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?

Thanks in advance,

SR