## Hybrid Calibrartion

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### Hybrid Calibrartion

Dr. Joshi and Fellow users,

I have the following situation in an Equity/IR hybrid calibration.

Model: Suppose, I am interested in an Equity/IR hybrid model and I have "Heston" type evolution for my Equity model and simple 1-F HW as my short rate model. I think there is a correlation btn the short rate (r) and the asset (S) and I calc this from the historical data. I assume zero correl btn r and v.
Calibration : I calibrate the IR model to usual instruments to get the HW parameters. The problem/question I have is regarding the equity calibration. Let's say, I calibrate the equity parameters to Equity option data using the Heston formula and get a set of parameters. Also let's assume by some divine intervention, I have perfect calibration .
Problem : If I use these calibrated parameters in my simulation along with the above mentioned correlation btn r and S, and price European options, I would surely not be able to recover the market-quoted option prices. The key point is that I have a stochastic IR whereas in Heston model, IR is deterministic. Essentially, I am using wrong formula.

The questions I have are:
1. What should I do in my particular instance? Do I use Monte Carlo to price Equropean options during my calibration procedure? In this set-up, at each iteration of my calibration optimization, I would run MC with correlation btn S and r and price European options. Keep on doing this till convergence.
2. What's the industry practice for Equity/IR hybrid models regarding calibration?

SR
sraks

Posts: 30
Joined: Tue Feb 16, 2010 3:31 am

### Re: Hybrid Calibrartion

I did this a long time ago so it's all a bit hazy.

I think there are some formulas in Brigo-Mercurio for hybrid short rate models but possibly not with SV.

You could
1) calibrate IR
2) work out the effective implied vols in a hybrid BS model for those IR params
3) then calibrate your Heston model to match those.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Hybrid Calibrartion

Thanks Dr. J. for the Brigo-Mercurio reference. Yes, there is an ugly BUT closed-form formula for the option price under HW 1-F in Brigo-Mercurio.

If I understand step 2 in your prescriton,

implied vol in hybrid BS-HW model = fn(HW params, market quoted BS vol)

RHS is all known. So, we are all good and use these new "effective" implied vols for Heston calibration. However, Heston calibration also involves "r". Do we now assume it is deterministic?

I guess this works if we can recover the option prices.

As an aside, what is your opinion on the calibration using MC? Obviously, it's much slower.

Regards,
SR
sraks

Posts: 30
Joined: Tue Feb 16, 2010 3:31 am

### Re: Hybrid Calibrartion

yes in stage 3, we use deterministic IR.

My view on MC calibration would be to get as close as you can without it, and then possibly use it to tighten up.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Hybrid Calibrartion

Thanks Dr. J. It was a very helpful and productive discussion.
sraks

Posts: 30
Joined: Tue Feb 16, 2010 3:31 am

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