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This formula will be useful in our later work.

(c) If there are n states and at least n independent securities with known prices, and no arbitrage possibility, then the risk-neutral probabilities can be found directly by solving the system of equations s

Example 9.10 (The film venture) We found the state prices of the full film venture (with three securities) to be

= g, V'2 = 5, ^3 = 5 Multiplying these by the risk-free rate 1.2, we obtain the risk-neutral probabilities

9 J = 2> <72 = -6, 03 = 2. Hence the price of a security with payoff (c/1, c/2, d3) is

Here again, this pricing formula is valid only for the original securities or linear combinations of those securities. The risk-neutral probabilities were derived explicitly to price the original securities.

The risk-neutral pricing result can be extended to the general situation that does not assume that there are a finite number of states (See Exercise 15 )

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