Pricing Of Financial Derivatives Research Articles

Preconditioned iterative methods for fractional diffusion models in finance

Most recent qualitative models for financial assets assume that the dynamics of underlying equity prices follows a jump or Lévy process. It has been evident that some most intricate characteristics of such dynamics can be captured by CGMY and KoBoL procedures. The prices of financial derivatives with such models satisfy fractional partial differential equations or partial integro-differential equations. This study focuses at aforementioned fractional equations and discretizes them via a monotone Crank–Nicolson procedure. A spatial extrapolation strategy is introduced to ensure an overall second-order accuracy in approximations. Preconditioned conjugate gradient normal residual methods are incorporated for solving resulted linear systems. Numerical examples are given to illustrate the accuracy and efficiency of the novel computational approaches implemented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1382–1395, 2015

A Journey into the Dark Arts of Quantitative Finance

A Journey into the Dark Arts of Quantitative Finance

Finite Difference Method: A Brief Study

This is a short article summarizing different finite difference schemes for the numerical solution of partial differential equation in application of pricing financial derivatives. The solution of partial difference equation (PDE) using finite difference method (FDM) with both uniform and non-uniform grids are presented here. This article mainly focuses on the solution of 1D and 2D PDE. The solution of 1D PDE using implicit, explicit and Crank-Nicolson scheme and 2D PDE using alternate direction implicit (ADI), alternate direction implicit operator splitting scheme (ADI-OS) and alternate direction explicit (ADE) is discussed here. There are various literature available for an extensive study of the finite difference methods. This article is intended to be written for a quick review of various finite difference schemes to solve a PDE.

Numerical Simulation of Black-Scholes Model by Finite Difference Method

Option is the typical representative of financial derivatives, and this paper is focused on the valuation problem of Option. Based on the Black-Scholes Pricing model which had far-reaching influence on the pricing of financial derivatives, researched its theoretical basis and derivation process, and then get the numerical solution via finite difference method and image simulation. And it also includes the part of empirical studies. In research, ZTR and HQ is chosen and analyzed, in order to get the pricing of European put option.

Intrinsic Prices of Risk

We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets. We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena. In a continuous time framework, we bring together the notion of intrinsic risk and the theory of change of measures to derive a probability measure, namely risk-subjective measure, for evaluating contingent claims. This paper is a modest attempt to prove that measure of intrinsic risk is a crucial ingredient for explaining these phenomena, and in consequence proposes a new approach to pricing and hedging financial derivatives. By adapting theoretical knowledge to practical applications, we show that our approach is consistent and robust, compared with the standard risk-neutral approach.

Good Path Generation Methods in Quasi-Monte Carlo for Pricing Financial Derivatives

The quasi-Monte Carlo (QMC) method is a powerful numerical tool for pricing complex derivative securities, whose accuracy is affected by the smoothness of the integrands. The payoff functions of many financial derivatives involve two types of nonsmooth factors: an indicator function (called jump structure) and a positive part of a smooth function (called kink structure). This paper develops a good path generation method (PGM) for recovering the superiority of the QMC method on problems involving multiple such structures. This is achieved by realigning these structures such that the associated nonsmooth surfaces are parallel to as many coordinate axes as possible. The proposed method has the advantage of addressing different structures according to their importance. We also offer a systematic measurement of different structures for quantifying and then ranking their importance. Numerical experiments demonstrate that the proposed method is more efficient than traditional PGMs for pricing exotic options, suc...

High Performance Computing for a Financial Application Using Fast Fourier Transform

Fast Fourier Transform (FFT) has been used in many scientific and engineering applications. In the current study, we have applied the FFT for a novel application in finance. We have improved a mathematical model of Fourier transform technique for pricing financial derivatives to help design an effective parallel algorithm. We have then developed a new parallel algorithm for FFT using a swapping technique that exploits data locality. We have analyzed our algorithm theoretically and have reported the significance of the new algorithm. We have implemented our algorithm on 20 node SunFire 6800 high performance computing system and compared the new algorithm with the traditional Cooley-Tukey algorithm both as stand alone comparison of the performance and in relation to our theoretical analysis and showed higher efficiency of our algorithm. We have presented the computed option values for various strike prices with a proper selection of strike-price spacing to ensure fine-grid integration for FFT computation as well as to maximize the number of strikes lying in the desired region of the asset price.

