Multivariate option pricing using copulae

The complexity of financial products significantly increased in the past 10 years. In this paper, we investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. Our approach assumes that the underlying assets evolve as dependent GARCH(1, 1) processes. The dependence among the assets is modeled using a copula based on pair‐copula constructions. Unlike most previous studies on this topic, we do not assume that the dependence observed between historical asset prices is similar to the dependence under the risk‐neutral probability. The method is illustrated with US market data on basket options written on two or three international indices. Copyright © 2012 John Wiley & Sons, Ltd.     

A jump-diffusion model for option pricing under fuzzy environments

Owing to fluctuations in the financial markets from time to time, the rate λ of Poisson process and jump sequence {V_i} in the Merton’s normal jump-diffusion model cannot be expected in a precise sense. Therefore, the fuzzy set theory proposed by Zadeh [Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338-353] and the fuzzy random variable introduced by Kwakernaak [Kwakernaak, H., 1978. Fuzzy random variables I: Definitions and theorems. Inform. Sci. 15, 1-29] and Puri and Ralescu [Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409-422] may be useful for modeling this kind of imprecise problem. In this paper, probability is applied to characterize the uncertainty as to whether jumps occur or not, and what the amplitudes are, while fuzziness is applied to characterize the uncertainty related to the exact number of jump times and the jump amplitudes, due to a lack of knowledge regarding financial markets. This paper presents a fuzzy normal jump-diffusion model for European option pricing, with uncertainty of both randomness and fuzziness in the jumps, which is a reasonable and a natural extension of the Merton [Merton, R.C., 1976. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125-144] normal jump-diffusion model. Based on the crisp weighted possibilistic mean values of the fuzzy variables in fuzzy normal jump-diffusion model, we also obtain the crisp weighted possibilistic mean normal jump-diffusion model. Numerical analysis shows that the fuzzy normal jump-diffusion model and the crisp weighted possibilistic mean normal jump-diffusion model proposed in this paper are reasonable, and can be taken as reference pricing tools for financial investors.     

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.     

Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.     

Robust option pricing

In this paper, we combine robust optimization and the idea of ?$?$-arbitrage to propose a tractable approach to price a wide variety of options. Rather than assuming a probabilistic model for the stock price dynamics, we assume that the conclusions of probability theory, such as the central limit theorem, hold deterministically on the underlying returns. This gives rise to an uncertainty set that the underlying asset returns satisfy. We then formulate the option pricing problem as a robust optimization problem that identifies the portfolio which minimizes the worst case replication error for a given uncertainty set defined on the underlying asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-asset, American and Asian options using linear optimization; and thus the computational complexity of our approach scales polynomially with the number of assets and with time to expiry and (b) modeling flexibility illustrated by our ability to model different kinds of options, various levels of risk aversion among investors, transaction costs, shorting constraints and replication via option portfolios.     

Cpk index estimation using fuzzy numbers

Process capacity indices (PCIs) were developed and have been successfully used by companies to compete in and dominate the high-profit markets by improving the quality and the productivity since the past two decades. There is an essential assumption, in the conventional application, wherein the output process measurements are precise and distributed as normal random variables. Since the assumption of normal distribution is untenable, errors can occur if the C_pk index is computed using non-normal data. In the present study, we address the situation that the output of data from measurement of the quality of a product is insufficiently precise or scarce. This is possible when the quality measurement refers to the decision-maker’s subjective determination. In such a situation, the linguistic variable that is easier to capture the decision-maker’s subjective perception is applied to construct the PCI C_pk. The present approach can mitigate the effect when the normal assumption is inappropriate and extends the application of C_pk index.     

Comparison of fuzzy numbers using a fuzzy distance measure

A new approach for ranking fuzzy numbers based on a distance measure is introduced. A new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first, and then it is used to formulate the distance measure for fuzzy numbers. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches.     

Bivariate option pricing with copulas

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Fréchet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.     

An improved fuzzy time series forecasting method using trapezoidal fuzzy numbers

One of the major drawbacks of the existing fuzzy time series forecasting models is the fact that they only provide a single-point forecasted value just like the output of the traditional time series methods. Hence, they cannot provide a decision analyst more useful information. The aim of this present research is to design an improved fuzzy time series forecasting method in which the forecasted value will be a trapezoidal fuzzy number instead of a single-point value. Furthermore, the proposed method may also increase the forecasting accuracy. Two numerical data sets were used to illustrate the proposed method and compare the forecasting accuracy with three fuzzy time series methods. The results of the comparison indicate that the proposed method can generate forecasting values that are more accurate.     

Comparing and ranking fuzzy numbers using ideal solutions

This paper presents a new approach for comparing and ranking fuzzy numbers in a simple manner in decision making under uncertainty. The concept of ideal solutions is sensibly used, and a distance-based similarity measure between fuzzy numbers is appropriately adopted for effectively determining the overall performance of each fuzzy number in comparing and ranking fuzzy numbers. As a result, all the available information characterizing a fuzzy number is fully utilized, and both the absolute position and the relative position of fuzzy numbers are adequately considered, resulted in consistent rankings being produced in comparing and ranking fuzzy numbers. The approach is computationally simple and its underlying concepts are logically sound and comprehensible. A comparative study is conducted on the benchmark cases in the literature that shows the proposed approach compares favorably with other approaches examined.     

