Section: New Software and Platforms

PREMIA

Keywords: Computational finance - Option pricing

Scientific Description

Premia is a software designed for option pricing, hedging and financial model calibration. It is provided with it's C/C++ source code and an extensive scientific documentation. The Premia project keeps track of the most recent advances in the field of computational finance in a well-documented way. It focuses on the implementation of numerical analysis techniques for both probabilistic and deterministic numerical methods. An important feature of the platform Premia is the detailed documentation which provides extended references in option pricing.

Premia is thus a powerful tool to assist Research & Development professional teams in their day-to-day duty. It is also a useful support for academics who wish to perform tests on new algorithms or pricing methods without starting from scratch.

Besides being a single entry point for accessible overviews and basic implementations of various numerical methods, the aim of the Premia project is: 1 - to be a powerful testing platform for comparing different numerical methods between each other, 2 - to build a link between professional financial teams and academic researchers, 3 - to provide a useful teaching support for Master and PhD students in mathematical finance.

Functional Description

• Participants: Mathrisk project team and contributors

• Partners: Inria - Ecole des Ponts ParisTech - Université Paris-Est - Consortium Premia

• Contact: Agnès Sulem

• URL: http://www.premia.fr

• AMS: 91B28;65Cxx;65Fxx;65Lxx;65Pxx

• License: Licence Propriétaire (genuine license for the Consortium Premia)

• Type of human computer interaction: Console, interface in Nsp, Web interface

• OS/Middelware: Linux, Mac OS X, Windows

• APP: The development of Premia started in 1999 and 16 are released up to now and registered at the APP agency. Premia 16 has been registered on 0303/2015 under the number IDDN.FR.001.190010.013.S.C.2001.000.31000

• Programming language: C/C++ librairie Gtk

• Documentation: the PNL library is interfaced via doxygen

• Size of the software: 280580 lines for the Src part only, that is 11 Mbyte of code, 130400 lines for PNL, 105 Mbyte of PDF files of documentation.

• interfaces : Nsp for Windows/Linux/Mac, Excel, binding Python, and a Web interface.

• Publications: [12] , [61] , [69] , [77] , [80] , [49] , [59] .

Content of Premia

Premia contains various numerical algorithms (Finite-differences, trees and Monte-Carlo) for pricing vanilla and exotic options on equities, interest rate, credit and energy derivatives.

1. Equity derivatives:

   The following models are considered:

   Black-Scholes model (up to dimension 10), stochastic volatility models (Hull-White, Heston, Fouque-Papanicolaou-Sircar), models with jumps (Merton, Kou, Tempered stable processes, Variance gamma, Normal inverse Gaussian), Bates model.

   For high dimensional American options, Premia provides the most recent Monte-Carlo algorithms: Longstaff-Schwartz, Barraquand-Martineau, Tsitsklis-Van Roy, Broadie-Glassermann, quantization methods and Malliavin calculus based methods.

   Dynamic Hedging for Black-Scholes and jump models is available.

   Calibration algorithms for some models with jumps, local volatility and stochastic volatility are implemented.

2. Interest rate derivatives

   The following models are considered:

   HJM and Libor Market Models (LMM): affine models, Hull-White, CIR$++$, Black-Karasinsky, Squared-Gaussian, Li-Ritchken-Sankarasubramanian, Bhar-Chiarella, Jump diffusion LMM, Markov functional LMM, LMM with stochastic volatility.

   Premia provides a calibration toolbox for Libor Market model using a database of swaptions and caps implied volatilities.

3. Credit derivatives: Credit default swaps (CDS), Collateralized debt obligations (CDO)

   Reduced form models and copula models are considered.

   Premia provides a toolbox for pricing CDOs using the most recent algorithms (Hull-White, Laurent-Gregory, El Karoui-Jiao, Yang-Zhang, Schönbucher)

4. Hybrid products

   A PDE solver for pricing derivatives on hybrid products like options on inflation and interest or change rates is implemented.

5. Energy derivatives: swing options

   Mean reverting and jump models are considered.

   Premia provides a toolbox for pricing swing options using finite differences, Monte-Carlo Malliavin-based approach and quantization algorithms.

Premia design

To facilitate contributions, a standardized numerical library (PNL) has been developed by J. Lelong under the LGPL since 2009, which offers a wide variety of high level numerical methods for dealing with linear algebra, numerical integration, optimization, random number generators, Fourier and Laplace transforms, and much more. Everyone who wishes to contribute is encouraged to base its code on PNL and providing such a unified numerical library has considerably eased the development of new algorithms which have become over the releases more and more sophisticated. J. Ph Chancelier, B. Lapeyre and J. Lelong are using Premia and Nsp for Constructing a Risk Management Benchmark for Testing Parallel Architecture [59] .

