IBM Research | Ponder This | August 2006 solutions
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# Ponder This

## August 2006

<<July August September>>

Answer:

Let the demand for the contract from A and B be a and b respectively. Negative values of a or b correspond to a desire to sell the contract short. Expected log wealth for A will is

p*log(X+a*(1-r))+(1-p)*log(X-a*r)

This has derivative with respect to a of

p*(1-r)/(X+a*(1-r)-(1-p)*r/(X-a*r)

Setting the derivative to 0 and solving for a we find

p*(1-r)*(X-a*r)=(1-p)*r*(X+a*(1-r))

( p*(1-r)-(1-p)*r)*X=((1-p)*r*(1-r)+p*r*(1-r))*a

(p-r)*X=r*(1-r)*a

a=(p-r)*X/(r*(1-r))

Similarly

b=(q-r)*Y/(r*(1-r))

Setting a+b=0 and solving for r we obtain

(p-r)*X/(r*(1-r)) + (q-r)*Y/(r*(1-r)) = 0

(p-r)*X + (q-r)*Y = 0

p*X+q*Y=(X+Y)*r

r=(p*X+q*Y)/(X+Y)

So the market clearing price is just the weighted average of the opinions of A and B as to the correct price.

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