Hi,
To answer this you will need to brush up on stochastic calculus.
I will give you a few pointers and then lets see (assuming that you check this forum before your interview) if you can figure it out...
OK, so let's use the following notations:
St0 = Stock price at time zero (now)
Sigma = Stock's volatility
Sat1= Stock price a (130) at time 1 (time 1 is 5 months from now)
Sbt1 = Stock price b (140) at time 1
The first question is how volatile we expect the stock to be between t0 and t1. If the Annualised volatility for the stock is 12% (stock volatilities are ALWAYS quoted on an annualised basis) then what is the 5 month volatility?
Because standard deviation (the measure we use for volatility) is not linear through time but variance (the square root of the standard deviation) is linear through time we must multiply the annualised volatility by the square root of 5 months to get the 5 month volatility.
As there are 12 months in a year we must look at 5 months as 5/12, so:
12%*sqrt(5/12) = 7.75%.
So, to a 1 standard deviation probability the stock will rise or fall by 7.75%.
As 1 standard deviation represents a probability of approx. 68.28% we can say that, to a 68.28% probability the stock will be between 107.75 and 92.25 in 5 months because:
100*(1+7.75%)=107.75
and
100*(1-7.75%)=92.25
So, extending this out to 2 standard deviations:
100*(1+(2*7.75%))= 115.49
and
100*(1-(2*7.75%))= 84.54
The probability of a move in price exceeding 2 standard deviation is approx 100% - 95.45% = 4.55%
So, a 3 standard deviation move up or down would be:
100*(1+(3*7.75%))= 123.24
and
100*(1-(3*7.75%))= 76.76
As the stock price has a 99.73% probability of remaining within 3 standard deviations we can say to that probability that the price will not exceed 123.24 on the up side.
The odds of it being between 130 and 140 are therefore slim.
One of the assumptions which we have made here is that volatility is constant. In reality volatility is itself stochastic (this can be empirically observed in the volatility of the vix). So if asked a question like this you would do well to ask about the stability of volatility during the 5 month period as this would need to be factored in.
Knowing nothing about your academic background it is tricky to know what to compare the rest of the calculation to so I will coin it in financial terms.
Think of a binary range option which pays out 100% if the stock is between 130 and 140 at expiry in 5 months, but 0% if it is not.
To price such an option follow the steps below. The price of the option will be the probability of the event occurring multiplied by the payout if it does occur.
A binary (call) is priced as $*exp(-r*t)*CND(d2)
where $ is the payout amount if the contingent event (130<Price<140) occurs, exp is the number e, CND is the cumulative normal distribution and d2 is the d2 formula from Black Scholes:
d2 = (Log(S / X) + (b - v ^ 2 / 2) * T) / (v * Sqr(T))
Where Log is the natural logarithm, X is the strike price, v is sigma, T is the time to expiry and Sqr is square root.
A binary put is priced as: $*exp(-r*t)*CND(- d2)
A binary range is therefore a package of 4 binarys:
Long a call at strike 1, short a put at strike 1, long a put at strike 2 and short a call at strike 2. Where strike 1 is the lower strike and strike 2 the upper.
So: the probability of the price being between 130 and 140 in 5 months is the same % as the price of a binary range paying 100 if the market is IN the range at expiry in 5 months.
There are several ways in which you could approach this problem and I tend to think in terms of options so the binary range option analogy is the one I would find the most helpful.
So, what do you think the answer is…?
Good luck!
NQR
.