An Interview Question

Stock price today is $20 and it follows a Geometric Brownian Motion. The beta of stock is 1. You are trying to model a stock price a year from today. Which outcome is more likely; stock price >$20 or stock price <$20?
Thanks!
 
mm should it be >$20 because the expected value of the stock price is 20*exp(r), where r = mu - sigma^2/2. But r should be the risk free rate (we all know very well that it isn't in reality) and thus > 0
 

Yike Lu

Finder of biased coins.
GBM implies no drift. The median of a lognormal distribution is less than the mean of the same lognormal distribution. But the mean is 20 since GBMs are martingales. That means it is more likely that the stock prices is less than $20.
 
The present value of a risk neutral GBM implies no "drift". A generic GBM is not necessarily a martingale...

The median of a generic GBM with drift alpha and volatility sigma is \( S(0)e^{(\alpha-\frac{1}{2}\sigma^2)t} \)
This is because \( log(S_t)=log(S(0))+(\alpha-\frac{1}{2} \sigma^2)t+\sigma W_t \)
Since the median of \( W_t \) is zero, the median of \( log(S_t) = log(S_0)+(\alpha-\frac{1}{2} \sigma^2)t \) , so the median of \( S_t \) is \( S(0)e^{(\alpha-\frac{1}{2}\sigma^2)t} \)

Since the beta is one, the stock is perfectly correlated with the market (iirc) and thus the question is whether the market is more likely to be above where it is currently at in a year...which I would think is the greater possibility, at least in normal economic times ;).
 

Diego Calderon

Grad Student
I would agree with DStahl on his last point. Since the beta is 1 (perfectly correlated with the market) and since the market rewards investors in equity capital for assuming the 'most' risk as shareholders (get paid last)...year over year, the trend is upward. So it will be > $20.
 

Yike Lu

Finder of biased coins.
You guys make good points, but at an interview what I would actually do is to clarify the purpose of the question if objections are raised to my martingale assumption/answer. Without knowing the purpose of the question, it is impossible to determine whose answer would be "accepted", although again DStahl and Diego both make valid poitns.
 
\(\beta=\frac{Cov(r, r_m)}{Var(r_m)}=\rho\frac{\sigma}{\sigma_m}\)

Therefore,

\(\sigma=\frac{\sigma_m}{\rho}\)

And the median stock price is above 20 iff

\(\mu-\frac{1}{2}\frac{\sigma_m^2}{\rho^2}>0\)

So the stock price is more likely to go up for a higher correlation and less likely for higher market volatility. We are not talking about risk-neutral drift here, so it is really a question about what mu is. I would probably say it is more likely to go up, since stocks go up on average in the long run.
 
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