2017 September Q5

[Darragh Kelly]

Hi,

Q5(iii)

So by solving the general solution for the discounted share price, e^-rt*St = St,squiggle (given we've already solved SDE dSt = mew*St+sigma*St*dWt, to get an expression for St), using Ito's lemma, and writing in terms of browian motion under the risk-neutral measure Q (using Cameron-Martin-Girsanov-Therom), we get the following expression:

dSt,squiggle = sigma*St,squiggle*dWt,squiggle

So the solution from the Acted exam paper concludes that because this process has 0 drift term, then it is a martingal (which I fully understand, the zero drift is a martingale property).

But the question has asked us to show (using parts (i) and (ii)) that Wt + lambda*t is Browian motion under the risk-neutral proability Q. I'm just struggling how this has been proven through showing that under risk-neutral probability measure Q that, the process is a martingale. Please could you shed some light on what I'm missing?

Q5(v)

Is this in the notes or should we have understood that one-factor stochastic processes based on the same factor have by definition the same price of risk?

Many thanks,
Darragh

 

[Steve Hales]

(iii)
You're correct in saying that just because the drift is zero that it doesn't necessarily make the process a standard Brownian motion. The result actually comes from part (i); it's the CMG theorem which says that a gamma exists which makes Wt + gamma*t a standard Brownian motion under some probability measure. The only thing left is to show is that when gamma is equal to the lambda suggested in the question that the CMG lands on the probability measure known as the risk-neutral probability measure.

(v)
No, this statement wasn't in the Core Reading in 2017. It's not present in 2022 either.

 

[Darragh Kelly]

(iii) Ok so what you are saying is because we've shown the SDE for the discounted asset process (e^-rt*St) is a martingale, this means that you are dealing with the risk-neutral probability measure by definition (this was proven in the Acted notes,page 12 chapter 16). ie as you said above you've landed on risk-neutral probability measure.

And because there the browian motion in the SDE of the discounted asset process dSt,squiggle = sigma*St,squiggle*dWt,squiggle, which is equal to dWt +lambda*dt (ie dWt,squiggle = dWt +lambda*dt) then we can say with certainty that Wt + lambda*t is browian motion under the risk-neutral probability measure, thus proven.

Please confirm I have interpreted above correctly.

Many thanks,

 

[Steve Hales]

Yes, you've got it.
Thanks

 

[Darragh Kelly]

Thanks for your help on this Steve.