- Implied Normal Volatility is a short communication on an analytical formula for the calculation of implied normal volatility (also known as Bachelier volatility) from vanilla option prices. (January 2017; Wilmott, pages 52-54, March 2017).
justifiable dividend modelling is a presentation on methods to incorporate a realistically chosen dividend
model (where the actually paid dividend depends on the concurrent share price) into a real market making system:
cash dividends that can be downsized if the share price collapsed, arbitrage-free and consistent with the market's
implied volatility surface, stochastic volatility parameter generation of implied volatility surfaces that are
consistent by construction with the dividend forecasts (no numerical/iterative calibration involved), and local
volatility dynamics that are also by construction, i.e., without any numerical calibration, exactly calibrated to
the chosen dividend process (ICBI Global Derivatives Conference, Amsterdam, May 2015, and WBS Fixed Income
Conference, Paris, October 2015).
- Ultra-Sparse Finite-Differencing For Arbitrage-Free Volatility Surfaces From Your Favourite Stochastic Volatility Model
[Joint work with colleagues at VTB Capital].
Inspired by the Hyperbolic-Hyperbolic parametric local-stochastic
volatility model, we present a practical method for an
arbitrage-free definition of an implied volatility surface with wide
ranging parametric flexibility, based on high-efficiency ultra-sparse
finite differencing techniques combined with arbitrage-free implied
volatility inter- and extrapolation
(ICBI Global Derivatives Conference, Amsterdam, May 2014, and WBS Fixed Income Conference, Barcelona, September 2014).
40 years of evolution of smile parametrisation
Commitment to Spatial Discreteness and the importance of being Metzler
Stencil Viability, Boundary Conditions, Continuous-Time stability, Ito's lemma of pure jump processes for Exact
Martingales, Algebraic Splitting, Anti-Diffusion Limiting, and other lessons to learn from 60 years of Computational
From Continuous Time to Numerical Integration
Box Transition Probability Translation
Cash dividends without arbitrage or approximations
- In Clamping Down on Arbitrage
(December 2013), the subject of arbitrage-free interpolation of implied Black volatility is addressed. The presented
method preserves a preselected (and thus preferred) interpolation method as much as possible, and only invokes
corrections where needed. All calculations are analytical without any numerical fitting that could so easily lead to
undesirable shapes of implied volatility profiles. The article also contains considerations on
arbitrage-free extrapolation, and an effectively analytical procedure for the inverse of the logarithm of the scaled
complementary error function (which is required for one of the analytical arbitrage-free extrapolation methods),
i.e., the solution to ln(Phi(x))+x²/2 = c for x. We mention that the same equation occurs in other context, e.g., as the maximum attainable
call option delta in the context of FX option quotations of volatilities over delta-with-premium.
- Let's Be Rational (November
2013; Wilmott, pages 40-53, January 2015) is a follow-up article on By
Implication (July 2006). In this newer article, we show how Black's volatility can be implied from option
prices with as little as two iterations to maximum attainable precision on standard (64 bit floating point) hardware for all possible inputs.
Open-source Reference Implementation of "Let's
Be Rational" (November 2013). Permission to use, copy, modify, and distribute this software is freely
granted, provided that the contained copyright notice is preserved. WARRANTY DISCLAIMER: The Software is
provided "as is" without warranty of any kind, either express or implied, including without limitation any
implied warranties of condition, uninterrupted use, merchantability, fitness for a particular purpose, or
- In A default
copula in a lattice-based credit model (June 2013), we add a default event copula to the framework of
a multi-factor credit model with correlated stochastic hazard rates. We find that the codependence structure of
default events in the limit of infinitesimal time steps in a finite-differencing framework is dominated by the
lower tail dependence coefficient of the employed copula. Collaboration article with Christian Kahl, Lee Wild,
and Ioannis Chryssochoos.
differencing schemes as Padé approximants (April 2013) is a summary of various numerical schemes for the
solution of parabolic partial differential equations with particular emphasis on their representation as Padé
approximants of the exponential function. Of applied interest are some of the higher order schemes that are
discussed, some of which go back to the 1960's, and several others to the 1980's, though they seem to be not in
the commonly known toolset of practitioners in financial mathematics. This note is an attempt make these powerful methods more widely known.
- Geodesic strikes
for composite, basket, Asian, and spread options (July 2012, updated February 2017) is a note on simple methodologies for the selection
of most relevant or effective strikes for the assessment of appropriate implied volatilities used for the
valuation of composite, basket, Asian, and spread options following the spirit of geodesic strikes in
Reconstructing volatility by M. Avellaneda, D. Boyer-Olson, J. Busca, and P. Friz, Risk, pages 91–95, October 2002.
