Comparison of Black–Scholes Formula with Fractional Black–Scholes Formula in the Foreign Exchange Option Market with Changing Volatility

In this paper, the authors discuss the fractional option pricing with Black–Scholes formula, deduce the Fractional Black–Scholes formula, show the empirical results by using China merchants bank foreign exchange call option price, and find when the volatility is smaller, the asymptotic mean squared error of Fractional Black–Scholes is bigger than the Traditional Black–Scholes’, while the volatility is bigger—the market mechanism has a full play, the result is reverse. Namely when the market mechanism is given a full scope, the estimating effect of Fractional Black–Scholes is better than Traditional Black–Scholes’.     

Pricing Black–Scholes options with correlated interest rate risk and credit risk: an extension

This article provides a closed-form valuation formula for the Black–Scholes options subject to interest rate risk and credit risk. Not only does our model allow for the possible default of the option issuer prior to the option's maturity, but also considers the correlations among the option issuer's total assets, the underlying stock, and the default-free zero coupon bond. We further tailor-make a specific credit-linked option for hedging the default risk of the option issuer. The numerical results show that the default risk of the option issuer significantly reduces the option values, and the vulnerable option values may be remarkably overestimated in the case where the default can occur only at the maturity of the option.     

Statistical arbitrage in the Black–Scholes framework

In this study, we prove the existence of statistical arbitrage opportunities in the Black–Scholes framework by considering trading strategies that consist of borrowing at the risk-free rate and taking a long position in the stock until it hits a deterministic barrier level. We derive analytical formulas for the expected value, variance and probability of loss for the discounted cumulative trading profits. The statistical arbitrage condition is derived in the Black–Scholes framework, which imposes a constraint on the Sharpe ratio of the stock. Furthermore, we verify our theoretical results via extensive Monte Carlo simulations.     

Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black–Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.     

The Instantaneous Volatility and the Implied Volatility Surface for a Generalized Black–Scholes Model

Takaoka (Asia–Pacific Financial Markets 11:431–444, 2004) proposed a generalization of the Black–Scholes stock price model by taking a weighted average of geometric Brownian motions of different variance parameters. The model can be classified as a local volatility model, though its local volatility function is not explicitly given. In the present paper, we prove some properties concerning the instantaneous volatility process, the implied volatility curve, and the local volatility function of the generalized model. Some numerical computations are also carried out to confirm our results.     

A new closed-form solution as an extension of the Black–Scholes formula allowing smile curve plotting

In this paper, as a generalization of the Black–Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J-model. This closed-form solution is based on a new stochastic process, called the J-process, which is an extension of the Wiener process satisfying the martingale property. The J-process is based on a new statistical law called the J-law, which is an extension of the normal law. The J-law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J-model can also serve to determine the implied volatility by inverting the J-formula and can be used to price other kinds of options such as American options.     

Superconvergence of the finite element solutions of the Black–Scholes equation

We investigate the performances of the finite element method in solving the Black–Scholes option pricing model. Such an analysis highlights that, if the finite element method is carried out properly, then the solutions obtained are superconvergent at the boundaries of the finite elements. In particular, this is shown to happen for quadratic and cubic finite elements, and for the pricing of European vanilla and barrier options. To the best of our knowledge, lattice-based approximations of the Black–Scholes model that exhibit nodal superconvergence have never been observed so far, and are somehow unexpected, as the solutions of the associated partial differential problems have various kinds of irregularities.     

Option pricing with Weyl–Titchmarsh theory

In the Black–Merton–Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schrödinger spectral problem, which has been well studied in quantum mechanics. With Weyl–Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.     

Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, the Black–Scholes PDE associated with a European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knock-out barrier moves to infinity under suitable conditions.     

New analytical option pricing models with Weyl–Titchmarsh theory

Abstract

In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal–dual linear Monte Carlo algorithm that allows for efficient simulation of the lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of the primal–dual Monte Carlo algorithm for standard Bermudan options proposed by Schoenmakers et al. [SIAM J. Finance Math., 2013, 4, 86–116] to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers. At each level, the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. The algorithm also provides approximate continuation functions that may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull and White model setup. The simple model choice allows for comparison of the computed price bounds with the exact price obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.     

First steps towards an equilibrium theory for Lévy financial markets

For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Lévy process. The agent is allowed to consume a lump at the terminal date; before that, only flow consumption is allowed. The agent’s utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require for our equilibrium existence result that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium asset price processes depend on the agent’s risk aversion (through the felicity functions). Even in our simple, straightforward economy, the equilibrium asset price processes will essentially only be (stochastic) exponential Lévy processes when they are already geometric Brownian motions. Our equilibrium asset pricing formulae can also be modified to obtain explicit equilibrium derivative pricing formulae.     

Prospect theory–based portfolio optimization: an empirical study and analysis using intelligent algorithms

The behaviourally based portfolio selection problem with investor’s loss aversion and risk aversion biases in portfolio choice under uncertainty is studied. The main results of this work are: developed heuristic approaches for the prospect theory model proposed by Kahneman and Tversky in 1979 as well as an empirical comparative analysis of this model and the index tracking model. The crucial assumption is that behavioural features of the prospect theory model provide better downside protection than traditional approaches to the portfolio selection problem. In this research the large-scale computational results for the prospect theory model have been obtained for real financial market data with up to 225 assets. Previously, as far as we are aware, only small laboratory tests (2–3 artificial assets) have been presented in the literature. In order to investigate empirically the performance of the behaviourally based model, a differential evolution algorithm and a genetic algorithm which are capable of dealing with a large universe of assets have been developed. Specific breeding and mutation, as well as normalization, have been implemented in the algorithms. A tabulated comparative analysis of the algorithms’ parameter choice is presented. The prospect theory model with the reference point being the index is compared to the index tracking model. A cardinality constraint has been implemented to the basic index tracking and the prospect theory models. The portfolio diversification benefit has been found. The aggressive behaviour in terms of returns of the prospect theory model with the reference point being the index leads to better performance of this model in a bullish market. However, it performed worse in a bearish market than the index tracking model. A tabulated comparative analysis of the performance of the two studied models is provided in this paper for in-sample and out-of-sample tests. The performance of the studied models has been tested out-of-sample in different conditions using simulation of the distribution of a growing market and simulation of the t-distribution with fat tails which characterises the dynamics of a decreasing or crisis market.     

