Arbitrage Pricing Flashcards
(21 cards)
Theory of arbitrage
When markets are working well and are efficient, there cannot be arbitrage opportunities.
An arbitrage is a portfolio with zero price and non-negative payoffs.
Law of one price
All portfolios with the same payoffs have the same price.
Under the law of one price, price determination of complicated products is possible by replicating the payoffs with assets with known prices.
Law of one price fails…
One asset is too expensive and another is too cheap, what should you do?
Short sell the expensive and buy the cheap.
Incorrect prices may lead to arbitrage opportunities.
Graphical Approach to Arbitrage: where is the line of zero price?
The line of zero price is orthogonal to the price vector.
Any portfolios with positive payoff below the line of zero price are arbitrage opportunities (i.e., when the cone of positive payoff portfolios goes past the zero price line).
Below the line of negative price are the portfolios with negative prices.
Cone of positive payoffs (graphical arbitrage approach)
The union of payoff vectors produces a cone of positive payoffs.
Is there a non-empty intersection between positive payoff portfolios and negative price portfolios?
If the intersection is zero, no arbitrage portfolios exist.
State Prices
States of the world have prices - if the market is complete, you can give a price to each state.
Arrow-Debreu Security
State prices are unique if the market is complete, i.e., there are sufficiently many independent assets to construct an Arrow-Debreu security for every state, a security that pays 1 in specific states of the world and zero in others.
Arrow-Debreu securities allow the break down of an asset into simpler state-contingent assets.
Why are prices obtained in a risk-neutral world?
Because prices cannot differ based on the risk appetites of different people.
Prices are the discounted expected values in the risk-neutral world.
Risk-neutral / Martingale Probabilities
Asset prices are given by the discounted (by the risk-free rate) expected payoffs matrix.
Expected payoffs use the risk neutral probability measure.
State prices are the discounted risk-neutral probabilities. These probabilities make pricing risk-neutral.
What are Factor Models used for?
Asset prices can be predicted using a small number of appropriate risk factors.
The idea is to compare assets and factors, and set prices based on such relationship so that mis-pricing is avoided.
Factor models evaluate assets’ prices from linear combinations of a small number of variables, related to factors with explanatory power.
What are some examples of factors?
Macro variables, e.g., GDP, inflation, etc.
Market variables, e.g., Gold price, oil price.
Other assets’ returns, e.g. market return in CAPM.
Firm characteristics (e.g., Size effect factor - Banz 1981 rate of return).
Firm financial characteristics (e.g., value factor by Fama and French, valuing growth).
Why do factor models not work after some time?
System changes over time mean that factor models don’t work after some time.
Pricing means trying to forecast the ‘result’…
How to forecast and is there feedback?
Can use historical data and extrapolate - try to forecast future patterns.
We can compare the forecasted price to the market price to see the difference.
Financial forecasts affect the very object they are trying to forecast…
- Aggregating expectations (if an asset is cheap, people buy it and the price goes up).
- There is always feedback in the market.
What are factor pricing models good for?
Allow identification of statistical arbitrages.
Whenever a market price is different from what is predicted by the model, there is an arbitrage opportunity.
When factors are assets, we can trade them, and the pricing model becomes a trading model for statistically hedged portfolios.
What are the caveats of factor pricing models?
These models can only spot statistical arbitrage opportunities when the model is good.
These are statistical arbitrages, so they are based on expectations which are based on historical data.
What is a forward contract?
An agreement to buy or sell an underlying asset for a certain price (the delivery price) at a future date.
Forwards are traded OTC and are fully customisable.
It is an OBLIGATION.
How does a futures contract differ from a forward contract?
Futures are settled daily. If the value of the future has changed (through changes in the underlying asset price)…
- if the change is in your favour, you are given the amount of cash proportional to the number of units you have.
- if the change is bad relative to your position, you must pay the difference.
Who organises daily settlement of futures?
The exchange.
To avoid contract defaults, i.e., not honouring the futures’ obligations by a counterpart.
What is a margin?
When an investor enters the futures contract, her broker requires an amount of cash, the margin, to be deposited in a margin account.
Daily settlement (marking-to-market) at the end of each trading day. The margin is adjusted to reflect the gain or loss on the futures account.
What determines margin size?
The margin is proportional to the Futures contract size and number of contracts. If the margin goes below a certain level, the broker requires it to be topped up.
What should a forward price be?
A forward price should exactly be the compounded spot price of the underlying asset.
At the start of the contract, the forward price should equal the delivery price.
