Why is the solution for the state price vector unique in the three-state market model?
Conditions
- The market model has three states and three traded assets.
- The pricing system is a 3x3 linear system.
- The coefficient matrix of the system is nonsingular (determinant is nonzero).
Reasoning, step by step
- Set up the linear pricing equations based on the asset payoffs and current prices.
- Extract the coefficient matrix from the system of equations.
- Check the determinant of the coefficient matrix.
- Conclude that because the determinant is nonzero, the matrix is invertible, ensuring a unique solution for the state price vector.
Example
The coefficient matrix for the three-state market is . The lecturer notes that switching the second and third rows makes the determinant easy to compute, and concludes it is nonzero, meaning the solution is unique.
Common misconceptions
- Misinterpreting the phrase 'the matrix is non-zero' as merely not being the zero matrix, rather than understanding it means the determinant is nonzero (nonsingular).
- Assuming that any system of linear equations with an equal number of equations and unknowns automatically has a unique solution, without checking the determinant.
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Conditions: One-period model with a finite set of states.; Prices of all traded assets are known today.; State-contingent payoffs for each asset are known.; The unknowns are the state prices (Arrow-Debreu security prices).
You convert a state price vector into risk-neutral probabilities by multiplying each state price by the gross risk-free factor , where is the per-period risk-free interest rate. This scaling transforms the monetary prices of Arrow-Debreu securities into a valid probability distribution that sums to 1.
Conditions: The state prices are already determined.; is the known per-period risk-free interest rate.; The market is a one-period finite-state model.
The arbitrage-free price of a European put option is calculated by first determining its state-contingent payoffs, decomposing this payoff vector into the Arrow-Debreu basis, and then taking the dot product of the payoff vector with the state price vector. This relies on the linearity of pricing in a finite-state market.
Conditions: The market is a one-period finite-state model.; The state price vector is known.; The option is European and expires at the end of the period.; The underlying asset's state-contingent payoffs and the strike price are known.
The state price vector for this specific three-state market model is . This vector is found by solving the linear system of pricing equations derived from the given asset payoffs and their common current price of 100.
Conditions: One-period securities market model.; Three possible scenarios (states).; Risk-free bank account with 10% interest (payoff 110 in all states).; Current price of each security is 100.; Security X payoffs: 220, 55, 0.; Security Y payoffs: 0, 0, 250.
The bank account contributes 110 in every state because it is a risk-free asset that pays 10% interest per year. Investing 100 today guarantees a return of 110 at time one, regardless of which of the three possible scenarios occurs.
Conditions: The bank account is risk-free.; The bank account pays 10% interest per year.; The initial investment (current price) is 100.; The model is a one-period setting with multiple states.
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