To create a portfolio X0, ∆0 that satisfies the given conditions, we need to understand the terms and their implications first.

Here, X0 is the initial wealth or investment, ∆0 is the number of shares of a risky asset to hold, X1(H) is the wealth at time 1 if the high state occurs, and X1(T) is the wealth at time 1 if the low state occurs. The parameters r, d, and u are the risk-free rate, the down state, and the up state of the risky asset, respectively.

The conditions are:

1. X0 ≤ 0: The initial investment is less than or equal to zero. This could mean that we are borrowing money or selling short.

2. X1 ≥ 0: The wealth at time 1 is non-negative, regardless of the state of the world. This means that we are not losing money.

3. Not all of X0, X1(H), and X1(T) are equal to zero: At least one of the initial investment, the wealth in the high state, and the wealth in the low state is non-zero. This means that there is some activity in the portfolio.

Given these conditions, we can create a portfolio as follows:

Step 1: Borrow money or sell short to get a negative initial investment. For example, we could borrow $100, so X0 = -$100.

Step 2: Invest the borrowed money in the risky asset. The number of shares to buy, ∆0, would be the borrowed money divided by the price of the asset. For example, if the price of the asset is $20, we would buy 5 shares, so ∆0 = 5.

Step 3: Calculate the wealth at time 1 in the high state and the low state. This would be the number of shares times the price of the asset in each state. For example, if the price of the asset in the high state is $30 and in the low state is $10, we would have X1(H) = 5 * $30 = $150 and X1(T) = 5 * $10 = $50.

This portfolio satisfies all the given conditions: X0 is negative, X1 is non-negative in both states, and not all of X0, X1(H), and X1(T) are zero.

Question

To create a portfolio X0, ∆0 that satisfies the given conditions, we need to understand the terms and their implications first.

Here, X0 is the initial wealth or investment, ∆0 is the number of shares of a risky asset to hold, X1(H) is the wealth at time 1 if the high state occurs, and X1(T) is the wealth at time 1 if the low state occurs. The parameters r, d, and u are the risk-free rate, the down state, and the up state of the risky asset, respectively.

The conditions are:

1. X0 ≤ 0: The initial investment is less than or equal to zero. This could mean that we are borrowing money or selling short.

2. X1 ≥ 0: The wealth at time 1 is non-negative, regardless of the state of the world. This means that we are not losing money.

3. Not all of X0, X1(H), and X1(T) are equal to zero: At least one of the initial investment, the wealth in the high state, and the wealth in the low state is non-zero. This means that there is some activity in the portfolio.

Given these conditions, we can create a portfolio as follows:

Step 1: Borrow money or sell short to get a negative initial investment. For example, we could borrow $100, so X0 = -$100.

Step 2: Invest the borrowed money in the risky asset. The number of shares to buy, ∆0, would be the borrowed money divided by the price of the asset. For example, if the price of the asset is $20, we would buy 5 shares, so ∆0 = 5.

Step 3: Calculate the wealth at time 1 in the high state and the low state. This would be the number of shares times the price of the asset in each state. For example, if the price of the asset in the high state is $30 and in the low state is $10, we would have X1(H) = 5 * $30 = $150 and X1(T) = 5 * $10 = $50.

This portfolio satisfies all the given conditions: X0 is negative, X1 is non-negative in both states, and not all of X0, X1(H), and X1(T) are zero.

Knowee AI · Accepted Answer