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BerndSchmitz
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Joined: August 16th, 2011, 9:48 am

### Girsanov's theorem and Heston's SV-model

Hey,just a quick question about Girsanov in Heston's SV-model:Now I assume that dZ1_t and dZ2_t are 2 independen BMs and dW1_t = dZ1_t and (or vica versa).Now I apply Girsanov (i.e. and for some "sufficiently nice" functions lambda1 and lambda2) to get to an equivalent martingale measure (there is at least on if I assume an arbitrage-free market).Now my question:Is Girsanov the only way to get to an EMM? Or put differently: Is there an EMM which cannot be "reached" through Girsanov? I think the answer is no but wanted to varify this.Thanks,Bernd

BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

### Girsanov's theorem and Heston's SV-model

BTW: Has the answer anything to do with the thoerem of Radon-Nikodym?

Alan
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### Girsanov's theorem and Heston's SV-model

My first reaction to what you wrote was that the stochastic process that generates the EMM only need preserve the covariance structure of the diffusion part of the process.That would seem to allow the addition of an arbitrary jump structure to both the stock price and the volatility,as long as you (i) compensate the stock price jumps to keep e^(-r t) S(t) a martingale, (ii) keep the state space positive,and likely (iii) avoid explosions. However, a counter-argument to what I just said is that, intuitively, the EMM must assign a zero value to betsthat are also impossible under the original measure. By an impossible bet, I mean a bet with a payoff that occursonly under an impossible event. If you could place a bet on the process (stock or volatility) having a discontinuityin some fixed interval of time, then that would seem to rule out the jumps. If this second argument is right, thenthat leaves your Girsanov transformation as the only possible one.Maybe someone on the forum can resolve it?
Last edited by Alan on September 28th, 2011, 10:00 pm, edited 1 time in total.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### Girsanov's theorem and Heston's SV-model

Girsanov theorem just says that a brownian motion under P is still a brownian motion under M when the Radon Nikodym derivative of the measure change is given by a exponential local martingale. Alan is totally right when he says that the most important is to guarantee a risk neutral measure for having a martingale for the discounted stock price, with a positive support and which avoid explosion (see feller test).It's important to check the Novikov condition when you use Girsanov theorem. http://en.wikipedia.org/wiki/Novikov%27 ... onRemember also that your discounted process under the risk neutral measure have to be a martingale and not a strict local martingale.Example a driftless GBM dX=sigma*X*dW is a pure martingale while a process like dX=sigma*X²*dW is a strict local martingale. Moreover if I remember well your measure change has to be a continuous martingale (so no jump in the Radon Nikodym derivative).
Last edited by frenchX on September 28th, 2011, 10:00 pm, edited 1 time in total.

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### Girsanov's theorem and Heston's SV-model

First of all what this has to do with RN theorem is if they re equivalent then there s a density.-"Is Girsanov the only way to get to an EMM?" The probability triplet is sometimes useful in structuring these ideas. If you take the natural algebra of W then yes every equivalent measure can be got through an exponential. If you take the canonical space, every equivalent measure comes from a Cameron-Martin transform; then the likelihood ratio can be defined without reference to W by using adequate projections, though technical care is required if the lambdas are unbounded-cf Dellacherie & Meyer 6.43. You could very well have a richer probability space, and define equivalent measures on that, that have nothing to do with W The only necessary and sufficient condition is that:!conditional on W the density has expectation 1! . -Alan: I'm a big fan offer me a faculty job please. Your first reaction is tied to the space of continuous paths: if the covs are different the laws are singular, eg using a.s. short time asymptotics. (By the way, for financial pricing and hedging it's the law of S that counts, and as such:!you only need to respect the projection of Vt onto S's past!) If you carry out your program i,ii,iii there is simply no chance of the two measures being equivalent. More precisely, we have defined (S,V) using Doob regularisation, and this means a.s. there is no discontinuity of type 1. You can write some jump dynamics realised by an SDE seen in the Skorokhod space, but still "has jump size * at time *" is a cylinder set so its law won't be equivalent to that of our (S,V).-frenchX: I am looking for tenure track,how is the job market in physics/chemistry for those with qfin interests? A BM under P won't always/exactly be a BM under M.. In certain cases you can "prove Girsanov" ie mass 1, but "Novikov" is not verified, like many cases around Heston, so in this case it's not such a great suggestion.. Well the RN derivative is martingale, and therefore due to right-continuity of the filtration, we can chose a continuous version (this is a very fascinating result of Doob).I'm hoping someone can chime in to add stuff about what the jump times of semimartingales can look like, and for example what is the relationship between the laws of VGs with different parameters.
Last edited by croot on September 29th, 2011, 10:00 pm, edited 1 time in total.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### Girsanov's theorem and Heston's SV-model

