expected value of the discounted payoff

Question

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?

Answer 1

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.

Answer 2

In Mathematical finance, the concept of "arbitrage-free" portfolios is usually introduced before risk neutral measure. No arbitrage theory is intuitive for student when formulated this way: "there is no strategy that can beat the market". (or NFLVR)

Then one can show that this implies the existence of a risk neutral measure (completeness of the market gives uniqueness of the market). Finally the reverse implication is mathematically showed, but has little less interest because one can grasp the idea with the first proof.

Why is this equivalent ? The value of an asset is not his expected value at the end of the contract because there is risk. No Artbitrage hypothesis implies there is a common value for bearing the risk. If you mathematically change your probability space to take this common risk bearing premium into account in your metric, the value of the asset will be the expected value in the new probability space. This is mainly a mathematical trick, it's just factoring the risk premium.

Why is this interesting ? It's easier to find the risk neutral probability once and for all assets and then pricing the assets taking their expected values than calculating each asset expected value corrected by the risk premium.