Financial Mathematics Probability Question

Question

It was crucial for our no arbitrage computations that there were only two possible values of the stock. Suppose that a stock is now at 100, but in one month may be at 130, 110, or 80 in outcomes that we call 1, 2, and 3. (a) Find all the (nonnegative) probabilities p1, p2, and p3 = 1 - p1 - p2 that make the stock price a martingale.

I am trying to solve the above problem but I keep going in a continuous loop trying to find the 3 probabilities. I am trying to use this method which they use in my textbook, but it does not explain what to do in the case of 3 probabilities and not 2.

Here is the example if you need help understanding: Textbook example

Answer 1

You want a solution to $$30 p_1 + 10 p_2 - 20 (1 - p_1 - p_2)=0$$ that is, $$50 p_1 + 30 p_2 = 20$$ satisfying the constraints $p_1, p_2 \ge 0$ and $p_1 + p_2 \le 1$. (There are infinitely many solutions.)

Answer 2

Since we have a martingale, we have: \begin{align*} 130p_1 + 110p_2 + 80p_3 &= 100 \\ 13p_1 + 11p_2 + 8p_3 &= 10 \\ 13p_1 + 11p_2 + 8(1 - p_1 - p_2) &= 10 \\ 13p_1 + 11p_2 + 8 - 8p_1 - 8p_2 &= 10 \\ 5p_1 + 3p_2 &= 2 \\ p_1 &= \frac{2 - 3p_2}{5} \,\,\,\,\,\,\,\,\,\,\, \text{(eq 1)} \\ \end{align*} Hence any $p_1$, $p_2$, $p_3$ satisfying eq 1 and the following conditions will work. $$ 0 \leq p_1 \leq 1 $$ $$ 0 \leq p_2 \leq 1 $$ $$ 0 \leq p_3 \leq 1 $$ $$ p_1 + p_2 + p_3 = 1 $$