B. Fitting a Forecast Distribution with the WTA Market
The WTA markets can be used to forecast the expected distribution of future market
capitalizations, not just a point estimate for the expected capitalization. In its simplest form, the WTA
price vector is a vector of (risk-neutral) probabilities of six events (and after August 4, eight events).
Knowledge of the CDF of a random variable allows one to calculate any moments of interest. However,
because the highest interval (greater than $40 billion prior to August 4 and greater than $50 billion
afterwards) is unbounded from above, some assumption must be made about the distribution of outcomes
in this range when this contract trades above a zero price. For this reason, we assume that at any point in
time, t, the future (unknown) capitalization is distributed log normally with mean µ t and standard
deviationσt . We further assume that the probability of no IPO equals zero.21
Intuitively, we assume that the normalized closing prices of contracts on date t reflect estimates of
t h e p r o b a b i l i t i e s o f o b s e r v i n g o u t c o m e s i n e a c h r a n g e e a c h d a y . F o r g i v e n t µ a n d t σ , i n t e g r a t i n g t h e
log normal distribution over each range yields predicted probabilities of being in each range. We derive
estimates of the distribution mean and standard deviation by minimizing the distance between observed
and predicted probabilities.
F o r m a l l y , a s s u m e t h e r e a r e K s e c u r i t i e s t r a d e d e a c h d a y a n d t h a t t h e y h a v e a p a y o f f , X i , o f
= $1if Zi−1 < Market Capitalization (MC) ≤ Zi = $0 otherwise
for i =1,..., K
For concreteness assume that Z0=0 and that ZK= ∞ . The probability that market capitalization (MC) lies in
interval i is
) | ( ) | ( ) ( 1 t i t i t t Z F Z F P θ θ θ − − =
where F is the cumulative distribution function of the random variable MC. One of these securities is
The log normal distribution is uncontroversial while assuming that the probability of no IPO is zero is consistent with Google’s stated strong intention to issue in the summer of 2004 and the long horizon on the contracts.