障碍平方期权的定价

本讨论了一种变异期权——收益结构为平方的障碍期权，在股票价格服从几何布朗运动的模型下，由带单侧吸收壁的布朗运动的密度和分布函数得到连续障碍平方期权的定价公式．     

On the theory of option pricing

The objective of this article is to provide an axiomatic framework in order to define the concept of value function for risky operations for which there is no market. There is a market for assets, whose prices are characterized as stochastic processes. The method consists of constructing a portfolio of these assets which will mimic the risks involved in the operation. We follow the terminology of the theory of options although the set-up goes beyond that particular problem.     

美式期权定价的数值方法

介绍了定价美式期权的几种常见数值方法.对最近几年的主要研究成果做了简单的介绍和比较,并给出了数值算例.特别回顾了美式期权定价的蒙特卡罗模拟加速方法.     

Markov-modulated jump-diffusions for currency option pricing

This paper introduces dynamic models for the spot foreign exchange rate with capturing both the rare events and the time-inhomogeneity in the fluctuating currency market. For the rare events, we use a compound Poisson process with log-normal jump amplitude to describe the jumps. As for the time-inhomogeneity in the market dynamics, we particularly stress the strong dependence of the domestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on the time-varying sovereign ratings in the currency market. The time-varying ratings are formulated by a continuous-time finite-state Markov chain. Based on such a spot foreign exchange rate dynamics, we then study the pricing of some currency options. Here we will adopt a so-called regime-switching Esscher transform to identify a risk-neutral martingale measure. By determining the regime-switching Esscher parameters we then get an integral expression on the prices of European-style currency options. Finally, numerical illustrations are given.     

Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform

We show that the generalized Fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi-dimensional option pricing. We derive a general algorithm, including a transformation of the Black–Scholes equation into the heat equation, that can be used in any number of dimensions. Numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes. Furthermore, the gain increases with the number of dimensions.     

Exponential Rosenbrock integrators for option pricing

In this paper, we are concerned with the time integration of differential equations modeling option pricing. In particular, we consider the Black-Scholes equation for American options. As an alternative to existing methods, we present exponential Rosenbrock integrators. These integrators require the evaluation of the exponential and related functions of the Jacobian matrix. The resulting methods have good stability properties. They are fully explicit and do not require the numerical solution of linear systems, in contrast to standard integrators. We have implemented some numerical experiments in Matlab showing the reliability of the new method.     

Equilibrium pricing bounds on option prices

We consider the problem of valuing European options in a complete market but with incomplete data. Typically, when the underlying asset dynamics is not specified, the martingale probability measure is unknown. Given a consensus on the actual distribution of the underlying price at maturity, we derive an upper bound on the call option price by putting two kinds of restrictions on the pricing probability measure. First, we put a restriction on the second risk-neutral moment of the underlying asset terminal value. Second, from equilibrium pricing arguments one can put a monotonicity restriction on the Radon-Nikodym density of the pricing probability with respect to the true probability measure. This density is restricted to be a nonincreasing function of the underlying price at maturity. The bound appears then as the solution of a constrained optimization problem and we adopt a duality approach to solve it. Explicit bounds are provided for the call option. Finally, we provide a numerical example.     

Matching asymptotics in path-dependent option pricing

The valuation of path-dependent options in finance creates many interesting mathematical challenges. Among them are a large Delta and Gamma near the expiry leading to a big error in pricing those exotic options as well as European vanilla options. Also, the higher order corrections of the asymptotic prices of the derivatives in some stochastic volatility models are difficult to be evaluated. In this paper we use the method of matched asymptotic expansions to obtain more practical values of lookback and barrier option prices near the expiry. Our results verify that matching asymptotics is a useful tool for PDE methods in path-dependent option pricing.     

Ranking multi-criterion river basin planning alternatives using fuzzy numbers

The methodology proposed by Anand Raj and Nagesh Kumar [5] to rank the river basin planning and development alternatives under multi-criterion environment using fuzzy numbers is applied to a case study. The purpose is to find the most suitable planning of reservoirs with their associated purposes aimed at the development of one of the major peninsular river basins (Krishna river basin) in India. A set of 7 alternative systems with 8 main objectives, which are further subdivided into 18 criteria, are considered for ordering or ranking them employing the opinion (preference structure) of three experts: an acadamician, a field engineer and an official from Ministry of Water Resources, using fuzzy numbers. The fuzzy weights (w_i) of alternatives (A_i) are computed using standard fuzzy arithmetic. The concepts of maximizing set and minimizing set are introduced to decide total utility or order value of each of the alternatives.