- Quanto Skew with stochastic volatility (March 2010)
is a continuation of the analysis in Quanto Skew to the presence of both
local and stochastic volatility for the underlying asset and the FX rate process.
- Quanto Skew (July 2009) presents an analysis
of the humble quanto vanilla option. A conventionally used quanto adjustment is compared with exact results using a
simple double displaced diffusion model. Arguably (not) surprisingly, it turns out that the conventional quanto
adjustment results in price and (quanto-) implied volatility differences that are negligible only for short-dated contracts.
- The following articles appeared in the
Encyclopedia of Quantitative Finance (John Wiley and Sons, 2010):
- A singular Variance
Gamma expansion (May 2009) is a note on an analytical expansion for option prices and Black implied
volatilities generated by the Variance Gamma model based on a singular expansion of the standard gamma density
in terms of the Dirac functions and its derivatives. The expansion is done up to fifth order in the Variance
Gamma kurtosis parameter ν using the open source computer algebra system Maxima. The Maxima
BlackVolatilityExpansionForVarianceGammaModel_macsyma.txt is straightforward and could easily be translated to any
other symbolic mathematics package.
- Positive semi-definite correlation matrix completion for stochastic volatility models
(joint paper with Christian Kahl, May 2009) outlines how one can, for any stochastic volatility model, given cross-asset, and asset-volatility correlations, fill in the remaining elements of the complete correlation matrix in a flexible way that is guaranteed to always give a positive semi-definite matrix.
- The Discrete Gamma Pool
model (February 2008; Wilmott Journal, 1(1):23-40, 2009) 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.
- 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.
- Hyp Hyp Hooray (June 2007,
joint paper with Christian Kahl; Wilmott, pages 70-81, March 2008).
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.
- 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.
- 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
- 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.
- Semi-analytic valuation of credit linked swaps in a Black-Karasinski framework
(Quant Congress Europe, London, October 2006) is a presentation on a simple model for the valuation of credit linked
swaps in a framework that allows for strictly positive default hazard rates and permits explicit control over the
market-observable skew of implied volatilities for options on the underlying swap. We discuss different aspects of
calibration depending on the nature of the underlying swap. For speedy numerical evaluation, the resulting pricing
equations are reduced to a dimensionality-pruned quadrature over a generic Ornstein-Uhlenbeck process path space.
- Not-so-complex logarithms in the Heston model (joint paper with Christian Kahl, Wilmott, pages 94-103, September 2005).
- 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.
- Options on Credit Default Index Swaps (joint paper with Yunkang Liu, Wilmott, pages 92-97, July 2005).
- A practical method for the valuation of a variety of hybrid products
(ICBI Global Derivatives Conference, Paris, May 2005) is a presentation on a flexible model framework that can be used to price products on multiple underlyings, from different asset classes, allowing for arbitrary volatility smiles. The model is effectively an approximate Markov functional model. Its numerical implementation allows for very fast pricing of fully smile dependent contracts similar to local volatility models, but without any numerical short time-stepping, and without any numerical calibration noise as is so often associated with local volatility models.
- A note on multivariate Gauss-Hermite quadrature (May 2005). 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
- A toy example for
weighted sampling for variance reduction (October 2004).
This is a demonstration how biasing the variates used for a Monte Carlo simulation can significantly reduce the variance
of the simulation result. As is so often the case with this technique, its applicability in practice depends on having a
good estimate for the optimal bias in a least-variance sense. In this example for a digital option in a Gaussian model,
I give analytical approximations for the optimal bias derived from its defining transcendental equation.
- 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
- 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.
- More likely than not.
In a nutshell: this is a collection of likelihood ratio formulae.
- 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!
- 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).
- Stochastic volatility models - past, present, and future.
(Presentation at the "Quantitative Finance Review" conference in November 2003 in London).
- The Future is Convex.
(Wilmott, pages 2-13, February 2005).
- The link between caplet and swaption volatilities in a
Brace-Gatarek-Musiela/Jamshidian framework: approximate solutions and
(The Journal of Computational Finance, 6(4), 2003, pages 41-59, submitted in 2000).
- Mind the Cap.
(Wilmott, pages 54-68, September 2003).
- The handling of continuous barriers for derivatives on many underlyings.
(Presentation at the Quantitative Finance Conference in London, November 2002).
- Errata in Monte Carlo methods in finance (John Wiley and Sons, February 2002).
These are already corrected in the latest print batch.