QuoteOriginally posted by: croot-frenchX: I am looking for tenure track,how is the job market in physics/chemistry for those with qfin interests? I can only say for my country. There is no tenure in France. Just do a PhD, 1 or 2 postdocs and then apply for a researcher position. If you are taken, congrats if not redo a more interesting postdoc and again. Actually I would say that nowadays it's easier to be a physicist than a quant. You can do a math PhD if you want, I have some in my school who got kicked out their banks after the 2008 crisis.
Last edited by frenchX on September 28th, 2011, 10:00 pm, edited 1 time in total.

Alan
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### Girsanov's theorem and Heston's SV-model

QuoteOriginally posted by: croot-Alan: I'm a big fan offer me a faculty job please. Your first reaction is tied to the space of continuous paths: if the covs are different the laws are singular, eg using a.s. short time asymptotics. (By the way, for financial pricing and hedging it's the law of S that counts, and as such:But still "has jump size * at time *" is a cylinder set so its law won't be equivalent to that of our (S,V).I'm hoping someone can chime in to add stuff about what the jump times of semimartingales can look like, and for example what is the relationship between the laws of VGs with different parameters.OK, you are appointed assistant prof. in the Student Forum.Re "still "has jump size * at time *" is a cylinder set. Does it make any difference to your conclusion (jumps are excludedfrom the EMM) if the jump times are unpredictable (like a Poisson process) or not?

mj
Posts: 3449
Joined: December 20th, 2001, 12:32 pm

### Girsanov's theorem and Heston's SV-model

if the probability of a jump before the measure change is zero, it will be zero after the measure change. Brownian motions are incredibly rigid. A measure change can really only change the drift.

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### Girsanov's theorem and Heston's SV-model

"mj:Brownian motions are incredibly rigid. A measure change can really only change the drift."It's all in the "really" : consider the canonical space and on it two prob measures, one is Wiener measure P the law of a brownian W, the other is (also the same Wiener measure) Ptilde the law of Wtilde:=-W . Well P and Ptilde are equivalent (RN=1..), while W and Wtilde are not related by any "drift change". Love your work btw, contact me if you have any opportunities down under,tyvm. Alan: To look at the law of (S,V) it's enough to consider a modification, and we can chose this modication so that P(t->(St,Vt) continuous on [0 T])=1. If Ptilde~P then it satisfies the like condition. frenchx: I already got the PhD and already got kicked out from my bank! Good to see I'm right on track.
Last edited by croot on September 29th, 2011, 10:00 pm, edited 1 time in total.

list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### Girsanov's theorem and Heston's SV-model

is there a fact that measures corresponding 2 processes q W ( t ) and d W ( t ) are singular on [ 0 , 1 ]?

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### Girsanov's theorem and Heston's SV-model

list: no there isn't, for instance if q²=d² .

list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### Girsanov's theorem and Heston's SV-model

is there a fact that measures corresponding 2 processes h + q W ( t ) and d W ( t ) contains a singular components on [ 0 , 1 ]?it seems that sdes with equal diffusion is a class which is proved that measures are abs continuous. i do not remember the necessary conditions for abs continuity 2 measures of the sde while Girsanov's techniques looks like sufficient ones.

list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### Girsanov's theorem and Heston's SV-model

for sqrt process we could not apply directly Girsanov transformation. We should first to prove something as far as diffusion coefficient reach 0. 1st we can use Girsanov for stopped solutions at h neighborhood. Next to show the possibility h goes to 0.

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