HEC Montréal, Canada, May 2 - 4, 2011

2011 Optimization Days

HEC Montréal, Canada, 2 — 4 May 2011

Schedule Authors My Schedule

TA8 Ingénierie financière I / Financial Engineering I

May 3, 2011 10:30 AM – 12:10 PM

Location: Nancy et Michel-Gaucher

4 Presentations

  • 10:30 AM - 10:55 AM

    Refining the Least-Square Monte Carlo Method by Imposing Structure.

    • Pascal Letourneau, presenter, HEC Montréal

    The Least-Squares Monte Carlo method of Longstaff and Schwartz (2001) is an option pricing numerical method with regressions. Using few regressors leads to biased results; increasing the number can lead to numerical problems. We show that by imposing structure we can improve the method by reducing the bias.

  • 10:55 AM - 11:20 AM

    Robust Option Pricing - An Epsilon-Arbitrage Approach

    • Chaithanya Bandi, presenter, Operations Research Center, MIT
    • Dimitris Bertsimas, Massachusetts Institute of Technology

    This research aims to provide tractable approaches to price options using robust optimization. The pricing problem is reduced to a problem of identifying the replicating portfolio which minimizes the worst case arbitrage possible for a given uncertainty set on underlying asset returns. We construct corresponding uncertainty sets based on different levels of risk aversion of investors and make no assumption on specific probabilistic distributions of asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-dimensional options and (b) modeling flexibility illustrated by our ability to model the "volatility smile". We also show the applicability of this pricing method in the case of exotic and multi-dimensional options, in particular, we provide formulations to price Asian options, Lookback options and also Index options.

  • 11:20 AM - 11:45 AM

    Data-Driven Robust Optimization with Application to the Portfolio Selection Problem

    • Selim Mankaï, presenter, Université Paris Ouest Nanterre La Défense

    This paper develops a data-driven robust optimization model where the uncertainty set contains a finite number of possible realizations based on a gradual break in the dataset. We formulate the problem for the different polyhedron-shaped uncertainty sets and analyze the sensitivity of the robust solution. Without making assumptions about the unknown data generating process, we propose an a priori non-parametric reliability measure. An application to the portfolio selection problem is discussed. Theoretical analysis reveals many interesting properties of the model compared to the budget of uncertainty approach. The risk aversion level determines the degree of conservatism, the protection threshold and the reliability level of the robust solution. Ex-post analysis shows the effectiveness of the proposed approach, especially for long-term horizons.

  • 11:45 AM - 12:10 PM

    A Spectral Method for Option Pricing Under Garch

    • Saad Serghini Idrissi, presenter, HEC Montréal
    • Michèle Breton, GERAD, HEC Montréal
    • Javier de Frutos, GERAD - Universidad de Valladolid

    We propose a numerical algorithm for pricing derivative under GARCH process. The procedure is based on dynamic programming coupled with spectral approximation, using Chebyshev polynomials, to compute the value at all observation dates and levels of the state vector. The hybrid interpolation approach exhibits spectral convergence, while avoiding localization errors and numerical instability. The approach is very flexible and can be used e.g. for models with non-gaussian innovations. Numerical experiments are presented for Bermudian put options under the standard Black-Scholes model and under GARCH specifications, comparing spectral interpolation, finite element interpolation and PDE methods.