First, two facts:
1. In a Local Volatility context for single underlying, it is possible to take a long first time step, by using the (pre-calculated) inverse of the density function to map (simulated) probabilities to logmoneynesses. (I.e. the volatility smile at time T is known --> we know the density function at T --> we can calculate the probability corresponding to a given logmoneyness --> we can inverse this and calculate the logmoneyness corresponding to a certain probability.) This means that one can take a long first time step, and doesn't have to bother about the often very steep Volatility Smiles for the shortest expiries. The Monte-Carlo simulation becomes much more accurate than if we had taken many short time-steps.
2. For multi-underlying with time-dependent volatility, the terminal correlations can be calculated from the instantaneous correlations, as described by e.g. Rebonato in chapter 5.3.4 and 5.4 in "Volatility and Correlation", 2nd edition.
Question:
Is there any similar technique to calculate the appropriate terminal correlation when the volatility is depending not only on time, but also on the underlying price, i.e. a Local Volatility context? I.e. a combination of the two cases described above? I'd like to do something like this, at the first time-step:
- Simulate two independent Rectangular distributed QRN's (or PRN's) i.e. probabilities
- Transform them to two uncorrelated Normal distributed RN's
- Transform them to two correlated Normal distributed RN's, using appropriate terminal correlation
- Transform them back to probabilities
- Transform these two probabilities to logmoneynesses
Perhaps one can use some kind of underlying-price-independent average, e.g. (time-dependent) corresponding theoretical Variance Swap Volatilities, calculated from the smiles at each point in time? Or some other technique?
Any ideas? Or are there any articles on this subject?
Regards,
/Samuel, Stockholm, Sweden