Don't drag me into your wikipedia logic drivel, please.

OK is Wolfram better? https://mathworld.wolfram.com/Validity.html

(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. Thanks. Are there any simpler ways to derive the Formula direct from the BS...

Mars: "which is not that there is no risk premium" OK so (3) below is false. (3)        E[S T ]  =  Se rT   We all agree (I hope) that (4) is true (4)    E[(S T  – K)+]  =  E[(S T ] N(d1) – KN(d2)   And we agree that (3) and (4) implies (5)    E[(S T  – K)+]  =  Se rt  N(d1) – KN(d2) However as I p...

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. So the BS Formula is true only when there is no risk premium? False. 

Does that help?

So you don't agree that it is impossible to derive a false statement from a true statement. Fair enough. Logicians would disagree.

Bearish answer was better, in my view.

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.   Yes. Black Scholes: “There is only one formula w(x, t) that satisfies the differential e...

"Why do people think it is generally true?" Why do you think that people think it is generally true? I apologise, I was being insufficiently precise in the OP. I should have distinguished two statements. (1)             Ê[S T ]  =  Se rT (3)             E[S T ]  =  Se rT   The first (1) is what log...

Being precise always helps. OK. So I asked how is the Girsanov Theorem, which is true, logically connected with this statement  E[S T ] = Se rt 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 argu...

You're very imprecise in your statements for an alleged logician. I wrote a statement of fact. It can't be false.

OK when I said 'the statement above' I was referring to the following statement:

    E[S_T] = Se^rt

Does that help?

It's merely a mathematical transformation which is used to make your integrals/valuations simpler. Unless you want a proof that the Girsanov Theorem does what it does   How is the Girsanov Theorem, which is true, logically connected with the statement above, which is false? https://en.wikipedia.org...

Actually, there is, in the general case. Ideally, you’d like to establish the equivalence between the absence of arbitrage opportunities and the existence of an equivalent martingale measure, as well as the equivalence between complete markets and the uniqueness of said martingale measure (for a gi...

My background is in logic, in which I have published work. An assumption or premiss is any statement made, together with other such statements, in support of a conclusion. The premisses support the conclusion iff the premisses cannot be true and the conclusion false. Then we have a valid argument.  ...

Paul: " Untrue, but used in proof by contradiction" But the proof of the Formula does not have the form of a reductio .  Consider (3)    E[S T ] Se rt which unlike (1) above is false when risk premia exist. Substitute this into  (4)    E[(S T – K)+]  =  E[(S T ] N(d1) – KN(d2) to give (5)    E[(S T ...

No. Over 30 years experience of speaking to people about this topic. It confuses people more than anything else. Do you think it is confusing? If not, can you give a clear explanation of (Hull's words, not mine) "the assumption that investors are risk-neutral"? Is it the same as (1) the assumption ...

Hull again (p.258) To apply risk-neutral valuation to the pricing of a derivative, we first calculate what the probabilities of different outcomes would be if the world were risk-neutral. We then calculate the expected payoff from the derivative and discount that expected payoff at the risk-free rat...