Risk-neutral Measure - The Origin of The Risk-neutral Measure (Arrow Securities)

The Origin of The Risk-neutral Measure (Arrow Securities)

It is natural to ask how a risk-neutral measure arises in a market free of arbitrage. Somehow the prices of all assets will determine a probability measure. One explanation is given by utilizing the Arrow security. For simplicity, consider a discrete world with only one future time horizon. In other words, there is the present (time 0) and the future (time 1), and at time 1 the state of the world can be one of finitely many states. An Arrow security corresponding to state n, A_n, is one which pays $1 at time 1 in state n and $0 in any of the other states of the world.

What is the price of A_n now? It must be positive as there is a chance you will gain $1; it should be less than $1 as that is the maximum possible payoff. Thus the price of each A_n, which we denote by A_n(0), is strictly between 0 and 1.

Actually, the sum of all the security prices must be equal to the present value of $1, because holding a portfolio consisting of each Arrow security will result in certain payoff of $1. Consider a raffle where a single ticket wins a prize of all entry fees: if the prize is $1, the entry fee will be 1/number of tickets. For simplicity, we will consider the interest rate to be 0, so that the present value of $1 is $1.

Thus the A_n(0) 's satisfy the axioms for a probability distribution. Each is non-negative and their sum is 1. This is the risk-neutral measure! Now it remains to show that it works as advertised, i.e. taking expected values with respect to this probability measure will give the right price at time 0.

Suppose you have a security C whose price at time 0 is C(0). In the future, in a state i, its payoff will be C_i. Consider a portfolio P consisting of C_i amount of each Arrow security A_i. In the future, whatever state i occurs, then A_i pays $1 while the other Arrow securities pay $0, so P will pay C_i. In other words, the portfolio P replicates the payoff of C regardless of what happens in the future. The lack of arbitrage opportunities implies that the price of P and C must be the same now, as any difference in price means we can, without any risk, (short) sell the more expensive, buy the cheaper, and pocket the difference. In the future we will need to return the short-sold asset but we can fund that exactly by selling our bought asset, leaving us with our initial profit.

But regarding each Arrow security price as a probability, we see that the portfolio price P(0) is the expected value of C under the risk-neutral probabilities. If the interest rate were positive, we would need to discount the expected value appropriately to get the price.

Note that Arrow securities do not actually need to be traded in the market. This is where market completeness comes in. In a complete market, every Arrow security can be replicated using a portfolio of real, traded assets. The argument above still works considering each Arrow security as a portfolio.

In a more realistic model, such as the Black-Scholes model and its generalizations, our Arrow security would be something like a double digital option, which pays off $1 when the underlying asset lies between a lower and an upper bound, and $0 otherwise. The price of such an option then reflects the market's view of the likelihood of the spot price ending up in that price interval, adjusted by risk premia, entirely analogous to how we obtained the probabilities above for the one-step discrete world.