Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but struggling.
I read through the thread here : Radon-Nikodym derivative and risk natural measure as well.
which is under the real-world measure.
We claim there is an equivalent probability measure (called the risk-neutral measure) where the return is the risk-free rate, with the Brownian Motion adjusted with a drift.
- So we have set the 'market-price of risk' $= (\mu - r)/\sigma$
- We set the risk-neutral Brownian motion $dW(Q) = (\mu - r)/\sigma * dt + dW$
My understanding of the Radon-Nikodym and Girsanov Theorem, is that we can now express any asset X (or S in this case) from one measure (real-world) into another measure (risk-neutral) by the Radon-Nikodym as per follows
where EN is the expectation under risk-neutral world, X is asset X, EU is expectation under real-world' and dN/dU is the Radon-Nikodym derivative