# expected value of the discounted payoff

I don't understand the following statement: The price of a contingent claim is the expected value of the discounted payoff value under the risk neutral probability measure Q deﬁned in complete markets with no-arbitrage. $$\mathbb E^Q\left[(S_T-K)_+e^{-\int_0^T r_s\, ds}|\mathcal F_0\right]$$

$\mathbb E^Q [.]$ is the expectation under the risk neutral measure $Q$, $S_T$ is the underlying strike price, and $r_s$ is the risk-free rate.

Through what argument does the existence of a risk neutral probability measure imply an arbitrage-free value of the portfolio?

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This is a statement. What is the question? –  Christian Fries Jun 29 '13 at 18:36
Where did you find the statement? In which context? –  vonjd Jul 1 '13 at 11:00

1) If some process $V_t$ is a martingale under some measure $Q$, we can always write $V_t = \mathbb{E}^Q_t[V_T]$. It is simply a definition of a martingale.

2) Next question is "in which measure would my process be a martingale"? How do people in textbooks answer? They say, "we will measure the performance of your portfolio relative to money market account". Now, MMA grows in steady pace $r(t)$. Hence, in order to keep its value constant, they discount it with the integral in your expectation.

Then they adjust the brownian motion your asset follows by changing probability measure. Thats where discounting comes into formula from. And voila! Your process is now martingale in that measure and you happily write the formula above.

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