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

### local vol with stochastic vol and stochastic rates

We have done a paper on how to calibrate local vol when both volatility and rates are stochastics.

I'd be interested in any comments.

http://ssrn.com/abstract=2840628

VivienB
Posts: 130
Joined: August 6th, 2012, 3:32 pm

### Re: local vol with stochastic vol and stochastic rates

Hi,

Your article seems interesting to get a good approximation for the call prices and to simulate in such a model, but how does your calibration methodology compare to the particle method calibration (speed and accuracy) described on Guyon, Henry-Labordère, The Smile Calibration Problem Solve (this article go deeper than the Henry-Labordère's paper that you cite on your article, in particular there is an extension to stochastic rates)?

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: local vol with stochastic vol and stochastic rates

I just was a little confused by the 2.1st assumption. The risk neutral asset dynamics is usually has the risk free drift coefficient. Whether $\mu ( t , x )$ is the risk free rate. For more formal reason it might make sense introduce vector of economic variables on original probability space as it is common in general theory of the stoch processes. Then explain the necessity to make the change original probability measure to arrive at the risk neutral world. The mathematical paper will be looks much better.

VivienB
Posts: 130
Joined: August 6th, 2012, 3:32 pm

### Re: local vol with stochastic vol and stochastic rates

I just was a little confused by the 2.1st assumption. The risk neutral asset dynamics is usually has the risk free drift coefficient. Whether $\mu ( t , x )$ is the risk free rate. For more formal reason it might make sense introduce vector of economic variables on original probability space as it is common in general theory of the stoch processes. Then explain the necessity to make the change original probability measure to arrive at the risk neutral world. The mathematical paper will be looks much better.
Lol.
X is the "vector of economic variables of interest", not necessarily the assets. For instance, a CIR variance can be one of the coordinates of X.
Furthermore "mu is the risk free rate" is only true on kid's model. In the real life, mu is at least r - q, and can even be more complex depending on your dividend model.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: local vol with stochastic vol and stochastic rates

Vivein, can you please specify why and how does the risk neutral measure is defined for  the vector of economic variables in case when $\mu$ does not risk neutral rate? And by the way I have not thought about this phenomena. If q is dividend rate on underlying which is  defined for the stock on the real world. In BS model we should replace it on risk free rate r which corresponds to risk free rate of return of the BS hedged portfolio. How does  q will appear in risk neutral world?

Leptokurtic
Posts: 7
Joined: October 1st, 2014, 1:46 pm

### Re: local vol with stochastic vol and stochastic rates

Hi Mark,
With P. Henry-Labordere we have explained how the particle method achieves this, see
- The Smile Calibration Problem Solved: ssrn.com/abstract=1885032
- Later published (Jan 2012) in Risk: http://www.risk.net/derivatives/2135540 ... alibration
- Chap 10, 11, 12 of our book Nonlinear Option pricing: https://www.crcpress.com/Nonlinear-Opti ... 1466570337
Hope this helps!
Best,
Julien Guyon