Prove that paying the fair price for an option always results in expected value of $0

Economics Dec 15, 2020

Prove that paying the fair price for an option always results in expected value of $0.
In other words, prove that if E(A)=$x, then E(Pay $x for A)=$0.

Expert Solution

Lets assume that with probability P(x) you win $x.

Assume with probability P(y) you win $y.

Let $H = Expected Value of this game = P(x)x + P(y)y

Consider that the fair price of this game is exactly equal to its expected value.

Hence, you pay $H every time you play the game.

Now, the expected value of the game is given as:

P(x)[x - H] + P(y)[y - H]

P(x)x - P(x)H + P(y)y - P(y)H

P(y) = 1 - P(x)

P(x)x - P(x)H + [1 - P(x)]y - [1 - P(x)]H

P(x)x - P(x)H + y - P(x)y - H + P(x)H

P(x)x + y - P(x)y - H

P(x)x + y - [1 - P(y)]y - H

P(x)x + y - y + P(y)y - H

P(x)x + P(y)y - H = Expected value of game - Fair Price

H - H

= $0

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