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redundant because both the normalized prices and actual probabilities of being in each range always sum

to 1.

On date t, the log normal distribution parameter vector is characterized completely by the mean,

µ t a n d t h e s t a n d a r d d e v i a t i o n σ t ( i . e . , θ t = ( µ t , σ t ) ) . B e c a u s e t h e r e a r e K > 2 s e c u r i t i e s t r a d e d , i t i s p o s s i b l e

t o e s t i m a t e t h e p a r a m e t e r v e c t o r t θ f o r e a c h t r a d i n g d a t e , t . T h e r e a r e s e v e r a l m e t h o d s t h a t c o u l d b e u s e d

t o e s t i m a t e θ t . W e c h o s e a m i n i m u m χ 2 c r i t e r i o n a s t h e m e t h o d , a l t h o u g h w e a l s o e s t i m a t e d t h e

parameters using generalized method of moments and maximum average log likelihood criteria to see

whether any significant differences existed. None were found.

S p e c i f i c a l l y , f o r e a c h d a y , d e n o t e t h e o b j e c t i v e f u n c t i o n a s ( ) t V θ a n d s o l v e t h e f o l l o w i n g f o r t h e

e s t i m a t e s o f µ t a n d σ t :

ˆ ( ) t t A r g M i n V θ θ = =

t θ

K

i =1

2 , ( ( ( ) i t i t i t p P P θ θ ) )

(3)

w h e r e p i , t i s t h e p r i c e o f s e c u r i t y i ( o r m a r k e t b a s e d p r o b a b i l i t y f o r e c a s t f o r r a n g e i ) o n d a t e t a n d P i ( t θ ) i s

its expected value according to the estimated log normal distribution. Note that this results in both an ex

ante forecast of the post-IPO market capitalization and a direct ex ante measure of uncertainty

surrounding this forecast.

4. Results

A. Market Performance

F i g u r e 1 s h o w s t h e n o r m a l i z e d p r i c e s o f t h e I P O _ U P c o n t r a c t . 2 2

Trading in the Google Linear

market was light.23

From July 8, the first day after which all contracts had traded, through August 17, the

22

23

Note that the price of IPO_UP should equal 1 minus the price of the IPO_DN contract. However, due to asynchronous trading and bid/ask bounce, prices of IPO_UP and IPO_DN do not necessarily sum to exactly $1 at any given point in time. To adjust for this, we use normalized prices. The normalized price of each contract is the price of the contract divided by the sum of contract prices. The graph starts with July 8, the first day by which all contracts had traded. While these markets are thin, this does not necessarily imply an inefficient market. Prediction market research

14

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