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Quantization and its numerical applications

Bastian Gross

German Mathquant
Hello Quantneties,

I want to ask, is someone interested in functional quantization?
And not only because of an word joke with QUANTization und QUANTnet. :-\"

Functional quantization is an very useful alternative solution for pricing options, especially pathdepent options, in contrast to Monte Carlo methods.

I know, that (dutch) ABN Amro use a software packet to pricing its Option by functional quantization.
Do anyone of you know much more about this mathematical method in financial engineering?

The Book of Graf and Luschgy:
Foundations of Quantization for Probability Distributions

And here are two articels about Quantization and its numerical applications to pricing option:
  1. path-depent European options (Quadratic optimal functional quantization of stochastic processes and numerical applications)
  2. Swing Options (Optimal quantization for the pricing of swing options)

I'm prepared to answer further questions.

Bastian Gross

German Mathquant
Okay, I see there are some question about FQ (= functional quantization).

At Monte Carlo methods you use some randomnumbers generated by pseudo-random number generator (PRNG) (like Mersenne-Twister or linear congruential generator (LCG)) to calculate expectation value. But problems with deterministic pseudo-random number generators avoid good performance, so efficient MC-methods range between 10.000 and 100.000 simulations.
To speed-up performance one uses variance reduction.

By calculation expectation value with FQ you must estimate some optimal codebooks (this is the "computational geometry and "probability spaces stuff") and uses these to compute for instance derivate-prices.

I remember you calculation expectation value is the main subject of pricings derivates.


Faculty (Undercover)
The foundational paper is incredibly general--that is, abstract. I don't have the time or really the background to process it.

However, I've experimented in a primitive way with Gaussian discretizations and can identify a potential problem with these N-codebooks: They are constructed to minimize expected discretization error in the underlying, whereas for a given option, the error you wish to minimize is the error in the expected value of the payoff.

To take an example consistent with my own primitive understanding, consider using a one-size-fits-all discretization to price a far OTM call option. The codebook that minimizes error in estimation of the terminal spot concentrates almost all computational effort in areas you don't care about, since the vast majority of the terminal spot density occurs in a region where the value of the option is zero. Given 100 points to work with, you'd by far rather use a single point with a large weight to stand in for all the situations where the option expires worthless, then concentrate your 99 other points above the option strike.

Obviously all this supposes you have some idea where you should be concentrating the computational effort, which may not be possible with any but the simplest payoffs. Still, it's worth remembering that small errors in the discretization of the underlying do not necessarily lead to small errors in the option price obtained with it.