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2015

1. Wiley

   Let’s be rational

   Peter Jäckel

   Wilmott, 2015

   Abs

   Improving on [Jäc06] and [Vog07], 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. The method is based on four rational function branches for the initial guess adapted to the log-moneyness, two of which are combined with nonlinear transformations of the input price, and the use of the convergence order four Householder method which comprises a rational function of the residual. Despite sounding difficult, the method is simple in practice, and a reference implementation is provided in [Jäc13]. As was perhaps previously underestimated, of crucial importance for the precision of the implied volatility is a highly accurate Black function that minimizes round-off errors and numerical truncations in the various parameter limits. We implement the Black call option price with the aid of Cody’s [Cod69, Cod90] rational approximation for the complementary error function erfcx(·) and its little known cousin, the scaled complementary error function erfcx(·). The source code of the reference implementation is available at www.jaeckel.org/LetsBeRational.7z.

2014

1. Wiley

   Jackknife Resampling

   Herwig Friedl, and Erwin Stampfer

   Wiley StatsRef: Statistics Reference Online, 2014

   Abs

   The jackknife is a particular resampling method that aims primarily at the calculation of the bias and the variance of estimates, without making very restrictive distributional assumptions. It is nonparametric and easy to apply in quite general settings. As with other resampling techniques, the jackknife can be computationally very intensive. In the simplest case jackknife resampling is generated by sequentially deleting single cases from the original sample (delete-one jackknife). A more generalized jackknife technique uses resampling that is based on multiple case deletion (delete-d jackknife). The main part of this article focuses on independently and (in the introductory part also) identically sampled data. It then discusses the basics of the jackknife and its applications to nonidentically distributed observations that can be described by linear, generalized linear, nonlinear, and Cox’s regression models. Finally, the article gives an account of jackknife estimates for dependent data, e.g. in the context of generalized estimating equations or special time series models.

2011

1. SSRN

   Fast Greeks by algorithmic differentiation

   Luca Capriotti

   The Journal of Computational Finance, 2011

   Abs

   We show how Algorithmic Differentiation can be used to implement efficiently the Pathwise Derivative method for the calculation of option sensitivities with Monte Carlo. The main practical difficulty of the Pathwise Derivative method is that it requires the differentiation of the payout function. For the type of structured options for which Monte Carlo simulations are usually employed, these derivatives are typically cumbersome to calculate analytically, and too time consuming to evaluate with standard finite-differences approaches. In this paper we address this problem and show how Algorithmic Differentiation can be employed to calculate very efficiently and with machine precision accuracy these derivatives. We illustrate the basic workings of this computational technique by means of simple examples, and we demonstrate with several numerical tests how the Pathwise Derivative method combined with Algorithmic Differentiation – especially in the adjoint mode – can provide speed-ups of several orders of magnitude with respect to standard methods.

2010

1. Wiley

   Delete-2 and delete-3 jackknife procedures for unmasking in regression

   Michael A Martin, Steven Roberts, and Letian Zheng

   Australian & New Zealand Journal of Statistics, 2010

   Abs

   Single-case deletion regression diagnostics have been used widely to discover unusual data points, but such approaches can fail in the presence of multiple unusual data points and as a result of masking. We propose a new approach to the use of single-case deletion diagnostics that involves applying these diagnostics to delete-2 and delete-3 jackknife replicates of the data, and considering the percentage of times among these replicates that points are flagged as unusual as an indicator of their influence. By considering replicates that exclude certain collections of points, subtle masking effects can be uncovered.

2008

1. Wolfram

   Stock Price Probability with Stable Distributions

   Bob Rimmer

   2008

   Abs HTML

   Lévy-stable distributions were introduced for financial returns by Mandelbrot in 1963. They offer an advantage over the log-normal assumption in that extreme events can exist in the model. They have not been in much use because of the difficulty in performing the calculations. Most models using stable distributions have also assumed that they are stationary; when parameters are fitted to a stationary stable model, the tails are consistently found to be too heavy. The parameters for the particular example were scaled from intraday observations at one-minute intervals. The parameters chosen for the example were obtained by maximum-likelihood fitting using an FFT approximation of the stable density from the sampled stable characteristic function over six months of data, after rescaling each day’s data by a value of γ derived from an empirical characteristic fit. This method removes most of the serial dependence seen in the data and gives a value of α that is significantly higher than would be found by fitting the raw data, assuming the distribution is stationary. The parameters γ and δ are selected from average days, but the user needs to be aware that γ may vary by a factor of 2 or 3 during volatile periods in the market. The user who downloads the source may experiment with his own parameters.