buckybeaver, from D=0.9091, how do you get 20% down for a year?
A buckybeaver: a) 20% is in fact the volatility, however this is already the one period volatility (the period in this example was clearly 3 months not one year), hence we don’t need to do anything to this number to get our U, which is just equal to the following: U=exp(.20*sqrt(1)), again T=1 hence we are multiplying by sqrt(1) since one period is 3 months (it does say that every period the stock can go up or down 20% percent correct? and it does say that there are two such periods prior to expiration correct? right then one period has to equal 3 months and there is no need to mess with the volatility). b) U=exp(.2*sqrt(1))=1.2214, which implies D=.81873, hence c) Prob(U)=[exp(rT)-D]/[U-D]=[exp(.12*.25)-.81873]/[1.2214-.81873]=.21172453/.40267=.5258, hence Prob(D)=.4742. These are the risk neutral probabilities of an up and down move for this problem. Note that here we do have to convert the risk free rate to our corresponding 3 month period by multiplying it by .25 since the risk free rate was given on an annual basis and for us 1 period=3months, as was explained above. Unfortunately, I do not remember the the starting underlying price or strike price and so I can’t use these probabilities to produce an initial value for this call (?). If someone remembers these details I can crunch the numbers.
adavydov7, I think it’s right that 20% is for one period. The problem is simple: U=1.2, D=0.8. P(U) = [exp(0.12/4) - 0.8]/(1.2 - 0.8) = 57.6%, which is exactly one of the answers. Following buckybeaver’s approach, I got the price of the option that is also exactly one of the answers. Simple question, you guys scared me last night.
@NYC: If you assume U=1.2 instead of computing U as exp(stdev*sqrt(T)), which by the way I don’t know why you would do since you are given all the inputs for computing U properly, then D cannot be .8, d has to be 1/U in which case it would equal .8333.
stdev wasn’t given, and from other CFA exams, I also remember if they give you up movement of +20%, then U is obviously 1.2. Hence D should be .833333. I honestly think FRM did some intermediate rounding here and used 0.8 instead – which is obviously wrong on their part. my final answer came down to like 53% or something (too lazy to do the math here), so I just guessed the b) answer choice, or 57.6%.
@adavydov7. They explicitly tell us U and D. BTW, no one says D must be 1/U. This is just convention. Please look at Hull or any elementary quant finance book.
@Adalfu and Rydex In the question as far as I can recall they had asked which one is not used to calculate frequency of an event (dont remeber verbatim but ’ frequency’ was there ), discrete dist is used to compute Frequency answer is a) Normal Dist
No, they don’t! They tell us the stock can increase or decrease 20% over a single period, this is not the same as an up move factor (U) or a down-move factor (D). These factors are functions of the volatility and length of time steps being used in the binomial model (not the return which is what 20% is)! They are calculated as follows: U=exp(stdev*sqrt(T)) D=exp-(stdev*sqrt(T)) Please notice that D must equal 1/U. Proof: D=exp-(stdev*sqrt(T))=1/exp(stdev*sqrt(T))=1/U QED Hence, this is not just convention! It a definitional property! So I would suggest that you look in Hull, or any elementary options book yourself before you start talking about something you clearly know nothing about. For your reference since you are so familiar with Hull and all: Hull 6ed, page 253! Oh, and here is the original source: http://fisher.osu.edu/~fellingham_1/seminar/CRR79.pdf And yet another: Schweser 2009, Book 2, page 241 Next time before you start talking at least have the courtesy to check the source which you are trying to cite in support of your own argument!
I hope for my sake, when they said the stock can go up by 20% and down by 20% each period, they were assuming that we would infer that the price 1 period from now would either be 1.2 times or .8 times the price at time zero. I feel like if they had wanted us to calculate the up and down movement factors, they would have provided std dev instead, but who knows. Would be nice to see how they score that one. I could see it going either way.
Rydex Wrote: ------------------------------------------------------- > > Would be nice to see how they score that one. I > could see it going either way. If i get it right, then let it count… else, please throw it out 
Because I’m so familiar with Hull and all that, I’m going to enter some equations. Around the same place in Hull, he has the following equation in order to match the VARIANCE of the stock: exp[r*dt]*(u+d) - u*d - exp(2r*dt) = sigma^2 * dt BE ALERT, we have 2 unknowns u and d, but only one equation. Three suckers called Cox, Ross and Rubenstein assumes u=1/d to get the equation you mentioned. BUT THIS IS A SPECIAL CASE. YOU can assume u=1.2 and d=0.8 and solve this equation for sigma. Let’s end the discussion for this question.
What matters is how FRM deals with it. I got many practice problems wrong trying to use D = 1/U, they for whatever reason would use the factors as 1.2 and .8 for an example like this so I used the same approach on the exam. In the end the points are all that matters, the reasoning, who cares…
People who don’t care about reasoning shouldn’t post here. I think @adavydov7 is with me on that.
Jesus christ, you just keep digging your hole deeper and deeper trying to sound smart, the formula is [exp(Mu*dt)]*(U+D)-U*D-epx(2*Mu*dt)=Var*dt Because as you point out U and D are two unknowns in a single equation many solutions exist, one of which is: U=exp(stdev*sqrt(T)) D=exp-(stdev*sqrt(T)) This would imply that by the proof I showed above D must equal 1/U CRR (1979) did not posit any other guesses nor did our assigned readings suggest any. You can assume U=1.2 and D=.8 only if you don’t have enough information to compute the proper up and down factors (i.e. U=exp(stdev*sqrt(T)) and D=exp-(stdev*sqrt(T))), which we clear had! Like I said check your reference before you open your mouth.
@joe: like I say in my prior post you can make that simplifying assumption if you aren’t given the volatilities, because without them you can no longer assume D=exp-(stdev*sqrt(T)), which is 1/U. However, in this example enough information was given to determine U and D properly.
Haaaa. Now I’m speechless. Let’s say how long you can keep your mouth open.
adavydov7 Wrote: ------------------------------------------------------- > @joe: like I say in my prior post you can make > that simplifying assumption if you aren’t given > the volatilities, because without them you can no > longer assume D=exp-(stdev*sqrt(T)), which is 1/U. > However, in this example enough information was > given to determine U and D properly. The volatility was not given. It only said that it can move up or down by 20%. It did not say the standard deviation was 20%. So explicitly stating that 20% is the standard deviation, it may be a bold step to assume that to be the case. 20% could be 4 standard devations. Who knows.
You can compute the st. dev. using everything that is given.
I stand by 100% that the up factor was supposed to be 1.2 and the down factor was supposed to be .8. They did not five us a standard deviation so there is no way to calculate the up and down factors in any other manner, because you cannot assume that 20% was the standard devation. Sure you can back into the standard deviation, but first you must have the up and down factors. If you do that, where does it get you, it is a circular argument.