# What is the call price in a two-stock two-period model (binomial)?

I am trying to wrap my head around the binomial model. In particular, how would the calculation go for deriving the price of a call option for one of the stocks expiring at time 2? You can pick any K.

I am usually used to just one stock, so this has me a bit confused.

(Each cell is a price, the arrows indicate change of time into new states, and the upper cell contains prices of stock 1, and lower cell prices of stock 2. No risk-free asset, no dividends).

## 1 Answer

You have to calculate the Risk-Neutral probability of the upmoves and downmoves. The key point to bear in mind is that every stock (and risk-free bond) have the same expected value at each time step.

Therefore calling $q_{01}$ the probability of an upmove from time 0 to 1, you would have:

$$120q_{01}+90(1-q_{01})=X$$ $$130q_{01}+80(1-q_{01})=X$$

Now solve for $q_{01}$ and $X$, which gives you:

$$q_{01}=0.5$$ $$X=105$$

Do the same at each node and you should find the probabilities. Finally to find the value of a call option just multiply the $q$s for the payoff at the in-the-money nodes (where the stock price is higher than $K$).

This payoff has to be discounted using the risk-free rate which you can easily find from the calculations above. For example at the first node: $$1+r_{01}=\frac{X}{100} \quad \Rightarrow \quad r_{01}=0.05$$ assuming a time step of one year and discrete compounding.

• I have not before heard of the idea that the expected values "should be the same", Could you expand? I do know of the formula that says $S_t = E^Q_t S_{t+1}/R_t$, i.e. $S_t$ is your vector of stock prices today, and that should equal the expectation of that vector in the next period, but under the $Q$-measure, and after discounting. – Imean H Jun 26 '17 at 12:16
• Actually, I got it! Thanks. What your method basically amounts to is that you write $R_tS_t = E_t^QS_{t+1}$, and then you can solve for $q$ and $X$. (you just wrote $X$ for $R_tS_t$). – Imean H Jun 26 '17 at 12:27