Operator Methods, Abelian Processes and Dynamic Conditioning

Abstract

A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can effectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions.

A backward Monte Carlo approach to exotic option pricing

“…The first one is based on the concept of optimal state-partitioning of a random variable (called stratification in Barraquand & Martineau (1995) and quantization in Bally & Pagès (2003)) and employs the RMQA (see Callegaro et al, 2015;Pagès & Sagna, 2015). The second approach provides a recipe to compute in an effective way the transition probability matrix between any two arbitrary dates for a piecewise time-homogeneous process (see Albanese, 2007;Reghai et al, 2012). More details on these two methods will be given in Section 3.…”

A Backward Monte Carlo Approach to Exotic Option Pricing

“…Once L Γ is constructed, one writes a (matrix) Kolmogorov equation for the transition probability matrix. In particular, using operator theory [34], the transition probability matrix between any two arbitrary dates u k and u k with 0 ≤ u k < u k ≤ T can be expressed as a matrix exponential. Remark 3.…”

Efficient Pricing of CPPI Using Markov Operators

“…Numerically, there are two ways of doing the computation. The first solution is to take the n th power of the matrix M by fast exponentiation [Alb07]: M 2 p are computed recursively for growing p by M 2 p = M 2 p−1 M 2 p−1 . The power n is decomposed in basis 2 as n = i 2 p i and M n is obtained by multiplying the corresponding M 2 p i :…”