Risk neutral probability for each of 3 states

Question

I need help to find the risk-neutral probability for states 1,2 and 3

I have two stocks: A and B.

The price of A today is 180 and in a year it will be worth 288 (S1), 180 (S2) or 120 (S3);

The price of B today is 100 and in a year it will be worth 94(S1), 134(2) or 54(S3)

The annual rf rate is 2%

I did the following for stock A: up : 288/180-1= 0,6 down :120/180-1 =-0,3333

Finding prob for s1: (2- (-33,33))/ 60- (-33,33)) = 0,3785

Then tried to solve for p2 and then p3:
PV= (0,3785* 288 + p2*180+ (0,6215 -p2 )* 120)/1,02= 180

These probabilities don't add up when I try to find the price of stock B, so they are clearly wrong. Can you tell me how to find the right probabilities?

Answer 1

I cannot really follow your formulas, what is the logic behind them?

Seems to me there is no way to divine three state risk-neutral probabilities from 1 financial instrument's prices. The equation $$ p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 = 180 $$ is underdetermined, there are infinitely many solutions to it. Now if you solved the equation system $$ \begin{align*} p_1 288 + p_2 180 + (1 - p_1 - p_2) 120 & = 180 \\ p_1 94 + p_2 134 + (1 - p_1 - p_2) 54 & = 100 \end{align*} $$ there you have two unknowns and two equations, so usually this would admit exactly one solution.

P.s.: The numbers in my example are not corrected for intertemporal differences, there has been no discounting.