4- Arbitrage pricing theory Flashcards
What will be the price of a portfolio?
Sum of product of asset prices and holdings:
p = Σpᵢxᵢ
What will be the payoff of a portfolio?
Sum of products of asset holdings and their payoffs in every possible state:
v = Vx
What is the Law of one price?
All portfolios with the same payoff have the same price:
px = px’
Define an arbitrage in terms of asset pricing
A portfolio with zero price and non-negative payoffs with a strict inequality for at least one state k:
Σpᵢxᵢ = 0
Vx ≥ 0
What constitutes a strong arbitrage?
An arbitrage with a strictly negative price
What are the 2 steps to finding a missing asset price (pₐ)?
-Impose 0 payoffs in each state of the payoff matrix to solve for holdings as proportion of asset holding (x/xₐ)
-Impose 0 portfolio price and divide equation by asset holding (xₐ) to solve for the asset’s price
What are state prices (q)?
Asset prices in each state multiplied by their respective payoff gives all asset prices:
p = qV
What is the main thing factor pricing models are good for?
Whenever a market price is different from what predicted by the model, we have an arbitrage opportunity
What is the main caveat of using factor pricing models?
The arbitrages are statistical, so they are based on expectations which are based on historical data
How does daily settlement of futures work?
At the end of each trading day, margin is adjusted to reflect the gain or loss on the Futures price
What is the formula for Forward price (F₀)?
F₀ = S₀eʳᵀ
What is the intuition behind forward price formula?
Forward seller has to borrow notional value to buy the underlying stock
What is the formula for forward value (f) during the contract’s lifetime?
f = (F₀ - K)e⁻ʳᵀ
How can you prove you can obtain an arbitrage from a matrix of asset prices?
-Find holdings matrix (x) for which portfolio price is 0
-Multiply holdings matrix by payoff matrix and show nonnegative payoffs
