3 Step Binomial Model (2004 FRM exam)

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Hi David,

Question:
An option on a stock has a payoff equal to the square of the positive excess of the stock price over the exercise price at expiration only if the stock exhibits an annual growth rate of 15% or more every year. Given the following assumptions and using a 3-step binomial model and rounding to the nearest USD, which of the following would be the option's price?
-Time to expiration is 3 years
-Current price is USD10
-The annual std deviation is 15%
-The risk free rate is 5% per annum
-The strike price is USD10
-Assume the stock pays no dividends

Asnswer: A USD7

u = EXP(sigma*sqrt(t)) = EXP(0.15*SQRT(1.0)) = 1.162 ------- "1" instead of "3" is used bcos it is annual std deviation???
d=EXP(-sigma*sqrt(t))=EXP(-0.15*1.0) =0.861
p= (EXP(r*t)-d) / (u-d) = (exp(0.05*1)-0.861) / (1.162-0.861) = 0.633

Until here, I think I still can follow. But, I have confusion on the following part.

Option price formula = f = exp(-r*t)[pfu +(1-p)fd]
But, answer continues as follow
Option price = exp(-r*t)*p^3*(current price*1.162^3 - strike price)^2 = 7.04

Your guidance, please

Thanks
Learning

 

[David Harper CFA FRM]

Hi Learning,

I input the problem into a version of the binomial XLS:
http://sheet.zoho.com/public/btzoho/nov17-3s-binomial

but, it's a good example (maybe) of where it might have been easier if i'd paused to examine the question before i launched into calcs (at first glance, i thought, a 3-step binomial would be too time consuming, but after, i realized the restriction is meant to simplify)...
you can see where i solved in orange; i.e., cell E25 has option value of $7.05

in regard to: Option price formula = f = exp(-r*t)[pfu +(1-p)fd]
you'll notice that if we apply that formula, the answer is: option value = $1.85
...per regular method, that discounts all future nodes

but here the option has a restriction: "only if the stock exhibits an annual growth rate of 15% or more every year"
and since u = 16%, there is only one path that implies an exercisable option: uuu

so the formula given (i.e., exp(-r*t)*p^3*(current price*1.162^3 - strike price)^2) is discounting the future intrinsic value^2 ($32.30) back three years--i.e., exp(-rt)--but also weights by probability: p^3. We have to multiply by p^3 because (1-p^3) is the probability of any other path, with payoff = 0
...and re: SQRT(1): the time to expire is 3 years, and it's a 3 step, so each step must be 3 years/3 = 1 year.

David