Pricing by Arbitrage
Lecture 2: Pricing by Arbitrage
Readings:
Ingersoll – Chapter 2
Dybvig & Ross – “Arbitrage,” New Palgrave entry
Ross – “A Simple Approach to the Valuation of Risky Streams,” Journal of Business, 1978
Here we will take a first look at a financial market using a simple state space model. We first develop some structure then examine the implications of the absence of arbitrage.
Often in finance problems, uncertainty is characterized by the use of a set of random variables with a particular joint distribution, perhaps something like ~ N(, ).
Here, we characterize uncertainty by considering a state space tableau of payoffs on the primitive assets. We assume that there are a finite number of states of nature and that each security has its payoffs written explicitly as a function of the realized state of nature.
We index states by s = 1, 2, …, S (not a problem for S = but intuition can be lost as we look at this for the first time) and assets by i = 1, 2, …, N.
The 2-date investment problem can be characterized by the tableau of per share dollar payoffs on the N assets in each of the S states at date 2 (Y) and a set of current prices (v).
Y ≡ S states and N assets S × N matrix
We want to impose some structure on Y right off. In the investment decision, the agent can make choices only over outcomes (states) which can be distinguished by different patterns of payoffs on the marketed assets. Thus, for the investment decision, if there are states with identical payoffs on all of the assets, then we cannot distinguish between the two so we can collapse them into a single state (that is, the payoff matrix should not have 2 identical rows).
Example: but, and are both fine from this perspective.
To complete the description of the “technology” or opportunities of the model, we use the vector v = to represent the current price per share of each asset.
The decision maker’s choice variable is a portfolio (an N × 1 vector)
Readings:
Ingersoll – Chapter 2
Dybvig & Ross – “Arbitrage,” New Palgrave entry
Ross – “A Simple Approach to the Valuation of Risky Streams,” Journal of Business, 1978
Here we will take a first look at a financial market using a simple state space model. We first develop some structure then examine the implications of the absence of arbitrage.
Often in finance problems, uncertainty is characterized by the use of a set of random variables with a particular joint distribution, perhaps something like ~ N(, ).
Here, we characterize uncertainty by considering a state space tableau of payoffs on the primitive assets. We assume that there are a finite number of states of nature and that each security has its payoffs written explicitly as a function of the realized state of nature.
We index states by s = 1, 2, …, S (not a problem for S = but intuition can be lost as we look at this for the first time) and assets by i = 1, 2, …, N.
The 2-date investment problem can be characterized by the tableau of per share dollar payoffs on the N assets in each of the S states at date 2 (Y) and a set of current prices (v).
Y ≡ S states and N assets S × N matrix
We want to impose some structure on Y right off. In the investment decision, the agent can make choices only over outcomes (states) which can be distinguished by different patterns of payoffs on the marketed assets. Thus, for the investment decision, if there are states with identical payoffs on all of the assets, then we cannot distinguish between the two so we can collapse them into a single state (that is, the payoff matrix should not have 2 identical rows).
Example: but, and are both fine from this perspective.
To complete the description of the “technology” or opportunities of the model, we use the vector v = to represent the current price per share of each asset.
The decision maker’s choice variable is a portfolio (an N × 1 vector)