Reference no: EM133069311

1. Consider an arbitrary single period model defined by a payoff matrix A (including a column for a risk free bond defining the interest rate r) and a price vector z. Write functions for this model which determine:

(a) Whether there is arbitrage in the market (see hint below).

(b) If the market is complete.

(c) Whether a given payoff Y is attainable in the market. If so, return its price. If not, return the arbitrage free bounds on its price.

Hint: One method for determining if there is arbitrage in the market is to solve the following linear program:

max λ q,λ

ATq = (1 + r)z qi ≥ λ i = 1, . . . , M where 1 is a vector of ones. The first constraint is the linear system defining risk neutral probabilities.

If the optimal value of the optimization problem is strictly positive (or the problem is unbounded), then there is a solution to this linear system with all entries positive (i.e. a vector of risk-neutral probabilities), otherwise not. Then apply the first Fundamental Theorem of Asset Pricing.