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# Risk-neutral probability

#### ttezcan

##### New Member
i have a question like that. I didn't understand how to find U and D values in that problem?

Problem
A risk manager for bank xyz, marki is considered writing a 6 month american put option on a non-dividend paying stock ABC. The current stock price is USD 50 and the strike price of the option is USD 52. In order to find the no arbitrage price of the option, Mark uses a a two step binomial tree model. The stock price can go up or down by 20% each period. Mark's view is that the stock price has an 80% probability of goinh up each period and a 20% probability of going down. The risk free rate is 12% per annum with continuous compounding. What is the risk neutral probability of the stock price going up in a single step?

#### ShaktiRathore

##### Well-Known Member
Subscriber
its two step model so that each step is of duration 6 months/2 = 3 months or 1/4 yrs
U and D value are given that
The stock price can go up or down by 20% each period.
so that U=1.2(up 20%) and D=.8(down 20%)

risk neutral probability of up is given by=e^rT-D/U-D=(e^.12/4-.8)/(1.2-.8)=1.03045-.8/.4=.576

thanks

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#### akrushn2

##### Member
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In all the practice problems so far we compute the up movement as u= e^(volatility)*sqrt(t). I understood the sentence " stock price can go up or down by 20% each period" to mean volatility. I don't know why I did not read the sentence " The stock price can go up or down by 20% each period" as literally just that and ended up complicating this. In the exam are we usually not expected to calculate this? Can anybody comment?

I didn't get this question correct b/c I assumed the sentence "he stock price can go up or down by 20% each period" to mean volatility. So, I computed up movement as e^(20%)*sqrt(0.25) and down movement e^(-20%)*sqrt(0.25) and then tried to Insert this into my risk neutral probability formula. I now see that I overcomplicated this. Just wanted to ask though in the exam- is it typical of them to give us the up and down probabilities/is It still useful to actually memorize the above formula?

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#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @akrushn2 You should know the formula that determines binomial up/down as a function of volatility, but the phrase "The stock price can go up or down by 20% each period" is a very common way to describe the (alternative) binomial step pattern; e.g., it's used 2 or 3 times in GARP's 2019 practice paper; e.g., Question 19 says "The stock price can go up or down by 15% each period" ... Thanks,