# Risk Neutral Probability

I read that an option prices is the expected value of the payout under the risk neutral probability. Intuitively why is the expectation taken with respect to risk neutral as opposed to the actual probabilty.

Intuitively why would risk neutral probability differ from actual probability?

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Why do two probability measures differ? Because of the way they are constructed. Do you ask why risk-neutral measure is constucted in a different way then real-world measure? Or why it is constructed at all? –  Rustam Oct 25 '13 at 6:48
I tried to answer but maybe you're missing something from my answer. Please clarify if that is the case. –  Bob Jansen Oct 25 '13 at 14:17

Consider a one-period binomial lattice for a stock with a constant risk-free rate.

Determine the initial cost of a portfolio that perfectly hedges a contingent claim with payoff xu in the upstate and xd in the downstate (you can do this so long as the up and down price are different in your lattice).

Assuming there exists no portfolio that yields a profit without downside risk (assume no arbitrage) and that your economy is frictionless and competitive, show that any other price for the contingent claim, other than the initial cost of the replicating portfolio you found, would lead to the existence of a portfolio that yields a profit without downside risk. Pause and reflect on the fact that you have determined the price of any contingent claim without any mention of probability. However, don't forget what you assumed! What did you actually need to do what you just did?

Now that you know that the price of the initial portfolio is the "arbitrage free" price of the contingent claim, find the number q such that you can express that price of the contingent claim as the !discounted payoff! in the up state times a number q plus the discounted payoff in the downstate times the number 1-q. Solve for the number q. Interpret the number q as a probability and compute the expected value of the discounted stock with this probability. This should be the same as the initial price of the stock. Pause and reflect on the fact that you have determined the unique number q between 0 and 1 such that the expected value (using q) of the discounted stock is the initial price and that you can compute the price of any contingent claim by computing its expected (using q) discounted payoff.

It is clear from what you have just done that if you chose any other number p between 0 and 1 other than the q and computed the expected (using p) discount payoff, then you would not recover the arbitrage free price (remember you have shown that any other price than the one you found leads to an arbitrage portfolio). This means that if you had a real world probability p for your initial lattice, it is not the correct probability to use to when computing the price.

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I highly recommend studying Folmmer and Schied's Stochastic Finance: An Introduction in Discrete Time. In my opinion, too many people rush into studying the continuous time framework before having a good grasp of the discrete time framework. –  user6384 Oct 28 '13 at 19:55
(+1) you could have used some spaces, but it is a very clear explanation. If you have also some clear views about real-world probabilities perhaps you can help me here: quant.stackexchange.com/questions/8274/… –  sets Oct 29 '13 at 7:27

I think the classic explanation (any other measure costs money) may not be the most intuitive explanation but it is also the most clear in some sense and therefore does not really require a intuitive explanation.

That is to say: you could use any measure you want, measures that make sense, measures that don't but if the measure you choose is a measure different from the risk neutral one you will use money. Therefore, don't.

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Risk neutral probability differs from the actual probability by removing any trend component from the security apart from one given to it by the risk free rate of growth. If you think that the price of the security is to go up, you have a probability different from risk neutral probability.

In very layman terms, the expectation is taken with respect to the risk neutral probability because it is expected that any trend component should have been discounted for by the traders and hence at any moment, there is no non-speculative reason to assume that the security is biased towards the upside or the downside.

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You're missing the point of the risk-neutral framework.

The idea is as follows: assume the real probability measure called $\mathbb{P}$. The thing is, because investors are not risk-neutral, you cannot write that $v_0 = E_\mathbb{P} [ e^{-rT} V_T]$.

Using the Fundamental Theorem of Asset Pricing, you know that if the market is arbitrage-free, then there exists a probability measure $\mathbb{Q}$ such that $v_0 = E_\mathbb{Q} [ e^{-rT} V_T]$.

So what you do is that you define the probability measure $\mathbb{Q}$ sur that $v_0 = E_\mathbb{Q} [ e^{-rT} V_T]$ holds.

Obviously, they are not the same thing.

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This is the best explanation. –  Student T Nov 21 '13 at 2:16