Question

Discussion in 'P1.T2. Quantitative Methods (20%)' started by sipanivishal, Oct 26, 2008.

1. sipanivishalManager-Corporate Banking

For a lognormal variable X, we know that ln(X) has a normal distribution with a mean of zero and a standard deviation of 0.5. What are the expected value and and the variance of X?
2. David Harper, CFA, FRM, CIPMDavid Harper

Sipani,

Based on properties of lognormal (see http://en.wikipedia.org/wiki/Lognormal)
mean = EXP[0+0.5^2/2] = 1.13
variance = (EXP[0.5^2]-1)*EXP[2*0+0.5^2] = 0.36

However, we did not study this in 2008; lognormal was a "key distribution" in prior years but is not in Gujarati. This was be an unfair question this year. Our only "lognormal" concern is properties of stock price and please note the potential confusion concerning +(1/2)variance versus -(1/2)variance:

(Hull Chapter 13)
ln (St/S0) ~ Normal [(u - variance/2)*T, volatility*SQRT(T)]

such that (per 13.7)
if St = S0*EXP[xT] implies
x ~ normal(u - variance/2, ...)

This difference (add one half variance vs subtract) has given difficulties in previous years. They are completely compatible, but it's not obvious. But GARP shouldn't quiz on lognormal properties except where we are concerned: under stock price process where "volatility erodes (subtracts) returns."

I emphasize this, additionally, because the Merton model (like d2 in Black-Scholes) is utilizing this same geometric average expected return (i.e., minus 1/2 variance) so the other, at this point, could be a bit of a distraction...David