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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.

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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?

user119615
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2 Answers2

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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.

RRL
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  • Thank you, I hope it is ok if I ask one more question, I don't know many financial terms. I don't fully understand this: "They are speaking of price in the context...", I am not really sure what the price here means?The model says that $V_t=H_0B_t+\Sigma H_n*S_n(t)$, t=0,1, where V is the value, B is the bank-account process($B_0$=1),$H_0$ is the amount starting in the bank, $B_1$ and all the $S_n(1)$ are stochastic variables for prices of all the securities.I assume that you mean that the price they are speaking about are not the S's, can you explain it in terms of the model please? – user119615 Jul 31 '14 at 13:36
  • The "price" is $V_0$. Think of call option that pays $\max(S_T-K,0)$ where $S_T$ is the stock price at time $T$. If I have two strategies $V_t$ and $\hat{V_t}$ that lead to that option payoff, then the initial price must be the same -- $V_0 = \hat{V_0}$ – RRL Jul 31 '14 at 13:50
  • Thank you very much. I have just one final question if you have the time. I see that you justified the "law of one price" by the fact that we can not have any arbitage, and Wikipedia also says that it is derived from this need. However even though we have the implications "no arbitrage $\rightarrow$ law of one price" the opposite implication does not hold. So is there an intuitive way of explaining in terms of the model why the law of one price should hold without using concepts like "no arbitrage" or "no dominant trading strategies."? – user119615 Jul 31 '14 at 14:12
  • What I am trying to say is that "no arbitrage" seems logical, but because of the implications there must be markets where you have arbitrage, but in one the law of one price holds, and in the other it doesn't, why is then the market with the law of one price most realistic? – user119615 Jul 31 '14 at 14:13
  • This is a good question but getting more subjective. I suggest trying Quantitative Finance StackExchange, but I'll give it some thought. – RRL Jul 31 '14 at 14:46
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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.

Matt B.
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