Asset Pricing
Assumptions:
- Markets are complete: For each state , there exists a state-contingent claim that pays $1 in state and nothing in any other state
- These state-contingent claims are known as state-price securities or Arrow-Debreu securities
- The value of all state-price
Approach
When markets are complete and arbitrage-free, the law of one price states that an asset with state-contingent payoff has price
where is the ratio of state price to probability for state . is called the stochastic discount factor (SDF).
Change of Measure
Problem: how to find the SDF for the option market.
Let and be probability measures on a finite sample space , assume and for every , and define the random variable by
Then we have . Besides, for any random variable , it follows
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