Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations

Since the novel study of Lintner (Am Econ Rev 46:97–113, 1956), it has become a widespread idea that US firms only gradually adjust dividend levels toward long-term targets (Fama and Babiak, J Am Stat Assoc 63:1132–1161, 1968; Mueller, Q J Econ 81:58–87, 1967; Brav et al., J Financ Econ 77:483–527, 2005; Leary and Michaely, Rev Financ Stud 24:3198–3249, 2011). Dividend smoothing helps firms mitigate problems that arise from information asymmetry (e.g., signaling and reduction of agency costs). Gugler (J Bank Finance 27:1297–1321, 2003) and Michaely and Roberts (Rev Financ Stud 25:712–746, 2012) show evidence supporting this idea by using data from the UK and Austria, respectively. However, single country analyses do not provide conclusive answers to the question of why firms smooth dividends. There are significant variations in agency relationships across countries which generate substantial differences in dividend smoothing behaviors. Shleifer and Vishny (J Finance 52:737–783, 1997) point out that in continental Europe and East Asian countries, corporate ownership structures are highly concentrated and there are less severe conflicts between controlling shareholders and management. This fact naturally leads to the idea that international data provides us with an appropriate research setting in which to address the question.

This chapter presents the binomial tree approach to the option pricing problem. We first illustrate the basic ideas of option pricing by considering the one-period binomial tree model and then extend to a multi-period binomial tree model. We then show that, by taking limits in an appropriate way, the binomial expression for the option price converges to the Black–Scholes option price and pricing equation. Alternatively, the continuous time model can be discretised in a way that yields the same expressions as obtained by the binomial tree approach.

This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

Learning Options and Binomial Trees

The option features embedded in many bonds and fixed-income securities have made the binomial interest rate tree approach to bond valuation a valuable model for pricing debt securities. Fundamental to the application of the binomial model to bond valuation is the assumption about the underlying stochastic process. There are two general approaches to modelling stochastic interest rate movements using a binomial model – the equilibrium model and the calibration model. Both models assume that the interest rate's logarithmic return is normally distributed. However, a number of empirical studies have provided evidence that the return distributions of a number of securities exhibit persistent skewness. In modelling interest rate patterns, the existence of skewness in a binomial process impacts not only the values of the up and down parameters, but also the probabilities of the underlying rate increasing or decreasing each period. The purpose of this study is to show how a binomial model of interest rates can be...

Line width at half maximum of the Voigt profile in terms of Gaussian and Lorentzian widths: Normalization, asymptotic expansion, and chebyshev approximation

We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.

Carbon option pricing based on uncertain fractional differential equation: A binomial tree approach

A comprehensive study on the $\kappa $ - $\mu $ phase statistics is conducted. A substantial number of new formulas, exact and approximate, closed form, and integral form, are presented. In particular, a tight close-form approximation for the phase probability density function is found that yields results almost indistinguishable from the exact integral formulation. Most strikingly, the approximate formulation comprises the exact Nakagami- $m$ phase as well as the exact Von Mises (Tikhonov) densities. Joint statistics involving combinations of the envelope, phase, and their time derivatives are derived in an exact manner. The exact phase crossing rate is then obtained in an integral form. A closed-form approximation is proposed that yields very good results as compared to the exact formulation. A Monte Carlo simulation plot is used to validate the formula of the exact phase crossing rate. The formulations presented here drastically facilitate the use of the phase statistics of the $\kappa $ - $\mu $ fading model.

Empirical Tests of Canonical Nonparametric American Option Pricing Methods

A method to reduce broadcast time in wormhole routed hypercube systems in described. The method takes advantage of the destance insensitivity of wormhole routing and the presence of multiple ports between processors and their routes. Performance results from an nCUBE-2 multicomputers are given that demonstrate the advantage of the method over the traditional spanning binomial tree approach.

By using the direct approach to the analysis of the serial-search spread-spectrum code acquisition, the author presents the approximate analysis of the cumulative probability density function (CDF) of the acquisition time. This approach is based on the binomial approximation to the distribution of the partial acquisition times, instead of the Gaussian one used previously. The binomial approximation gives very accurate results but, more importantly, it shows that the analysis of the arbitrary detection/verification scheme can be reduced to the much simpler analysis of the equivalent single-dwell system, whose parameters depend only on the mean test times and not on their distribution. Based on the exact results, simple closed-form approximations are derived for the CDF in the case of the straight-line serial search with uniform a priori distribution, which might be useful in practice.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We consider electric conductance through a narrow nanochannel in the thick-double-layer limit, where the space-charge Debye layers adjacent to the channel walls overlap. At moderate surface-charge densities the electrolyte solution filling the channel comprises mainly of counterions. This allows to derive an analytic closed-form approximation for the channel conductance, independent of the salt concentration in the channel reservoirs. The derived expression consists of two terms. The first, representing electromigratory transport, is independent of the channel depth. The second, representing convective transport, depends upon it weakly.

This paper presents several analytic closed-form approximations of the aggregated interference statistics within the framework of uplink massive machine-type-communications (mMTC), taking into account the random activity of the sensors. Given its discrete nature and the large number of devices involved, a continuous approximation based on the Gram–Charlier series expansion of a truncated Gaussian kernel is proposed. We use this approximation to derive an analytic closed-form expression for the outage probability, corresponding to the event of the signal-to-interference-and-noise ratio being below a detection threshold. This metric is useful since it can be used for evaluating the performance of mMTC systems. We analyze, as an illustrative application of the previous approximation, a scenario with several multi-antenna collector nodes, each equipped with a set of predefined spatial beams. We consider two setups, namely single- and multiple-resource, in reference to the number of resources that are allocated to each beam. A graph-based approach that minimizes the average outage probability, and that is based on the statistics approximation, is used as allocation strategy. Finally, we describe an access protocol where the resource identifiers are broadcast (distributed) through the beams. Numerical simulations prove the accuracy of the approximations and the benefits of the allocation strategy.

The Fokker–Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein–Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are ‘far’ from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.