OK. So I asked how is the Girsanov Theorem, which is true, logically connected with this statementBeing precise always helps.
I apologise, I was being insufficiently precise in the OP. I should have distinguished two statements."Why do people think it is generally true?"
Why do you think that people think it is generally true?
The right hand side of (5) is the forward price of a call option in the BSM model. That can be (and indeed was) derived without any reference to expectations at all, risk neutral or otherwise, by PDE methods.
Thanks to the Girsanov theorem you can replace the physical measure with such an equivalent (risk-neutral) measure r that the expectation can be simply expressed as E[S_t] = S*exp(r*t). Bearish answer to my question completes the general problem formulation.OK. So I asked how is the Girsanov Theorem, which is true, logically connected with this statementBeing precise always helps.
E[ST] = Sert
which is false? Logicians say that it is impossible to prove a false statement from a true statement. That is because a proof by definition is a valid argument, and a valid argument by definition is one whose conclusion cannot be false with the premisses true. Do you agree with logicians?
So you don't agree that it is impossible to derive a false statement from a true statement. Fair enough. Logicians would disagree.Does that help?
So the BS Formula is true only when there is no risk premium? False.When they solve the PDE the equation (4) only stand when [$]E[ \ ][$]
means expectation with a probability measure where the spot log normal diffusion as a drift coefficient [$]r S_t dt[$], i.e. what is called risk neutral measure.
No, B&S formula is true with the model assumptions (which is not that there is no risk premium), but solution of PDE is expectation only with the SDE compliant with the PDE coefficients (Feynmann-Kac theorem). And under the same risk neutral measure you got [$]E[S_T] = S_t \exp^{r (T-t)}[$], while in the same model on historical measure with drift [$] \mu S_t dt [$] you would have [$]E^{hist}[S_T] = S_t \exp^{\mu (T-t)}[$].So the BS Formula is true only when there is no risk premium? False.When they solve the PDE the equation (4) only stand when [$]E[ \ ][$]
means expectation with a probability measure where the spot log normal diffusion as a drift coefficient [$]r S_t dt[$], i.e. what is called risk neutral measure.
Thanks. Are there any simpler ways to derive the Formula direct from the BS PDE? And how to derive without any appeal to expectations at all?(3) is not a premiss. As I said B&S model => PDE => expectation with a specific measure coming from Feynmann-Kac theorem => (3) is true with the expectation using the same measure.
And in that model there are risk premiums.
Bearish casts pearls upon this forum - which get trampled sometimes. I just come here because I like Paul quoting Lewis Carroll.So you don't agree that it is impossible to derive a false statement from a true statement. Fair enough. Logicians would disagree.Does that help?
Bearish answer was better, in my view.