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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.
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?
You can solve it by change of variable to get heat equation which give you the B&S solution. Indeed In what I said we got PDE first then "Expectations" comes from Feynmann-Kac Theorem. And then come your equation(1), which use expectation without being clear on what it means. Using Feynmann-Kac make it crystal clear in my opinion.

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

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

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

Paul
Posts: 11261
Joined: July 20th, 2001, 3:28 pm

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

Girsanov isn't particularly useful, in my opinion.

Change of measure = change of variables. The latter you do in high school. The former is just a complicated presentation of the latter in a very narrow field to make it palatable for university professor probabilists to teach.

Assumptions hint at a model.

Randomness in model makes people worried about how to value risky things. E.g. expectations might not play a simple role.

Assumptions +model +hedging + no arbitrage => BS PDE, Binomial model, ...

Trivial observation about BS PDE or binomial model shows that there is a role for simple expectations.

That role is to pretend that assets all have the same rate of return. Even though they don't. This is risk neutrality.

Probabilists get all aroused. Trivial things gets messy.

As to derivation of BS formulae, there are many ways that are equally simple.

The role of expectations also has the result that values can be found by simulations, Monte Carlo. Very useful.

But risk neutrality distracts from reality. It's a very tempting dead end, but with lots of drink, drugs, free love.

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

But risk neutrality distracts from reality. It's a very tempting dead end, but with lots of drink, drugs, free love.

Sounds very 70s.
"Can't buy me love,no, no, no,no,no"

// You can stare at an SDE all day long $^*$ (some people do) but when you stare at a PDE then at some stage the solution jumps off the page.

$^*$ converging a.s./p.p.to Euler method. Every time. Without fail.
Last edited by Cuchulainn on December 21st, 2020, 8:23 pm, edited 2 times in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

katastrofa
Posts: 10069
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

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

OK is Wolfram better? https://mathworld.wolfram.com/Validity.html
No online resources can replace a book. For you, Shreve.

Paul: "Change of measure = change of variables."

In infinite number of dimensions.

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

dbl
Last edited by Cuchulainn on December 21st, 2020, 8:30 pm, edited 1 time in total.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

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

In infinite number of dimensions.
Bad news, kats. The only Borel measure on a Banach space is the trivial measure.

https://en.wikipedia.org/wiki/Infinite- ... ue_measure

That's an Ecumenical matter.

Not that anyone would want to do that.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

Paul
Posts: 11261
Joined: July 20th, 2001, 3:28 pm

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

OK is Wolfram better? https://mathworld.wolfram.com/Validity.html
No online resources can replace a book. For you, Shreve.

Paul: "Change of measure = change of variables."

In infinite number of dimensions.
I actually meant change of numeraire. But am very happy that you didn’t find fault with anything else!!!

katastrofa
Posts: 10069
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

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

I didn't mean to indicate any faults. My intention was to complete your abstract statement. I appreciate that you talk about mathematics at such a high level, especially in the context of an applied quantitative field.

Paul
Posts: 11261
Joined: July 20th, 2001, 3:28 pm

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

My intention is always to speak at the optimal level!

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

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

Of course. That's why you jump on the desk.

Paul
Posts: 11261
Joined: July 20th, 2001, 3:28 pm

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

You’ve seen me lecture on finite differences then? I climb up on furniture to explain the forward, backward and central differences.

I also find that wives are a great source of useful analogies.

Cuchulainn
Posts: 64397
Joined: July 16th, 2004, 7:38 am
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### Re: Proof of the risk-neutral assumption

You’ve seen me lecture on finite differences then? I climb up on furniture to explain the forward, backward and central differences.

I also find that wives are a great source of useful analogies.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

katastrofa
Posts: 10069
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

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

I saw a cute Maxwell demon increasing the Helmholtz energy to fuel his teaching efforts. And Cuchulainn is a buzzing Brownian computer. I'll always see the world with physicist's eyes!

Isn't it funny that the same mathematical theory and tools (theory of stochastic processes, Girsanov, Feynman-Kac, Fokker-Planck, Lie Trotter ...) work for a simple ferromagnet, DNA replication, derivative proving and more. Too beautiful to be true.

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

Lie Trotter .

Just so happens that I am implementing  LT as we speak. It's first-order. or 2nd order, go for Strang-Marchuk operator Splitting. Nice thing is splitting {diffusion, convection}, {potential, kinetic energy} and not {x,y}.

Wow! I didnie know you physicists knew BCH et al.

https://www.asc.tuwien.ac.at/~ewa/semin ... ag_Exl.pdf
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl