Having some difficulties with Neftci 2.10, first exercise. Under the assumption that there is no arbitrage, we should have:
$$\begin{bmatrix} 1 \\ 280 \end{bmatrix} = \begin{bmatrix} 1.0125 && 1.0125 \\ 320 && 290 \end{bmatrix} \begin{bmatrix} \phi_1 \\ \phi_2 \end{bmatrix}$$
Straight from the conditions given in the exercise.
Where the rate has been converted into three months: $0.05 \frac{3}{12} = 0.0125$.
However, when we proceed to solve these, we get that $\phi_1 = -6.42 / 30$, i.e. a negative value. And this cannot be true. What is incorrect in the above?
For those who do not have access to the exercise:
You are given the price of a nondividend paying stock $S_t$, and a European call option $C_t$ in a world where there are only two possible states: $S_t =$ $320$ if $u$ occurs, and $290$ if $d$ occurs. The true probabilities of the two states are given by $\{P_u = 0.5, p_d = 0.5 \}$. The current price of the $S_t$ = $280$. The annual rinterest rate is constatnt at $r = 5$%. The time is discrete, with $\Delta = 3$ months. The option has a strike price of $K = 280$ and expires at time $t + \Delta$. (a) Find the risk-neutal martingale measure $Q$ using the normalization by risk-free borrowing and lending.