Future Stock Prices

[scottc85]

David-thanks for the great site! How would the log formula in 'Lognormal Distribution. Part 3: Future Stock Price' (http://www.bionicturtle.com/learn/article/lognormal_distribution_part_3_future_stock_price/) be revised to solve for the expected price of the stock at the end of the term (as opposed to its relationship to the strike price)? I have a feeling it's right in front of me...

Thanks,
Scott

 

[David Harper CFA FRM]

Scott,

Thanks for liking us! Aren't the hardest, best things always right in front of us
You can just replace strike with initial stock price (S0) and it is good to go
(I cannot claim originality on any of this, I learned this all from John Hull's books)

It starts with stripped down cost of carry
S(future) = S0*EXP[(risk free rate)(time)]

Note: not realistic, imcomplete, just a framework, makes assumptions (notably: no systemic risk so no equity premium). For example, you can improve with:
S(future) = S0*EXP[(rate + equity premium = risky rate)(time)]

then take natural log of both sides (of the first, simple):
LN(S1) = LN(S0*EXP[(r)(T)] = LN(S0)*LN[EXP[(r)(T)], so
LN(S1/S0) = (r)(T)

So, the challenging part to this is that this equation says, if the return (r) is normally distributed, then price level (S1) or the wealth ratio (S1/S0) is LOGNORMALLY DISTRIBUTED: the future price does not have a normal, but rather a lognormal distribution. This is why there are TWO VALID EXPECTED FUTURE STOCK PRICES. Because, to quote Hull, "expected return is ambiguous." If you think about the lognormal distribution, turned on its side, prices can go up infinitely but are limited by zero on downside. The lognormal distribution has a MEAN that is different than its MEDIAN; specifically, it's mean will always be higher than it's median (positive skew: mean > median). So we have either:

S1 = S0 * EXP[(r)(T)] which is the mean of the lognormal distribution of future stock price, or
S1 = S0 * EXP[(r - variance/2)(T) which is the median of the lognormal distribution of future stock price; i like to say "this is your future wealth but eroded by volatility"

This corresponds to a classic debate about which is better, geometric or arithmetic returns. Sorry for length, it is just an common stumbling block and maybe this can be helpful...

David