I need help with this exercise:
Consider the market model with two assets risky represented by:
$S_0=220, S_u=230, S_m=210, S_d=190, D_0=105, D_u=115, D_m=105, D_d=100$
The market interest rate is 1%.
Show that the model has opportunities of arbitrage. Describe an opportunity of arbitrage
I tried the condition $u<(1+r)<d$ in each tree, but the result with the asset D is ok.
With no-arbitrage, the expectation of an asset price at the end of the period must equal the forward price. We should be able to find one set of risk-neutral probabilities $p_u, p_d \in [0,1]$ that allow the forward price condition to hold for each asset:
$$p_u S_u + p_d S_d + (1 - p_u - p_d)S_m = S_0(1 +r), \\ p_u D_u + p_d D_d + (1 - p_u - p_d)D_m = D_0(1 +r). $$
Substituting values and solving we find $p_u = -0.40$ and $p_d = -1.01.$ Since these probabilities are not valid the market model admits arbitrage opportunities.
An arbitrage opportunity that is easy to spot is to buy $\Delta = S_0/D_0$ units of asset D and sell short 1 unit of asset S at zero cost.
The inititial portfolio value is $V_0 = \Delta D_0 - S_0 = 0$. However, in each final state the portfolio has a positive value:
$$V_u = \Delta D_u - S_u = \frac{220}{105}115 - 230 \approx 11, \\ V_m = \Delta D_m - S_m = \frac{220}{105}105 - 210 = 10, \\ V_u = \Delta D_u - S_u = \frac{220}{105}100 - 190 \approx 20$$
