Selected documents by Peter Jäckel

• Errata in Monte Carlo methods in finance (John Wiley and Sons, February 2002). These are already corrected in the latest print batch.

• The handling of continuous barriers for derivatives on many underlyings. (Presentation at the Quantitative Finance Conference in London, November 2002)

• Mind the Cap. (Wilmott, pages 54-68, September 2003).

• The link between caplet and swaption volatilities in a Brace-Gatarek-Musiela/Jamshidian framework: approximate solutions and empirical evidence. (The Journal of Computational Finance, 6(4), 2003, pages 41-59, submitted in 2000).

• The Future is Convex. (Wilmott, pages 2-13, February 2005).

• Stochastic volatility models - past, present, and future. (Presentation at the "Quantitative Finance Review" conference in November 2003 in London).

• Valuing American options in the presence of user-defined smiles and time-dependent volatility: scenario analysis, model stress and lower-bound pricing applications. (The Journal of Risk, 4(1), pages 35-61,2001).

• Splitting the core. Ever wondered how to (approximately) decompose the correlation matrix used in the semianalytical pricing of CDOs in the default-time-copula model into the factor weights of a single systemic factor with a really simple formula, i.e. without the need for iterations or principal components analysis? Here is how!

• More likely than not. In a nutshell: this is a collection of likelihood ratio formulae.

• Stabilised multidimensional root finding. Underdetermined fitting and root finding problems can be stabilised by the addition of quantifiable desirable features to the task. Simply defining a weighted objective function containing the original problem and the desiderata function is generally not robust. By adding a Lagrange-multiplier weighted Newton-Raphson step condition to the desiderata function, however, even very large problems can be solved surprisingly efficiently.

• The practicalities of Libor Market models. There are many publications on the theory of the Libor market model and its extensions. There are very few sources on the issues a pracitioner faces during implementation and opertion of the model. This presentation (~160 slides) is the material for a one-day training course (first given in 2005) on the subject of how to make a Libor Market Model work in practice.

• A note on multivariate Gauss-Hermite quadrature. Univariate Gauss-Hermite quadrature is a very powerful and well understood tool in numerical analysis. In this document, I discuss some of the choices we have when it comes to more than one dimension. I also provide an explanation how polar coordinates can be used in two dimensions for which an unusual kind of one-dimensional quadrature is required: radial Gauss-Hermite quadrature. This is essentially the same as standard Gauss-Hermite, only that the integration domain starts at zero. I have precomputed the required roots and weights up to order 40. They are tabulated in RootsAndWeightsForRadialGaussHermiteQuadrature.cpp.

• Fast strong approximation Monte-Carlo schemes for stochastic volatility models (joint paper with Christian Kahl, September 2005, published in the Journal of Quantitative Finance, Vol. 6, No. 6, 2006, pp. 513-536). Fast numerical integration methods for stochastic volatility models in financial markets are discussed. We use the strong convergence behaviour as an indicator for the approximation quality.

• Not-so-complex logarithms in the Heston model (joint paper with Christian Kahl, Wilmott, pages 94-103, September 2005).

• By Implication (July 2006; Wilmott, pages 60-66, November 2006). Probably the most complicated trivial issue in financial mathematics: how to compute Black's implied volatility robustly, simply, efficiently, and fast.

• An asymptotic FX option formula in the cross currency Libor market model (joint paper with Atsushi Kawai, October 2006; Wilmott, pages 74-84, March 2007). Libor market models are becoming more and more popular, and approximate formulae for swaptions and caplets in aid of fast calibration are available. This article is about plain vanilla FX option approximations in a cross currency Libor market model with explicit (displaced diffusion) control over the skew of both domestic and foreign interest rates, as well as the spot FX process.

• Hyperbolic local volatility (November 2006). A parametric local volatility form based on a hyperbolic conic section is introduced, and details are given as to how this alternative local volatility form can be used as a drop-in replacement for the popular Constant Elasticity of Variance local volatility, and what parameter restrictions apply.

• Hyp Hyp Hooray (June 2007). A new stochastic-local volatility model is introduced. The new model's structural features are carefully selected to accommodate economic principles, financial markets' reality, mathematical consistency, and ease of numerical tractability when used for the pricing and hedging of exotic derivative contracts. Also, we present a generic analytical approximation for Black volatilities for plain vanilla options implied by any parametric-local-and-stochastic-volatility model, apply it to the new model, and demonstrate its accuracy.

• The Gamma Loss and Prepayment model (November 2007, published in Risk Magazine in September 2008, pp 134-139). We present a model for the dynamics of fractional notional losses and prepayments on asset backed securities for the valuation and risk management of derivatives such as the so-called waterfall structures and other structured debt obligations.

• The Discrete Gamma Pool model (February 2008) is a model for the dynamics of losses and spreads on portfolios for the purpose of pricing exotic variations of synthetic collateralised tranche obligations such as Loss Triggered Leveraged Super-Senior notes, multi-callable CDOs, and, by implication of the latter, options on forward starting CDOs. Also discussed are features such as the counterparty's right to deleverage upon a loss trigger event in a leveraged super senior can be understood as an embedded Bermudan swaption, and how this can be catered for in a numerical implementation.

• Implementation of the Discrete Gamma Pool model (February 2008) gives details as to how the numerical quadratures required for the valuation of contracts within the Discrete Gamma Pool model framework can be done.