tw


Total Posts: 4 
Joined: May 2017 


Hi
This is a question for the fans or gurus of Martingale pricing. I am coming at this in terms of a derivative pricing problem with a more complex price process in the drift in the real world measure, with two factors. The problem is not in interest rates. However, I am confused by a simple aspect of the basic problem.
As I read it (in, say Bjork's book), the task of pricing a contingent claim is to find a change of measure to convert the price process to an identical one where the drift is the risk free rate. Then calculate the expected payoff of the claim in this measure, discount the expectation and that is your answer. The key aspect is that the dynamics of the asset divided by the price of the bond is a Martingale.
So where the real world price process drift is a nontrivial function of the asset price, the risk neutral measure (i.e. Q measure as people seem to call it) will not be a simple adjustment to the constant drift.
Looking to find an instance where people look at this sort of problem, the most obvious examples seemed to be short rate models e.g. Vasicek, CIR etc.
It seems here the *Q measure* models are asserted to have forms where the drift has various forms of explicit dependence on the rate.
My question is, don't these forms of drift with r dependence destroy the Martingale property and hence the validity of the pricing methodology?
To put it explicitly If you were trying to price claims on a real world process that had this form
dS/S = (a + f(S)).dt + sigma.dW
where f(S) is some elementary function. Would you look to transform to a risk neutral process that looked like
dS/S = (r + f(S)).dt + sigma.dW (choice A)
or
dS/S = r.dt + sigma.dW (choice B)
The rates people seem to do choice A and I don't really understand why. Many thanks for any pointers!




pj


Total Posts: 3331 
Joined: Jun 2004 


From the top of my head, the short rate isn't tradable so it is not needed to be a martingale. HTH 
The older I grow, the more I distrust the familiar doctrine that age brings wisdom
Henry L. Mencken 

Cheng


Total Posts: 2836 
Joined: Feb 2005 


So where the real world price process drift is a nontrivial function of the asset price, the risk neutral measure (i.e. Q measure as people seem to call it) will not be a simple adjustment to the constant drift.
Not sure whether this is really an issue since Girsanov's theorem guarantees you that you only have to adjust the drift (although maybe in a nontrivial way). It also says that you cannot "choose"... the adjustment is given once you got the new measure and vice versa. See also Girsanov theorem. 
"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man) 


tw


Total Posts: 4 
Joined: May 2017 


Thanks for the response (also to pj).
Just to follow up (and apologies if this is laboured reasoning), I get the fact that the risk neutral measure is unique hence it's not a choice. I am just wondering what you would do practically in the example I was looking at. i.e. the real world looks like this: dS/S = (a + f(S)).dt + sigma.dW
To my mind, you want a change of measure where the process looks like this: dS/S = r.dt + sigma.dW'
Then you can price away in this measure.
Girsanov's theorem is the mechanism to do this. However, I can't quite get my head around the best plan of attack then there is more complex dependency in the drift. i.e. you seem to need to evaluate
exp( \int [rmu(S)]/sigma dW )
which is problematic. If you find a transformed price snew=snew(S) that removes the price dependence of the drift, then how does the measure change mechanics look from a transformed variable?
Also, to pj's point. If these short rate models (CIR/Vasicek etc) do not need to be martingales as the short rate is not traded, what do valuations of futures/options using them represent? Doesn't the whole equivalent martingale idea need them to be martingales to work?
Thanks again. 


ronin


Total Posts: 216 
Joined: May 2006 


@tw,
Shouldn't this be in Basics?
For a cash asset like shares, you can generate a risk free forward by buying the share today and funding it to when ever you need to deliver it. Hence the risk free forward rate which is the difference between your funding rate and the implied dividend rate.
An interest rate is not an asset  you can't buy it today and fund it to maturity. You replicate the appropriate forward rate by lending money to the appropriate longer term and borrowing it to the appropriate shorter term. Hence the forward rate which is implied from the ratio of discount factors (or bond prices).
For your process, it depends on what you are modelling and how you can fix the cost of delivering it at some future point.
You shouldn't be thinking in terms of mathematics  it never helps.

"People say nothing's impossible, but I do nothing every day" Winnie The Pooh 


Cheng


Total Posts: 2836 
Joined: Feb 2005 


There is only one measure that removes the drift, the so called riskneutral measure. Girsanov says this measure is unique outside null sets. If you transform the price (say you modeled a call and now look at a call on a call) you run the same procedure again (of course the payoff is different) and arrive at a riskneutral measure.
No rocket science, really. 
"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man) 

tw


Total Posts: 4 
Joined: May 2017 


Thanks Cheng. Didn't know there was a "basics" part of the site. Will post a followup question there. 


