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stampeding
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Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 2nd, 2019, 1:47 pm

Hi

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:
  1. Simulate two independent Rectangular distributed QRN's (or PRN's) i.e. probabilities
  2. Transform them to two uncorrelated Normal distributed RN's
  3. Transform them to two correlated Normal distributed RN's, using appropriate terminal correlation
  4. Transform them back to probabilities
  5. Transform these two probabilities to logmoneynesses
The problem here is which terminal correlation to use. For this case, it can't be calculated the way Rebonato suggests, since the volatility is also dependent on the underlying price (in addition to time).

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
 
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Alan
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 2nd, 2019, 3:31 pm

Just a few random thoughts, assuming two assets, although I may not understand it. 

1. Why not bite the bullet, don't take a long first step, but actually calibrate two local volatility functions [$]\sigma_i(t,S_i(t)), \,\,(i=1,2)[$] for [$]0 < t \le T[$]? Then, with the assumption that the two Brownian motion drivers have constant instantaneous correlation, you can simulate the calibrated risk-neutral bivariate process and estimate the terminal correlation. Once you have that, go back to doing whatever it is you want to do here. 

2. Alternatively, use the joint historical time series of the two assets to estimate your terminal correlation. Would be a sanity check on whatever you get with 1. You could also fit some joint 'wide-tailed' distribution to (daily) history and then 'risk-adjust' to get the risk-neutral distribution.   

3. Alternatively, in principle, if you have enough market information on the putative two-asset product, you can use a Breeden-Litzenberger type argument to infer the joint risk-neutral terminal distribution.
 
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stampeding
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 3rd, 2019, 12:52 pm

Hi, thanks for answer!

This is not for any particular product, but for a 3rd party system which must be consistent in the valuation. (I.e. analytical formulas, PDE & Monte Carlo must agree, when applied to same instrument.)

The two Local Volatility functions are known, i.e. the individual Distributions (for any time-step size, as long as we start in time 0) are known. Also, the Instantaneous Correlation is known.

Now, if the Volatilities would be time-dependent only, the relation between Instantaneous and Terminal Correlation would be easy to calculate. (As described by Rebonato.) So for a fairly simple instrument for which it exists an analytical formula, this analytical formula would basically yield exactly the same result as a PDE solver or a Monte-Carlo simulation using Instantaneous Volatilities & Correlation.

However, when we have "full Local" Volatility, i.e. the Volatility is also depending in Underlying Price, then the relation between Instantaneous and Terminal Correlation cannot be calculated with Rebonato's simple method. The individual Distributions are known, the Instantaneous Correlation is known -- but if we want to take a "long" first time step, it's thus unclear how to calculate the terminal correlation.

One idea I thought of was to use Rebonato's method, with the time-dependent Volatilities replaced by the time-dependent Variance Swap Volatilities, as some "average over all Strike/Underlying prices".

I also assume one could just use Fokker-Planck to calculate the full joint distribution, which would solve the problem, but I'm not sure how fast & how stable that method would be.

Regards,
/Samuel, Stockholm, Sweden
 
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Alan
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 3rd, 2019, 2:23 pm

I looked at Rebonato. 

Now that I understand it a little better, I have a conceptual problem with your question. 

In the case of deterministic volatility (purely time-dependent, no asset dependence), the terminal joint distribution of the two log-prices [$](x_1(T),x_2(T))[$]  is bivariate normal. So, indeed, it suffices to know the covariance matrix of the terminal variables (ignoring the drift for the moment) to do a "one long-step simulation".

But, in the case you are interested in, the terminal distribution is *not* bivariate normal. So, even if you had a black-box that output the terminal correlation that you seek, what would you do with it? You can't use it to do a "one long-step simulation" because of the non-normality. ??
 
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stampeding
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 3rd, 2019, 3:22 pm

Something like this. Definitions:

    Logmoneyness = x
    "Probability" p = CDF(x) = N(z)
    Corresponding Normally Distributed Variable z = N^-1(CDF(x))   i.e. Inverse Normal Distribution
    x = CDF^-1(p) = CDF^-1(N(z))

Simulation in One Dimension, Long Step:
    Simulate p, rectangular distribution [0, 1]
    x = CDF^-1(p)

Idea for Simulation in Two Dimensions:
    Simulate pA, rectangular distribution [0, 1] 
    Simulate pB, rectangular distribution [0, 1]
    z1, z1 = correlated Bivariate Normal Distribution using "copula technique", and PA, pB, rho, where rho is an approximation
    p1 = N(z1)
    p2 = N(z2)
    x1 = CDF^-1(p1) 
    x2 = CDF^-1(p2)

In best case, there actually exists this "terminal correlation approximation" rho, i.e. when Normal Distributed z1 and z2 are simulated as described above using this rho, and then transformed to Logmoneynesses x1 and x2 using the known distributions for x1 and x2, the joint distribution of x1 and x2 could perhaps in best case be "close enough" to what we would have gotten if we had gone "full PDE", e.g. solving Fokker-Planck with many time-steps and applying the (known) instantaneous correlation in each time-step?

