Serving the Quantitative Finance Community

 
User avatar
complyorexplain
Topic Author
Posts: 175
Joined: November 9th, 2015, 8:59 am

Re: Proof of the risk-neutral assumption

December 19th, 2020, 8:27 pm

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 that no risk premium exists, (2) the assumption that the expected return on an asset is the risk-free rate, or something else?
 
User avatar
Paul
Posts: 11276
Joined: July 20th, 2001, 3:28 pm

Re: Proof of the risk-neutral assumption

December 19th, 2020, 9:30 pm

Rabbit hole.

Examples of assumptions:

1. There is a finite number of primes. Untrue, but used in proof by contradiction.
2. Volatility is constant. Probably untrue. But makes life simpler. Assuming it will give wrong answers. But one has to be pragmatic.
3. Investors are risk neutral. Not true. But it doesn't matter. You will get the same option values regardless of veracity. As bearish says (and this is probably the best way of thinking of it), it is a computational device.
 
User avatar
katastrofa
Posts: 10082
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Proof of the risk-neutral assumption

December 19th, 2020, 9:32 pm

I have a suggestion of an answer to this question (a layman perspective):
Eq 2 is an empirical formula for non-arbitrage pricing. Mind that it contains only observed values and is gauge invariant.
Eq 1 is a model with the risk neutral measure assumption (which is wrong but may be useful), which can be used to price F in Eq 2 (and no arbitrage happens to be a necessary condition for risk neutral measure - at least at the first sight).
CMIIW
 
User avatar
complyorexplain
Topic Author
Posts: 175
Joined: November 9th, 2015, 8:59 am

Re: Proof of the risk-neutral assumption

December 19th, 2020, 10:17 pm

Paul: "Untrue, but used in proof by contradiction"

But the proof of the Formula does not have the form of a reductioConsider

(3)    E[ST] Sert

which unlike (1) above is false when risk premia exist. Substitute this into 

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

to give

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

Now (5) is true, but this is not a reductio, rather an unsound argument, namely and argument which is valid ( (3) and (4) together imply (5) ), but one or more of whose premisses are false (namely (3)). From which true premisses do we derive (5). Also, what does the PDE have to do with (3) - (5)?
 
User avatar
Paul
Posts: 11276
Joined: July 20th, 2001, 3:28 pm

Re: Proof of the risk-neutral assumption

December 19th, 2020, 11:03 pm

I am giving vastly different examples of the concept of “assumption.”

You are starting in the middle of something. One of the problems is what you meant by “assumption.”

I am guessing that your math background is of the more abstract, pure form. That can often lead to problems. A little knowledge etc. Personally I prefer to teach those who haven’t tasted the Pierian spring. Otherwise things often go badly.
 
User avatar
complyorexplain
Topic Author
Posts: 175
Joined: November 9th, 2015, 8:59 am

Re: Proof of the risk-neutral assumption

December 20th, 2020, 10:43 am

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. 

The argument (3)-(5) above is a valid argument, as is any argument using substitution. But it is not a 'sound' argument because one of the premisses i.e. (3) is false https://en.wikipedia.org/wiki/Soundness.
 
User avatar
Cuchulainn
Posts: 64439
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

Re: Proof of the risk-neutral assumption

December 20th, 2020, 1:42 pm

"Hills peep o’er hills, and Alps on Alps arise !"
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl
 
User avatar
katastrofa
Posts: 10082
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Proof of the risk-neutral assumption

December 21st, 2020, 1:15 am

I don’t think you’ll get there as long as you view it as an assumption. It’s a derived result, and not a particularly easy one either. On the rare occasion that I try to teach this, I start with the one step binomial. That contains most of the finance content of the argument. In the other extreme, you can go back to the original literature on this from ca 1980, summarized in these presentation notes from one of the old masters himself:

https://www.fields.utoronto.ca/programs ... liska2.pdf
Is there more to it than the Girsanov theorem? Not that Girsanov was the first with that one.
 
User avatar
bearish
Posts: 6449
Joined: February 3rd, 2011, 2:19 pm

Re: Proof of the risk-neutral assumption

December 21st, 2020, 1:52 am

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 given numeraire asset). That turned out to be a lot harder than some very smart people thought back in the 80’s.
 
User avatar
Paul
Posts: 11276
Joined: July 20th, 2001, 3:28 pm

Re: Proof of the risk-neutral assumption

December 21st, 2020, 9:10 am

Rabbit hole continued.

"Begin at the beginning," the King said gravely, "and go on till you come to the end: then stop."
 
User avatar
complyorexplain
Topic Author
Posts: 175
Joined: November 9th, 2015, 8:59 am

Re: Proof of the risk-neutral assumption

December 21st, 2020, 10:17 am

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 given numeraire asset). That turned out to be a lot harder than some very smart people thought back in the 80’s.
So to be clear, are you saying that it is possible to prove the following statement?
 
(3)    E[ST] = Sert
 
I don’t think you are saying that, but just asking.

[EDIT] Again, to be clear, I am not denying that (3) is true if we use the risk-neutral expectation operator. Then it is trivially true, for it states that in a world without risk premia, the expected return is the risk free rate. But (3) above doesn't state that. It states that it is unconditionally true that the expected return is the risk free rate. Which is false.
 
User avatar
katastrofa
Posts: 10082
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Proof of the risk-neutral assumption

December 21st, 2020, 10:48 am

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 (Bing it yourself).


I used it on multiple occasions in physics, to give one of the more complex ones which curiously involved the name of one of the brightest WF stars:

"Continuous quantum jumps and infinite-dimensional stochastic equations"
Dariusz Gatarek
Systems Research Institute, Polish Academy of Sciences, 01-447 Warszuwa, Newelska 6, Poland
Nicolas Gisin
Groupe de Physique Appliquee, Universite de Geneve, 20 rue de I’Ecote de Medecine, 1211 Geneve 4,
Switzerland


(I was implementing the Quantum State Diffusion method at a London uni college and I remember I naively tried to deliver the maths to people who struggled to understand the difference between an expectation and a mean - which didn't prevent them from putting the most statistically advanced physical theories as specialisations in their curricula ;-) They hated me for that.)
Attachments
qsd.pdf
(83.47 KiB) Downloaded 25 times
GatarekGisin.pdf
(665.06 KiB) Downloaded 20 times
 
User avatar
complyorexplain
Topic Author
Posts: 175
Joined: November 9th, 2015, 8:59 am

Re: Proof of the risk-neutral assumption

December 21st, 2020, 10:52 am

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/wiki/Validity_(logic)
 
User avatar
katastrofa
Posts: 10082
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Proof of the risk-neutral assumption

December 21st, 2020, 11:56 am

You're very imprecise in your statements for an alleged logician. I wrote a statement of fact. It can't be false.
 
User avatar
complyorexplain
Topic Author
Posts: 175
Joined: November 9th, 2015, 8:59 am

Re: Proof of the risk-neutral assumption

December 21st, 2020, 12:04 pm

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[ST] = Sert

Does that help?