Randomized Binomial Tree and Pricing of American-Style Options

Randomized binomial tree and methods for pricing American options were studied. Firstly, both the completeness and the no-arbitrage conditions in the randomized binomial tree market were proved. Secondly, the description of the node was given, and the cubic polynomial relationship between the number of nodes and the time steps was also obtained. Then, the characteristics of paths and storage structure of the randomized binomial tree were depicted. Then, the procedure and method for pricing American-style options were given in a random binomial tree market. Finally, a numerical example pricing the American option was illustrated, and the sensitivity analysis of parameter was carried out. The results show that the impact of the occurrence probability of the random binomial tree environment on American option prices is very significant. With the traditional complete market characteristics of random binary and a stronger ability to describe, at the same time, maintaining a computational feasibility, randomized binomial tree is a kind of promising method for pricing financial derivatives.

A Modified Empirical Martingale Simulation for Financial Derivative Pricing

This article extends the empirical martingale simulation (EMS) method from using a risk-neutral measure to using a dynamic measure for financial derivative pricing. Although the EMS is shown to be capable of obtaining consistent estimate of financial derivative prices in a more efficient way than the standard Monte Carlo simulation procedure, it can proceed only under a risk-neutral framework. In practice, however, it is cumbersome to obtain the explicit expression of a risk-neutral model when dealing with a complex model. To alleviate this difficulty, we compute the financial derivative prices under the dynamic model and impose the martingale property on the simulated sample paths of both the change of measure process and the underlying asset prices under the dynamic P measure. Hence, we call this modification the empirical P-martingale simulation (EPMS). The strong consistency of the EPMS is established and its efficiency is performed by simulation in the GARCH framework. Simulation results shows that EPMS has the similar variance reduction as the EMS method in option pricing if the risk-neutral model can be obtained, and is more efficient than the standard Monte Carlo simulation in most cases.

Exact simulation pricing with Gamma processes and their extensions

Exact path simulation of the underlying state variable is of great practical importance in simulating prices of financial derivatives or their sensitivities when there are no analytical solutions for their pricing formulas. However, in general, the complex dependence structure inherent in most nontrivial stochastic volatility (SV) models makes exact simulation difficult. In this paper, we present a nontrivial SV model that parallels the notable Heston SV model in the sense of admitting exact path simulation as studied by Broadie and Kaya. The instantaneous volatility process of the proposed model is driven by a Gamma process. Extensions to the model including superposition of independent instantaneous volatility processes are studied. Numerical results show that the proposed model outperforms the Heston model and two other L\'evy driven SV models in terms of model fit to the real option data. The ability to exactly simulate some of the path-dependent derivative prices is emphasized. Moreover, this is the first instance where an infinite-activity volatility process can be applied exactly in such pricing contexts.

A new hybrid Monte Carlo simulation for Asian options pricing

The aim of this paper is to present a new hybrid algorithm for pricing financial derivatives in the arithmetic Asian options. In this paper, two variance reduction techniques are combined, the multiple control variates (MCV) and the antithetic variates (AV). We propose an efficient algorithm for pricing arithmetic Asian options based on the AV and the MCV procedures. A detailed numerical study illustrates the efficiency of the proposed algorithm.

Fast and accurate long-stepping simulation of the Heston stochastic volatility model

In this paper, we present three new discretization schemes for the Heston stochastic volatility model two schemes for simulating the variance process and one scheme for simulating the integrated variance process conditional on the initial and the end-point of the variance process. Instead of using a short timestepping approach to simulate the variance process and its integral, these new schemes evolve the Heston process accurately over long steps without the need to sample the intervening values. Hence, prices of financial derivatives can be evaluated rapidly using our new approaches.

