# 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

deviationσ_{t }. 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

X_{i }

= $1if Z_{i−1 }< Market Capitalization (MC) ≤ Z_{i }= $0 otherwise

(1)

for i =1,..., K

For concreteness assume that Z_{0}=0 and that Z_{K}= ∞ . 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 − − =

(2)

# where F is the cumulative distribution function of the random variable MC. One of these securities is

21

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.

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