Development of the PNL in 2015 (J. Lelong) . Release 1.70 and 1.71, PNL Library (http://pnl.gforge.inria.fr ).

1. Release 1.72. of the PNL library (http://pnl.gforge.inria.fr/ ).

   1. Addition of a CMake module to include the library in other projects.

   2. Improvement of the pnl_basis module.

   3. Addition of the non central chi squared distribution to the random number generation toolbox.

   4. Addition of new functions in the linear algebra toolbox to build views.

Algorithms implemented in Premia in 2015

Premia 17 has been delivered to the consortium members in March 2015.

It contains the following new algorithms:

Commodities, FX, Insurance, Credit Risk

• Variables Annuities GLWB pricing in the Heston and Black-Scholes/Hull-White models with finite difference techniques.

• Variables Annuities GMAB, GMDB, GMMB pricing with Fourier-cosine techniques.

• A numerical scheme for the impulse control formulation for pricing variable annuities with a Guaranteed Minimum Withdrawal Benefit (GMWB) Z.Chen P.Forsyth

  Numerische Mathematik 109, 2008

• Managing Gap Risks in iCPPI for life insurance companies: A risk/return/cost analysis. A.Kalife S.Mouti L.Goudenege

  Insurance Markets and Companies: Analyses and Actuarial Computations, Issue 2 2014

• Simulating CVA on American Options. L. Abbas Turki, M.Mikou

Equity Derivatives

• Being particular about calibration. J.Guyon and P. Henry-Labordère.

  Risk magazine, Jan 2012.

• The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation. A.W. van der Stoepb, L. A. Grzelakb, C. W. Oosterlee

  International Journal of Theoretical and Applied Finance, to appear

• On the Heston model with stochastic interest rates. L. Grzelak C.W.Oosterlee

  SIAM J. Fin. Math. 2,2011.

• Alternating direction implicit finite difference schemes for the Heston Hull-White partial differential equation.

  The Journal of Computational Finance Volume 16/Number 1, Fall 2012

• Pricing American options in the Heston Hull-White and Hull-White2d Models: an hybrid tree-finite difference approach. M.Briani, L.Caramellino, A.Zanette

• Efficient pricing of Asian options under Lévy processes based on Fourier cosine expansions. Part I: European-style products. B.Zhang C.W.Oosterlee.

  SIAM J. Financial Math., 4(1)

• Low-bias simulation scheme for the Heston model by Inverse Gaussian approximation. S. T. Tse J. W. L. Wan.

  Quantitative Finance, Volume 13, Issue 6, 2013

• Simple Simulation Scheme for CIR and Wishart Processes P. Baldi, C.Pisani

  International Journal of Theoretical and Applied Finance Vol. 16, No. 08, 2013

• Importance sampling for jump processes and applications to finance. L. Badouraly Kassim, J. Lelong and I. Loumrhari.

  Journal of Computational Finance, to appear

• A Wiener-Hopf Monte Carlo simulation technique for Lévy process. A. Kuznetsov, A.E.Kyprianou J. C. Pardo and K. van Schaik.

  The Annals of Applied Probability, Volume 21, Number 6, 2011.

• A Wiener-Hopf Monte Carlo simulation approach for pricing path-dependent options under Lévy process. O. Kudryavtsev

  Preprint.

• An Efficient Binomial Lattice Method for Step Double Barrier Options. E.Appolloni, M.Gaudenzi A.Zanette.

  International Journal of Applied and Theoretical Finance Vol.17, Issue No. 6, 2014.

The algorithms

• “Pricing American-Style Options by Monte Carlo Simulation : Alternatives to Ordinary Least Squares” by Stathis Tompaidis and Chunyu Yang

• “Value Function Approximation or Stopping Time Approximation : A comparison of Two Recent Numerical Methods for American Option Pricing using Simulation and Regression” by Lars Stentoft

implemented in 2015 by CélineLabart will be included in the following release.

Moreover, Jérome Lelong has performed the following tasks:

1. Add an importance sampling based code for jump diffusion models.

2. Improve the internal enumeration mechanism (PremiaEnum).

3. Update gnuplot files generation for reports.

4. Everyday maintenance to fix various bugs.

5. Clean the generation of the documentation process.