Curbing domestic violence: instantiating C–K theory with formal concept analysis and emergent self-organizing maps

We propose a human-centred process for knowledge discovery from unstructured text that makes use of formal concept analysis and emergent self-organizing maps. The knowledge discovery process is conceptualized and interpreted as successive iterations through the concept–knowledge (C–K) theory design square. To illustrate its effectiveness, we report on a real-life case study of using the process at the Amsterdam–Amstelland police in the Netherlands aimed at distilling concepts to identify domestic violence from the unstructured text in actual police reports. The case study allows us to show how the process was not only able to uncover the nature of a phenomenon such as domestic violence, but also enabled analysts to identify many types of anomaly in the practice of policing. We will illustrate how the insights obtained from this exercise resulted in major improvements in the management of domestic violence cases. Copyright © 2010 John Wiley & Sons, Ltd.     

Testing the Prebisch–Singer hypothesis since 1650: Evidence from panel techniques that allow for multiple breaks

In this paper, we re-examine two important aspects of the dynamics of relative primary commodity prices, namely the secular trend and the short run volatility. To do so, we employ 25 series, some of them starting as far back as 1650 and powerful panel data stationarity tests that allow for endogenous multiple structural breaks. Results show that all the series are stationary after allowing for endogenous multiple breaks. Test results on the Prebisch–Singer hypothesis, which states that relative commodity prices follow a downward secular trend, are mixed but with a majority of series showing negative trends. We also make a first attempt at identifying the potential drivers of the structural breaks. We end by investigating the dynamics of the volatility of the 25 relative primary commodity prices also allowing for endogenous multiple breaks. We describe the often time-varying volatility in commodity prices and show that it has increased in recent years.     

Multi-asset portfolio optimization and out-of-sample performance: an evaluation of Black–Litterman,mean-variance,and naïve diversification approaches

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems of mean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman model and implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices, covering the period from January 1993 to December 2011. We test its out-of-sample performance relative to other asset allocation models and find that Black–Litterman optimized portfolios significantly outperform naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better than mean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios, even after controlling for different levels of risk aversion, investment constraints, and transaction costs. The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversification across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixed return estimates that incorporate the reliability of return predictions, smaller estimation errors, and lower turnover.     

A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro-differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman–Kac representation of variational solutions to partial integro-differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes. We allow a wide range of underlying stochastic processes, comprising processes with Brownian part as well as a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and \(\alpha\)-semistable Lévy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial integro-differential equation, our results provide a rigorous basis for numerous applications in financial mathematics and in probability theory. We implement a Galerkin scheme to solve the corresponding pricing equation numerically and illustrate the effect of a killing rate.     

Monetary control with an exchange rate objective: The bank of Japan, 1973–86

This paper analyzes the switch in monetary control procedures by the Bank of Japan toward a so-called ‘money-focused’ monetary policy in the mid-1970s. The extent to which monetary control under the new regime has been limited by an exchange rate objective is examined. Through estimation of an explicit Bank of Japan (BoJ) reaction function, we find evidence that the BoJ operating instrument has been systematically manipulated with a view to maintain short-term money control but that this objective has often been dominated by an attempt to moderate yen– dollar exchange rate fluctuations.     

Base independence in the analysis of tax policy effects: with an application to Norway 1992–2004

The analysis contrasts results of two recently expounded microlevel data approaches to derive robust intertemporal characterizations of redistributional effects of income tax schedules; the fixed-income procedure of Kasten et al. (Tax progressivity and Income Inequality, Cambridge University Press, 1994) and the transplant-and-compare method of Dardanoni and Lambert (J. Public Econ. 86:99–122, 2002). Our study is normative in that the Blackorby and Donaldson (Can. J. Econ. 17:683–694, 1984) index of tax progressivity is employed. This enables contributions from vertical redistribution and horizontal inequity also to be assessed, using for the latter one classical measure and one no reranking measure. When the competing methodologies are applied to Norwegian data for 1992–2004, their respective strengths and weaknesses are revealed. The transplant-and-compare procedure is found to have a number of advantages.      

A generalised Gerber–Shiu measure for Markov-additive risk processes with phase-type claims and capital injections

In this paper we consider a risk reserve process where the arrivals (either claims or capital injections) occur according to a Markovian point process. Both claim and capital injection sizes are phase-type distributed and the model allows for possible correlations between these and the inter-claim times. The premium income is modelled by a Markov-modulated Brownian motion which may depend on the underlying phases of the point arrival process. For this risk reserve model we derive a generalised Gerber–Shiu measure that is the joint distribution of the time to ruin, the surplus immediately before ruin, the deficit at ruin, the minimal risk reserve before ruin, and the time until this minimum is attained. Numeral examples illustrate the influence of the parameters on selected marginal distributions.