method is

Sell the 60 put for 14.29 and but the 50 put for 5.41, net inflow is 8.88

invest the 8.88 at rf it becomes 9.76 when the option expires

if the stock is at 60 or above you owe nothing to option holder so you end up with 9.76 which is the same thing as in the first method

if the stock is at 50, you will owe 10 but you got 9.76, so you end up paying 0.24 which is the same thing as in first method

there are some slight differences which i assume are due to rounding and the fact the BS model uses continuous rate while I did not when I did the loans, easy to change I just dont feel like it…

Anyway, it is interesting to know that the exact same thing can be done, any comments from experts ?

Nice, I wasn’t even aware of how to replicate a bull spread using puts. I thought it would just be the opposite of the calls (i.e. buy high-strike put, sell low-strike put), but after seeing this thread, I see it’s the same concept. Bull spread can be accomplished by buying call at X1 and selling call at X2, or by buying put at X1 and selling put at X2. X1 < X2. Thanks for the post.

^ also the bear can be done with calls in similar manner, and I am sure it can be proven correct as well

however unless you are very solid in everything else, i suggest you dont worry about replicating the bull or the bear, cfai seems to want bull with calls and bear with puts, know that it is doable, i doubt it would be on exam.

good luck mate.

Sounds good, thanks and good luck to you as well man.