Basic question mathematical finance, law of one price in single period markets.

Question

I am reading Pliskas Introduction to mathematical finance. And I am at single period models. It is the law of one price I am having a hard time of understanding.

I have some questions about this:

On Wikipedia I read the the law of one price means that there is only one price for each item, and it is fixed. However in the model introduced in the book there is allready a price process defined, so isn't there already one price for each item?

Also why is itnot realistic that two persons who start with a different amount of money(different $V_0$) will end up with the same amount of money?

Answer 1

They are speaking of price in the context of a strategy that replicates the payoff of a claim contingent on the price of an underlying asset. An example is a call option on a stock.

If there are two strategies that lead to the same payoff (with probability $1$) the amount of money required to initiate each strategy must be the same. Otherwise there would be an arbitrage -- you could for example use one strategy to replicate a long position in an option, use the other to replicate a short position on the same option and gain a sure profit.

In real markets, such opportunities seldom exist and are fleeting as they are quickly exploited by participants causing prices to adjust and eliminate the opportunity.

Answer 2

The reasoning goes as follows: - you know the price of your claim at the end of the period, because there is a formula for it for instance (options, futures, etc.).

• you can replicate with a self-financing portfolio that payoff at $T=1$, that is you found a way to deliver that claim

• then, you know that the value of the portfolio at any point in time has to be the value of the claim. It's the basic principle of arbitrage theory. If somebody could do it cheaper then they would buy it from them and sell it to you and make a profit.