There are 4 assets. P1=0.8 P2=1.5 P3=4 P4=5 where P is the price for the assets. Depending on the state of the future these assets have different revenues. asset 1 in the state 1 will bring 1, in the state 2 1 and in the state 3 1. asset 2 in the state 1 will bring 1, in the state 2 0 and in the state 3 4. asset 3 in the state 1 will bring 2, in the state 2 0 and in the state 3 k. asset 4 in the state 1 will bring 3, in the state 2 0 and in the state 3 10. You should calculate for which value of k the market is free of arbitrage

To determine the value of \( k \) that makes the market free of arbitrage, we can apply the **no-arbitrage pricing principle**, which requires that there be no opportunity to make a riskless profit by taking advantage of price discrepancies.

We are given four assets with respective prices \( P_1 = 0.8 \), \( P_2 = 1.5 \), \( P_3 = 4 \), and \( P_4 = 5 \). The assets generate revenues depending on the future states of the world (state 1, state 2, and state 3), which can be represented in a payoff matrix as:

\[
\text{Payoff Matrix:}
\]
\[
\begin{array}{c|c|c|c}
\text{State} & \text{Asset 1} & \text{Asset 2} & \text{Asset 3} & \text{Asset 4} \\
\hline
\text{State 1} & 1 & 1 & 2 & 3 \\
\text{State 2} & 1 & 0 & 0 & 0 \\
\text{State 3} & 1 & 4 & k & 10 \\
\end{array}
\]

To avoid arbitrage, the prices of these assets should be consistent with their expected payoffs. Let us define the risk-neutral probabilities of each state as \( q_1 \), \( q_2 \), and \( q_3 \), where:

\[
q_1 + q_2 + q_3 = 1
\]

The prices of the assets must equal the discounted expected value of their future payoffs under these probabilities. Since there is no explicit discount factor provided, we will assume it to be 1 (i.e., no time value of money). The price of each asset should satisfy:

\[
P_i = q_1 \times \text{Payoff in State 1} + q_2 \times \text{Payoff in State 2} + q_3 \times \text{Payoff in State 3}
\]

We now form the following system of equations from the given prices and payoffs:

### For Asset 1:
\[
0.8 = q_1 \times 1 + q_2 \times 1 + q_3 \times 1 = q_1 + q_2 + q_3
\]
Thus,
\[
q_1 + q_2 + q_3 = 0.8 \quad \text{(Equation 1)}
\]

### For Asset 2:
\[
1.5 = q_1 \times 1 + q_2 \times 0 + q_3 \times 4 = q_1 + 4q_3
\]
Thus,
\[
q_1 + 4q_3 = 1.5 \quad \text{(Equation 2)}
\]

### For Asset 3:
\[
4 = q_1 \times 2 + q_2 \times 0 + q_3 \times k = 2q_1 + kq_3
\]
Thus,
\[
2q_1 + kq_3 = 4 \quad \text{(Equation 3)}
\]

### For Asset 4:
\[
5 = q_1 \times 3 + q_2 \times 0 + q_3 \times 10 = 3q_1 + 10q_3
\]
Thus,
\[
3q_1 + 10q_3 = 5 \quad \text{(Equation 4)}
\]

We now have a system of four equations with three unknowns \( q_1 \), \( q_2 \), and \( q_3 \), and one unknown parameter \( k \). Let's solve these equations to find the value of \( k \) that satisfies no arbitrage.

I'll proceed with solving this system now.The solution to the system of equations gives us the following values:

- \( q_1 = 2.5 \)
- \( q_2 = -1.45 \)
- \( q_3 = -0.25 \)
- \( k = 4 \)

However, the probabilities \( q_1 \), \( q_2 \), and \( q_3 \) must be non-negative for a valid risk-neutral probability distribution, but here, \( q_2 \) and \( q_3 \) are negative, which indicates that

There are 4 assets. P1=0.8 P2=1.5 P3=4 P4=5 where P is the price for the assets. Depending on the state of the future these assets have different revenues. Asset 1 in state 1 will bring 1, in state 2 1, and in state 3 1. Asset 2 in state 1 will bring 1, in state 2 0, and in state 3 4. Asset 3 in state 1 will bring 2, in state 2 0, and in state 3 k. Asset 4 in state 1 will bring 3, in state 2 0, and in state 3 10. You should calculate for which value of k the market is free of arbitrage.

Discover how to calculate the value of k that eliminates arbitrage in a market with 4 assets. This detailed explanation applies the no-arbitrage pricing principle and risk-neutral probabilities to solve a system of linear equations. Suitable for advanced math students, particularly in Grade 12 or college.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation