Thanks for the great video (+1 like from me). I think that i do no fully understand the change of measure using the Radon-Nikodym derivative under the expected value. I know that you can define the Radon-Nikodym derivative of Q wrt. P (sometimes it's even called a Likehood process, L, but when?) as L(x) := dQ(x) / dP(x). However, how exactly does this work when taking expected values? I suppose it is somehow using the definition of the expected value being an integral but some details would be appreciated, thanks! My suggestion/ idea is something like this, where w (lowercase omega) is taken over the probability space W (uppercase Omega) E_Q[ X ] = int_W X(w) dQ(w) = Int_W X(w) dQ(w)/dP(w) * dP(w) = E_P[X dQ(w)/dP(w)]
The math is exactly how you’ve written it here, and what was showed in the importance sampling equation and change of measure for 25 std dev example. The expectation is the integral of the random variable over the domain space under the probability measure. When you change the probability measure to make your observed payoffs ect more likely, than as you’ve written in the 2nd line, you now are left with the ratio between the two probability distributions when integrating over the new probability measure space. When it comes to Monte Carlo, and taking the expectation, this translates as you’ve written in the 3rd line with the expectation with simulated variables under the new probability space multiplied by the likelihood of the simulated variables between the old distribution divided by the new one.
IIRC (and not totally sure if it’s applicable anymore with more recent versions of python & scipy, but ) I remember reading somewhere that using an internal function _pdf was Z times faster than the public function .pdf. Will Google later to see if it’s true, but I recall it was significantly faster.
Hey man, I just wanted to say that you are very good at explaining those concepts and your code looks very good. Keep up the good work!!
Asked and delivered - great vid, thanks!
Thanks for the great video (+1 like from me).
I think that i do no fully understand the change of measure using the Radon-Nikodym derivative under the expected value.
I know that you can define the Radon-Nikodym derivative of Q wrt. P (sometimes it's even called a Likehood process, L, but when?) as L(x) := dQ(x) / dP(x).
However, how exactly does this work when taking expected values?
I suppose it is somehow using the definition of the expected value being an integral but some details would be appreciated, thanks!
My suggestion/ idea is something like this, where w (lowercase omega) is taken over the probability space W (uppercase Omega)
E_Q[ X ]
= int_W X(w) dQ(w)
= Int_W X(w) dQ(w)/dP(w) * dP(w)
= E_P[X dQ(w)/dP(w)]
The math is exactly how you’ve written it here, and what was showed in the importance sampling equation and change of measure for 25 std dev example.
The expectation is the integral of the random variable over the domain space under the probability measure. When you change the probability measure to make your observed payoffs ect more likely, than as you’ve written in the 2nd line, you now are left with the ratio between the two probability distributions when integrating over the new probability measure space.
When it comes to Monte Carlo, and taking the expectation, this translates as you’ve written in the 3rd line with the expectation with simulated variables under the new probability space multiplied by the likelihood of the simulated variables between the old distribution divided by the new one.
IIRC (and not totally sure if it’s applicable anymore with more recent versions of python & scipy, but ) I remember reading somewhere that using an internal function _pdf was Z times faster than the public function .pdf. Will Google later to see if it’s true, but I recall it was significantly faster.
Muito bom!
Hi, could you share the code with us?
Hi Jonas, yes working on updating my website, code tutorials will be out later next week