I get a bit confused here, how come that it is a No? I get super confused because we are always calculating the rd and ru to get a risk neutral probability…
MODULE QUIZ 77.1
To construct a one-period binomial model for valuing an option, are probabilities of
an up-move or a down-move in the underlying price required?
A. No.
B. Yes, but they can be calculated from the returns on an up-move and a downmove.
C. Yes, the model requires estimates for the actual probabilities of an up-move
and a down-move.
The question is asking if you need the actual probabilities, but you only need the risk-neutral probabilities.
To construct a one-period binomial model for valuing an option, are actual probabilities of
an up-move or a down-move in the underlying price required?
A. No, the model requires risk-neutral probabilities of an up-move and a down-move.
B. Yes, but they can be calculated from the returns on an up-move and a downmove.
C. Yes, the model requires estimates for the actual probabilities of an up-move
and a down-move.
Aha alright, so we don’t have the actual probabilities we just assume those and that is what the question is really asking? If it has asked for the risk-neutral probabilities then it would have been B right?
You can construct a replicating portfolio which doesn’t use probabilities at all.
Suppose the initial stock price is S_{0} and the strike price of a call is X.
Assume there are 2 possible stock prices at the end of the period S_{u} and S_{d} with S_{u}>X>S_{d}.
You can construct a replicating portfolio hS-Ae^{-rt} which has the same payoff as the option:
(i) if the stock price goes down to S_{d}<X, the payoff is zero so hS_{d}-A=0
(ii) if the stock price goes up to S_{u}>X, the payoff is S_{u}-X so hS_{u}-A=S_{u}-X
If you subtract these 2 equations, you can eliminate A: h(S_{u}-S_{d})=S_{u}-X so h=\frac{S_{u}-X}{S_{u}-S_{d}}
and then A=hS_{d}=\frac{h(S_{u}-X)}{S_{u}-S_{d}}
An example with numbers:
call with strike 50 and initial stock price 50 and up price of 60 and down price of 40
replicate with hS-Ae^{-rt}
S=40 need payoff=0
S=60 need payoff=10
so replicating portfolio is S/2-20e^{-rt}
it came up in a question on the L2 forum.
It seems to only be used for the one-period binomial model (and also in brain teasers in job interviews)
Just from googling, see e.g. this link from NYU Stern (with whom I have no connection, it just came near the top of the results) Options: Valuation and (No) Arbitrage
and scroll down to
IV. The Binomial Pricing Model
Yeah alright cool, I think I will touch that area more in lvl 2 and lvl 3 I guess. Thank you so much once more for putting in the effort and answering my questions! It helps me a lot in my CFA journey.