On the Acceleration of Explicit Finite Difference Methods for Option Pricing

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant–Friedrichs–Lewy (CFL) condition which limits the latter class’s efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black–Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes

Derivatives Pricing with Accelerated Trinomial Trees

Accelerated Trinomial Trees (ATTs) are a derivatives pricing lattice method that circumvent the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs) in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps each of which contains an inner lattice/trinomial tree with N non-uniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, are constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk neutral extended probabilities; extended in the sense that values can lie outside the range [0; 1] on the substep scale but aggregate to probabilities within the range [0; 1] on the superstep scale. Hence it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient than competing implicit methods which use L time steps in pricing Black-Scholes American put options and 2-dimensional American basket options. Crucially this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs which may be relatively complex in their implementation

Path dependent option pricing under Lévy processes applied to Bermudan options

A model is developed that can price path dependent options when the underlying process is an exponential Lévy process with closed form conditional characteristic function. The model is an extension of a recent quadrature option pricing model so that it can be applied with the use of Fourier and Fast Fourier transforms. Thus the model possesses nice features of both Fourier and quadrature option pricing techniques since it can be applied for a very general set of underlying Lévy processes and can handle exotic path dependent features. The model is applied to European and Bermudan options for geometric Brownian motion, a jump-diffusion process, a variance gamma process and a normal inverse Gaussian process. However it must be noted that the model can also price other path dependent exotic options such as lookback and Asian options

Parameter uncertainty in Kalman filter estimation of the CIR term structure model

The Cox, Ingersoll and Ross (1985) term structure model describes the stochastic evolution of government bond yield curves over time using a square root Orstein-Uhlenbeck diffusion process, whilst imposing cross-sectional no-arbitrage restrictions between yields of different maturities. A Kalman filter approach can be used to estimate the parameters of the CIR model from panel data consisting of a time series of bonds of different maturities. The parameters are estimated by optimising a quasi log-likelihood function that results from the prediction error decomposition of the Kalman filter. The quasi log-likelihood function is usually optimised with a deterministic gradient based optimisation technique such as a quadratic hill climbing optimiser. This paper uses an evolutionary optimiser known as differential evolution (DE) to optimise over the parameter space. The DE optimiser is more likely to find the global maximum than a deterministic optimiser in the presence of a non-convex objective function which may be the case in multifactor term structure models with non-negativity constraints and parameter constraints. The method is applied to estimate parameters from a one and two-factor Cox, Ingersoll and Ross (1985) model. It is shown that in the two factor model the problem of local maxima arises whereby a number of different parameter vectors perform equally well in the estimation procedure. Fixed income derivative prices are particular sensitive to term structure parameters such as the volatility, the rate of mean reversion, and the market price of risk of each factor. The effect of different optimal parameter vectors on fixed income derivatives is examined and is found to be significant

A simple recursive numerical method for Bermudan option pricing under Lévy processes

A numerical method is developed that can price options, including exotic options that can be priced recursively such as Bermudan options, when the underlying process is an exponential Lévy process with closed form conditional characteristic function. The numerical method is an extension of a recent quadrature option pricing method so that it can be applied with the use of fast Fourier transforms. Thus the method possesses desirable features of both transform and quadrature option pricing techniques since it can be applied for a very general set of underlying Lévy processes and can handle certain exotic features. To illustrate the method it is applied to European and Bermudan options for a log normal process, a jump diffusion process, a variance gamma process and a normal inverse Gaussian process

Atlantic-dip: raised maternal body mass index (bmi) adversely affects maternal and fetal outcomes in glucose-tolerant women according to international association of diabetes and pregnancy study groups (iadpsg) criteria

Context: Raised maternal body mass index (BMI) in association with hyperglycemia is associated with adverse pregnancy outcome. The contribution of raised BMI as an independent risk factor for adverse pregnancy outcome is of growing concern and increasing prevalence. Objective: The aim of this study was to investigate the effects of raised maternal BMI on pregnancy outcome in glucose-tolerant women using the International Association of Diabetes and Pregnancy Study Groups criteria. Participants and Setting: We studied a cohort of glucose-tolerant, pregnant women (n = 3656) who were attending antenatal obstetric clinics and were recruited to a universal screening program for gestational diabetes under the ATLANTIC-DIP partnership. Design: We conducted a prospective observational study of pregnancy outcome. Maternal outcomes include glucose, delivery mode, pregnancy-induced hypertension, preeclampsia, antepartum hemorrhage, and postpartum hemorrhage. Fetal outcomes included birth weight, congenital malformation, fetal death, neonatal jaundice, hypoglycemia, and respiratory distress. Results: Increasing maternal BMI was associated with adverse pregnancy outcomes: higher cesarean section rates, preeclampsia, pregnancy-induced hypertension, increased birth weight, and congenital malformation. The association of glucose with adverse pregnancy outcome was weak and did not interact with raised BMI. A BMI threshold of 28 kg/m(2) was associated with a significant rise in adverse pregnancy outcome. Conclusions: Raised maternal BMI, within the overweight range, is associated with adverse pregnancy outcomes. These adverse effects of BMI occur independently of maternal glucose. It is apparent that pregnancy unmasks an underlying unhealthy metabolic milieu in obese and overweight women. (J Clin Endocrinol Metab 97: E608-E612, 2012

Short-chain fatty acids: microbial metabolites that alleviate stress-induced brain–gut axis alterations

There is a growing recognition of the involvement of the gastrointestinal microbiota in the regulation of physiology and behaviour. Microbiota‐derived metabolites play a central role in the communication between microbes and their host, with short‐chain fatty acids (SCFAs) being perhaps the most studied. SCFAs are primarily derived from fermentation of dietary fibres and play a pivotal role in host gut, metabolic and immune function. All these factors have previously been demonstrated to be adversely affected by stress. Therefore, we sought to assess whether SCFA supplementation could counteract the enduring effects of chronic psychosocial stress. C57BL/6J male mice received oral supplementation of a mixture of the three principle SCFAs (acetate, propionate and butyrate). One week later, mice underwent 3 weeks of repeated psychosocial stress, followed by a comprehensive behavioural analysis. Finally, plasma corticosterone, faecal SCFAs and caecal microbiota composition were assessed. SCFA treatment alleviated psychosocial stress‐induced alterations in reward‐seeking behaviour, and increased responsiveness to an acute stressor and in vivo intestinal permeability. In addition, SCFAs exhibited behavioural test‐specific antidepressant and anxiolytic effects, which were not present when mice had also undergone psychosocial stress. Stress‐induced increases in body weight gain, faecal SCFAs and the colonic gene expression of the SCFA receptors free fatty acid receptors 2 and 3 remained unaffected by SCFA supplementation. Moreover, there were no collateral effects on caecal microbiota composition. Taken together, these data show that SCFA supplementation alleviates selective and enduring alterations induced by repeated psychosocial stress and these data may inform future research into microbiota‐targeted therapies for stress‐related disorders