Stochastic Volatility One-Period Tree

We have three securities (money market account, stock, call option on the stock with K = 100). Today their prices are [100, 100, 8]. Suppose one period from now the stock can end up being either in the high-volatility (in which case S = 120 or 80) or low-volatility (in which case S = 110 or 90) regime.The MMA would still be 100. Now consider an exotic option X. The payoff of X is (30, 0, 0, 15) corresponding to (high vol stock up, high vol stock down, low vol stock up, low vol stock down). You can imagine it as a derivative that becomes a call when volatility is high and a put otherwise. What is today's value of X?

Ans: $12

Note: This problem is solvable because the matrix that represents basis securities payoffs has a left inverse. When would this be not true? -> When some basis security is the linear combination of others.