Consider a market consisting of one risk-free asset with \(A(0)=10\) and \(A(1)=11\) dollars, and one risky asset such that \(S(0)=10\) and \(S(1)=\) 13 or 9 dollars. On the \(x, y\) plane draw the set of all portfolios \((x, y)\) such that the one-step strategy involving risky position \(x\) and risk-free position \(y\) is admissible.

A risk-free asset is an investment that is expected to return its value in full, with no risk of financial loss. In the problem, the risk-free asset has an initial value of \(A(0)=10\) dollars. This means that at the beginning of the period, one unit of this asset costs 10 dollars.
The asset grows to \(A(1)=11\) dollars by the end of the period. Because it's risk-free, this growth is certain, with no variance. This is like putting money in a savings account with a guaranteed interest rate, offering a predictable return. Understanding risk-free assets is important because they provide stability within a portfolio and can offset risks from other investments.

A risky asset, on the other hand, does not have guaranteed returns. Its future value can vary, which implies the potential for both gains and losses. In this exercise, the risky asset starts with an initial value of \(S(0)=10\) dollars.
The future price of the risky asset can either be \(S(1)=13\) dollars or \(S(1)=9\) dollars. This variability introduces risk, as it reflects the uncertainty of the future price.
Investing in risky assets can potentially offer higher returns compared to risk-free assets, but they come with higher risk. In a portfolio, a mix of risky and risk-free assets can help balance out potential returns against risk.

A portfolio strategy involves selecting the proportions of different assets in a portfolio to achieve a financial goal. Here, the portfolio consists of positions in one risk-free asset and one risky asset, denoted by \(y\) and \(x\) respectively. The initial value of this portfolio is calculated as \(10x + 10y\) dollars.
The future value of this portfolio can be calculated based on the outcomes of the risky asset. If the risky asset's future value is 13, the portfolio value is \(13x + 11y\) dollars. If the future value is 9, then it becomes \(9x + 11y\) dollars.
The portfolio strategy must be planned such that it aligns the initial value with potential future values for the strategy to be considered admissible. This requires solving the given equations for \(x\) and \(y\).

Admissible portfolios are those that meet predefined criteria for initial and future values. In this context, admissibility involves finding the values of \(x\) and \(y\) such that the portfolio's initial value matches one of its future values.
This is done by solving the given equations: \(10(x + y) = 13x + 11y \) and \(10(x + y) = 9x + 11y \). When solving these, we find:

• For the scenario \(S(1) = 13\), we get \(0 = 3x + y\).
• For \(S(1) = 9\), we get \(x + y = 0\)

From these, we conclude \(y = -3x/11\) and \(y = -x\).
These results form the set of admissible portfolios, ensuring that the initial value of the portfolio aligns correctly with possible future outcomes. This ensures that the strategy reflects both possible states of the world by maintaining a balance between risky and risk-free assets.