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katastrofa
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### Re: Proof of the risk-neutral assumption

Being precise always helps.

complyorexplain
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Joined: November 9th, 2015, 8:59 am

### Re: Proof of the risk-neutral assumption

Being precise always helps.
OK. So I asked how is the Girsanov Theorem, which is true, logically connected with this statement

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?

Mars
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### Re: Proof of the risk-neutral assumption

"Why do people think it is generally true?"

Why do you think that people think it is generally true?

complyorexplain
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### Re: Proof of the risk-neutral assumption

"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[ST]  =  SerT

(3)             E[ST]  =  SerT

The first (1) is what logicians call necessarily true, because true in virtue of its meaning. It states that in a world without risk premium, the (non-income producing) asset grows on average at the risk free rate. It is true because ‘not having a risk premium’ means ‘grows on average at the risk free rate’.

The second is contingently true, i.e. not necessarily true. Its truth depends on the state of the world. In a a world without risk premium, (3) is true. Otherwise it is false.

To the question “Why do you think that people think it is generally true?”. If ‘it’ refers to proposition (1), then they think it is true because it is true in virtue of its meaning. If ‘it’ refers to proposition (3), well I’m not really sure whether people think it true or not. Here is one reason people might think it true. I think many think the following is true:

(4)    E[(ST – K)+]  =  E[(ST] N(d1) – KN(d2)

And everyone thinks this is true

(5)    E[(ST – K)+]  =  Sert N(d1) – KN(d2)

But (3) follows logically from (4) and (5).

bearish
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Joined: February 3rd, 2011, 2:19 pm

### Re: Proof of the risk-neutral assumption

Hmm. I think I had promised to give up, but taking one more crack. 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. Or, my personal favorite method, as the limiting value of a binomial model. Probabilistic techniques like measure changes are useful for calculating option values and their sensitivities, but usually just that. The undeniable fact that a lot of people get confused about what it all means and forget about the “risk neutral” modifier just generates noise that is best ignored.

complyorexplain
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### Re: Proof of the risk-neutral assumption

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 equation (7) subject to the boundary condition (8). This formula must be the option valuation formula” (1973 p643).

So what they did was to find the BSME by no-arbitrage considerations, then solve that equation by tinkering with the formula derived by Sprenkle, which is (4) above, namely this:

(4)    E[(ST – K)+]  =  E[(ST] N(d1) – KN(d2)

This was set out in a 1970 paper before they met Merton.

katastrofa
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Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Proof of the risk-neutral assumption

Being precise always helps.
OK. So I asked how is the Girsanov Theorem, which is true, logically connected with this statement

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?
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.

You could equally well ask people if MC importance sampling is generally true.

Does that help?

complyorexplain
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Joined: November 9th, 2015, 8:59 am

### Re: Proof of the risk-neutral assumption

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.
Last edited by complyorexplain on December 21st, 2020, 3:30 pm, edited 1 time in total.

Mars
Posts: 115
Joined: November 13th, 2002, 5:10 pm

### Re: Proof of the risk-neutral assumption

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.

complyorexplain
Topic Author
Posts: 176
Joined: November 9th, 2015, 8:59 am

### Re: Proof of the risk-neutral assumption

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.

Mars
Posts: 115
Joined: November 13th, 2002, 5:10 pm

### Re: Proof of the risk-neutral assumption

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.
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 $E^{hist}[S_T] = S_t \exp^{\mu (T-t)}$.

complyorexplain
Topic Author
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Joined: November 9th, 2015, 8:59 am

### Re: Proof of the risk-neutral assumption

Mars: "which is not that there is no risk premium"

OK so (3) below is false.

(3)        E[ST]  =  SerT

We all agree (I hope) that (4) is true

(4)    E[(ST – K)+]  =  E[(ST] N(d1) – KN(d2)

And we agree that (3) and (4) implies
(5)    E[(ST – K)+]  =  Sert N(d1) – KN(d2)

However as I pointed out a few times, (3) - (5) is an unsound argument, i.e. a valid argument with a false premiss.

Mars
Posts: 115
Joined: November 13th, 2002, 5:10 pm

### Re: Proof of the risk-neutral assumption

(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.

complyorexplain
Topic Author
Posts: 176
Joined: November 9th, 2015, 8:59 am

### Re: Proof of the risk-neutral assumption

(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 PDE? And how to derive without any appeal to expectations at all?

katastrofa
Posts: 10084
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Proof of the risk-neutral assumption

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.
Bearish casts pearls upon this forum - which get trampled sometimes. I just come here because I like Paul quoting Lewis Carroll.

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