<p>I cannot really follow your formulas, what is the logic behind them?</p>

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

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

CC BY-SA 4.0
2020-04-06T07:30:11.370Z
2020-04-06T07:36:23.710Z
2020-04-06T07:36:23.710Z
Giskard
<p>I need help to find the risk-neutral probability for states 1,2 and 3 </p>

<p>I have two stocks: A and B. </p>

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

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

<p>The annual rf rate is 2%</p>

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

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

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

<p>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? </p>

<finance><financial-economics><corporate-finance><probability><risk-aversion>
Risk neutral probability for each of 3 states
alatriste
# Risk neutral probability for each of 3 states

(Asked by: username_0 on 2020-04-06)

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?

**Tags:** finance, financial-economics, corporate-finance, probability, risk-aversion

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## 1 answers

### Answer by username_1 on 2020-04-06. Score: 3

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.


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