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complyorexplain
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Proof of the risk-neutral assumption

December 19th, 2020, 10:40 am

Here is the risk-neutral assumption as I understand it:

(1)             Ê[ST]  =  SerT

It states that the expected value 'in a risk neutral world' of the price of a non-income paying asset at time T is equal to the current (i.e. time = 0) asset price compounded at the risk free rate r. The assumption is clearly equivalent to the claim that there is no risk premium: that on average the price of the asset will grow at the risk free rate. 

The claim is true, but only in a 'risk neutral world', i.e. a world where there is no risk premium. Otherwise it is false. Why do people think it is generally true? I have looked at many presentations and discussions of the assumption, and they all start with the following assumption

(2)      F  =  SerT

This says that the price F we would rationally agree now, to pay for the asset at T, is equal to the current asset price compounded at risk free. That is clearly true, given the 'rationally'. If I agree a price now, I take on the price risk from the seller. But the seller receives no money until T. So the seller will charge interest at a rate r. Call this the no-arbitrage/rational pricing assumption.

However (2) is not equivalent to (1). Why then do people think that we can derive (1) from (2)? Or perhaps we can? How?
 
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bearish
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 1:32 pm

It is not an assumption, at least in the standard theory. You start by deriving the valuation results from replication arguments (or, in weaker versions of the theory, some kind of equilibrium model). Then, as a computational device, you observe that you can obtain the same results by taking expectations of payoffs discounted at the pathwise return of some (non dividend paying) numeraire asset under a probability measure where the ratio of any (non dividend paying) asset price to the price of the numeraire asset is a martingale. In a constant interest rate model, the easiest example of this is to pick the risk free asset as numeraire, and we refer to the corresponding martingale measure as risk neutral.

In your example, the forward price is derived by the argument that borrowing money and buying the asset today, costlessly storing it until the maturity date and paying back the loan at maturity will have the same cashflows in all states of the world if and only if the forward price is the one you state.
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 1:35 pm

In your example, the forward price is derived by the argument that borrowing money and buying the asset today, costlessly storing it until the maturity date and paying back the loan at maturity will have the same cashflows in all states of the world if and only if the forward price is the one you state.

I totally understand that, which would be assumption (2), the no-arbitrage assumption. But how is that mathematically (or logically) connected with assumption (1)?
 
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bearish
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 2:16 pm

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
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 2:25 pm

To be clear, both (1) and (2) are necessarily true. (1) states that, in a world where there is no risk premium, the price of a risky asset will grow on average at the risk free rate. Its truth follows from the definition of 'risk premium'. If exposure to an asset has a (positive) risk premium, then by definition the expected return is greater than risk free. Likewise (2) follows from the definition of 'arbitrage free'. 

On the one step binomial, I understand that, and I first came across it many years ago. 33 I think. But how is it logically connected with proposition (1)? 
 
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bearish
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 3:32 pm

If we normalize the one step binomial to have initial prices of both the stock and the bond to equal 1, then the model is fully parameterized by U>R>D, and the requirement that the probability of the up-state satisfies 0<p<1. You can now use replication and no arbitrage arguments to solve for the present value of any state contingent cash flow, including the two pure state contingent claims. You can also ask yourself, if I want to calculate such values by discounting expected cash flows at the risk free rate, what (risk neutral) probability q do I need to assign to the up-state? There is a unique answer to this question. And, using this answer, which was derived without any explicit reference to forward pricing, you can observe that (1) holds.
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 4:20 pm

Right, but I already said that (1) necessarily holds, proof of which does not require your long argument above. (1) states that in a world where there is no risk premium, the expected price of the asset at T equals the price now, compounded by risk free rate. It is a definition of the concept of risk premium. 
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 4:31 pm

Repeating the question in the OP. "Why then do people think that we can derive (1) from (2)? Or perhaps we can? How?"

The question is why people think something. 
 
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bearish
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 5:20 pm

Who are these people anyway? I’m not going to try anymore, but neither (1) nor (2) is an assumption in the standard theory. They are both results derived from more fundamental assumptions. So, unlike for assumptions, there is in fact a “proof”. I can’t be the only one to cringe a little when reading the topic title...
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 5:39 pm

Since we are clearly at cross-purposes, you are probably right not to try any more. I clearly didn't explain the point very well.
 
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Paul
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 6:09 pm

When I saw the title I cringed too. But I also thought “Here we go again” and “This should be fun”!
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 6:41 pm

When I saw the title I cringed too. But I also thought “Here we go again” and “This should be fun”!
It was not obvious that the title was ironic? If (1) is a definition then it cannot be proved.

>>But I also thought “Here we go again” 

Referrring to the sig root t thread, yes?
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 6:48 pm

Hull, 8th edition p.257:
We are now in a position to introduce a very important principle in the pricing of derivatives known as risk-neutral valuation. This states that, when valuing a derivative, we can make the assumption that investors are risk-neutral. This assumption means investors do not increase the expected return they require from an investment to compensate for increased risk.
In mathematical terms, how do we express this assumption?
 
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complyorexplain
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 7:36 pm

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 rate of interest.
My emphasis. So we first assume that there is no risk premium, then calculate the probability corresponding to that, in order to prove that there is no risk premium? Isn’t that circular?
 
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Paul
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Re: Proof of the risk-neutral assumption

December 19th, 2020, 8:00 pm

When I saw the title I cringed too. But I also thought “Here we go again” and “This should be fun”!
It was not obvious that the title was ironic? If (1) is a definition then it cannot be proved.

>>But I also thought “Here we go again” 

Referrring to the sig root t thread, yes?
No. Over 30 years experience of speaking to people about this topic. It confuses people more than anything else.