The alternative, I assume, is to solve Fokker-Planck to get the joint distribution of x1 and x2 and then simulate a point [x1, x2] in each simulation, derived from e.g. probabilities pA and pB which have Rectangular Distribution [0, 1].


This discussion has actually convinced me that this is what I need to do: Start with a couple both very different and very "fluctuating" Local Volatility Surfaces and (using Fokker-Planck) calculate the Joint Distribution (and do this for different Instantaneous Correlation). Transform the two Logmoneynesses to two Normally Distributed Variables, and finally check if the joint distribution of these two Normally Distributed Variables is (normally) "close enough" to a Bivariate Normal Distribution with some ("Terminal") Correlation rho.

If this is the case, then of course the Terminal Correlation I'm looking for is the output rho of these calculations, and if I'm even luckier I'll figure out some simpler approximation to calculate this rho...

Unfortunately, I don't think this will be a success, since, as you point out, the distribution is not a Bivariate Normal, and it needs to be very close to a Bivariate Normal to be of any use. (Note, though, that it's the "individually" Normally Distributed z1 and z2 which need to be "almost" jointly Bivariate Normal, not the Logmoneynesses x1 and x2, so there might still be a small possibility that the approximation will be close enough...)

I'll post my results, when I've tested this!

Regards,
/Samuel, Stockholm, Sweden
 
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Alan
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 4th, 2019, 1:57 pm

Yes, that will be interesting. Seems there are many related, but distinct, questions here:

Q1. How well does the [$](S^1_T,S^2_T)[$]-distribution generated from a (smile-consistent) bivariate Gaussian copula match the distribution from a local-vol model:
(*) [$] dS^i_t \sim \sigma_i(t,S^i_t) \, S^i_t \, dW^i_t,\quad (i=1,2), \quad dW^1_t dW^2_t = \rho \, dt[$],
where the [$]\sigma_i[$] are calibrated to match market option smiles? So, we have various distinct smile-consistent models.

Q2. How well do any of the models in Q1 match various two-asset option products that might trade?

According to this Q&A at stackexchange, there is discussion in Bergomi's book. Just checked and indeed there is in Sec 2.10. I believe it resolves a lot of the issues about this "one-step" simulation business. Apparently key is this article by Carr & Madan
 
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stampeding
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 7th, 2019, 6:28 pm

Yes, Bergomi section 2.10 discusses exactly this problem, it seems, but it doesn't look promising since he explicitly says "the model is unusable"...

Haven't read Carr & Madan yet, but hopefully tonight!

Thanks for the links!
/Samuel, Stockholm, Sweden
 
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Alan
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Re: Terminal correlations in Monte-Carlo Long-First-Step Local Volatility context?

May 7th, 2019, 11:00 pm

Yes, Bergomi section 2.10 discusses exactly this problem, it seems, but it doesn't look promising since he explicitly says "the model is unusable"...

Haven't read Carr & Madan yet, but hopefully tonight!

Thanks for the links!
/Samuel, Stockholm, Sweden

You're welcome.
I don't see "the model is unusable". But I do see:

MFMs can be used for European options on a basket of equities [$]S^i[$]. Let us call [$]T[$] the option's maturity: one draws the (correlated) Gaussian random variables [$]W^i_T[$], applies the mapping in (2.128) and evaluates the payoff. This is exactly equivalent to using the marginal densities supplied by the market smile for maturity [$]T[$] for each asset, and the using a Gaussian copula function to generate the multivariate density for the [$]S^i_T[$].

It seems this is exactly the procedure you are seeking. To confirm Bergomi's statement, you need to work out the multivariate generalization of the 4 equations on pg 86, starting from [$](*) \vec{S}_t = f(t,\vec{W}_t)[$]. (Have not done so myself). A by-product is exactly the local vols to feed to a PDE solver to get exactly the same option values as (i) one-step Monte Carlo using (*), and (ii) Gaussian copula with smile-consistent marginals. At that point, you have three equivalent procedures that each produce the same option values: you write up your report, submit your bill, and declare victory.  :D