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today I had an oral exam about Stochastic Finance. With one of the questions I was pretty helpless. We were talking option pricing in a scenario where we have Portfolio with n-assets and k-states. But then he asked me:

How do you derive the risk neutral probability by using the arrow debreu price vector?

Maybe I'm just lost in the terminology, but any help is appreciated!

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I will try to give a simple explanation. So please add a comment if you have any specific questions.

Let's assume there are k states, and let $p_s$ be the price of a state s. So p is the state price vector $ p=\left(p_1, p_2, \dots, p_k\right)$. Let

$ \pi_s =\frac{p_s}{\sum_{i=1}^k{p_i}}$

Then $ \pi_s $ as defined above can be interpreted as probabilities (they sum to one, are positive etc), and state space as probability space. Additionally, the denominator is the sum of the price of all k securities, so it is the price of an asset that pays 1 in every state, and therefore we can interpret it as the price of a risk free asset, $R_f$. Hence we can write the above equation as follows:

$ \pi_s =\frac{p_s}{R_f}$

Now, let's say we have a security bundle with payoff $x=\left( x_1, x_2, \dots, x_k\right)$ in the k-states. So its price can be written as follows:

$Price(x)=\sum_{i=1}^k{x_i p_i}$

Which we can easily re-write in terms of the probabilities above ($p_i=\pi_i*R_f$):

$Price(x)=\sum_{i=1}^k{x_i \, \pi_i \, R_f}=R_f \sum_{i=1}^k{x_i \, \pi_i}$

Now, the probability weighted payoff is exactly how one writes expectation, so this becomes:

$Price(x)=R_f E \left[ x\right] $

Which is just the risk neutral valuation formula. You can see from the above that risk neutral probabilities are just forward state prices.

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