risk neutral probability measure Q

[maz1987]

From what I understand, the risk neutral probability measure Q is the probability measure under which investors are assumed to be neutral to any risk.

Under Q we can then determine the fair price for an option whose price depends on the value of an underlying stock. Using the one-period Binomial model, we say that the investor does not require an additional risk premium for the risk involved in the movement of the share price.

However, I don't understand why we can make that assumption. What stops me from making a ludicrous assumption that the investor is, for example, hates being wealthy and will pay a high sum for an option that all but guarantees he will make zero, and pricing an option under that probability measure? We know that investors do require an additional risk premium, so why can we use a probability measure that assumes they do not?

My only possible explanation is that there is zero risk in the replicating portfolio matching the value of the derivative. But then there is still risk in what the value of the portfolio actually is at T=1.

Thanks

 

[Oxymoron]

However, I don't understand why we can make that assumption. What stops me from making a ludicrous assumption that the investor is, for example, hates being wealthy and will pay a high sum for an option that all but guarantees he will make zero, and pricing an option under that probability measure? We know that investors do require an additional risk premium, so why can we use a probability measure that assumes they do not?

My only possible explanation is that there is zero risk in the replicating portfolio matching the value of the derivative. But then there is still risk in what the value of the portfolio actually is at T=1.

Simplicity!

Under the risk neutral measure, the market price of risk {[Mu-r]/sig} is 0. So Mu = r. So if St ~ LogNor( ln(S0) + (r - sig^2/2)*t, sig*dt). So, the discounted expected value of the stock process under risk free measure is a martingale - and the portfolio can be replicated - bam!

If you're assuming a risk lover, he will have a negative MPR. Mu = r - sig*MPR. Discounted value of this process under r is NOT a martingale - so you'll need to find something else to discount it with. This will vary for each instrument and person, and will be impossible to strike a balance ... and all this makes things more complicated.

Additionally, risk neutral measure theoretical sense too - given you'll have risk lovers and risk averse people sort of evenly distributed, with a mean risk taking ability of 0 at a portfolio level...

 

[Whippet1]

If we find can find a portfolio of shares and cash that replicates the derivative payoff under all future circumstances, then we can argue that in an arbitrage-free market, this current cost of this portfolio must equal the current value of the derivative.

However, we can then go on to rearrange the pricing formula we obtain in terms of a discounted cash flow looking formula involving 0<q<1 and 0<1-q<1. As "q" and "1-q" are constants between zero and one, we can think of them as probabilities. Whereas the real-world "p" probabilities are the weights on the tree branches that make the expected return on the underlying share equal to what it actually is in the real-world (ie risk-free rate + risk premium), the "q" "probabilities" are the weights on the tree branches that make the expected return on the underlying share equal to the risk-free rate.

We can then argue that in a pretend world in which all investors were risk-neutral and hence cared only about expected returns (and not risk) when choosing investments, the expected return on all investments must be the same, ie the risk-free rate - as there would be no premium for accepting additional risk.

It is because of this feature (only), that we can choose to call them risk-neutral probabilities.

 

[maz1987]

Thanks for the replies.

I'm still not entirely comfortable with the assumption, although thanks to your replies I am much more accepting of it and am able to apply it to exam questions.

For those interested, I also found this paper useful:

http://www.bus.lsu.edu/academics/finance/faculty/dchance/instructional/tn96-02.pdf

 

[Whippet1]

I think there are a number of different probabilities we could use in practice and not just the real-world p's and risk-neutral q's, provided we adjust the expected rate of geometric return or drift accordingly.

It is a bit similar to valuing an increasing annuity in which the payments grow at "g" per annum compound. We could either discount the increasing series of payments at a discount rate "i". Or equivalently, we could discount a level series of payments at "i-g". Either way we obtain the same present value.

Similarly, valuing an asset whose using real-world p probabilities and a risk discount rate equal to risk-free rate + risk premium, gives the same answer as obtained using risk-neutral q probabilities and and a discount rate equal to just the risk-free rate.