A new tree method for pricing financial derivatives in a regime-switching mean-reverting model

This paper develops a new tree method for pricing financial derivatives in a regime-switching mean-reverting model. The tree achieves full node recombination and grows linearly as the number of time steps increases. Conditions for non-negative branch probabilities are presented. The weak convergence of the discrete tree approximations to the continuous regime-switching mean-reverting process is established. To illustrate the application in mathematical finance, the recombining tree is used to price commodity options and zero-coupon bonds. Numerical results are provided and compared.

Analyzing Second and Third Moments of Electricity Prices

The volatility of electricity prices is the important information for the risk management of electricity markets and the pricing of electricity financial derivatives. A multicycle GARCH-M model based on Gram-Charlier series expansion of the normal probability density function is proposed, in which the time trend, time-varying variance, time-vaying skewness, multicycles and the relationship among load and spot price can be fully taken into account. The numerical example based on the historical data of the Pennsylvania-New Jersey-Maryland electricity market shows that the time-varying variance and squared system load have a significant effect on the average daily electricity prices, there exist volatility clustering and weekly, semi-monthly, monthly, bimonthly, quarterly and semi-annual periods, and the second and third moments of electricity price series manifest the clear synchronous time-varying characteristics.

Numerical pricing of financial derivatives using Jain’s high-order compact scheme

Purpose: This paper develops new fast and accurate computational schemes for pricing European and American bond options under generalised Chan-Karoyli-Longstaff-Sanders term structure models. Methods: We use high-order compact discretisations of the pricing equations and an operator splitting method for American options. Results: Highly accurate numerical solutions can be computed using relatively coarse grid sizes and numerical solutions exhibiting fourth-order convergence are obtained for bond and bond option prices. The scheme is also stable and efficient for pricing financial problems with time dependent parameters. Conclusions: The new schemes are efficient alternatives to schemes based on the Crank-Nicolson discretisation for the pricing of interest rate derivatives.

Parallel random number generators in Monte Carlo derivative pricing: An application-based test

Parallel pseudorandom number generators (PPRNG) that satisfy classical statistical tests may still demonstrate intra-stream and inter-stream correlations in real life applications. In order to investigate the suitability of a PPRNG for use in Monte Carlo pricing of financial derivatives, an application-based test is proposed to evaluate the bias and the standard error of the mean (SE) associated with the PPRNG as a gauge of intra-stream and inter-stream correlations respectively. This test involves estimating the price of a vanilla European call option via Monte Carlo simulation, where the asset price at maturity is estimated by propagating the Black–Scholes stochastic differential equation via the Euler–Maruyama discretization scheme. The mean and SE profiles of the numerical results based on three PPRNG libraries (RngStream, TRNG and SPRNG) that implement parallel random numbers via sequence splitting strategies (RngStream and TRNG) and parameterization strategy (SPRNG) are compared. In terms of the bias and SE profiles, the best performing PPRNG constructed using the sequence splitting strategy is comparable to that constructed using parameterization, both use multiple recursive generators in their kernel.

Model risk of the implied GARCH-normal model

Model risk causes significant losses in financial derivative pricing and hedging. Investors may undertake relatively risky investments due to insufficient hedging or overpaying implied by flawed models. The GARCH model with normal innovations (GARCH-normal) has been adopted to depict the dynamics of the returns in many applications. The implied GARCH-normal model is the one minimizing the mean square error between the market option values and the GARCH-normal option prices. In this study, we investigate the model risk of the implied GARCH-normal model fitted to conditional leptokurtic returns, an important feature of financial data. The risk-neutral GARCH model with conditional leptokurtic innovations is derived by the extended Girsanov principle. The option prices and hedging positions of the conditional leptokurtic GARCH models are obtained by extending the dynamic semiparametric approach of Huang and Guo [Statist. Sin., 2009, 19, 1037–1054]. In the simulation study we find significant model risk of the implied GARCH-normal model in pricing and hedging barrier and lookback options when the underlying dynamics follow a GARCH-t model.

Non-parametric partial importance sampling for financial derivative pricing

Importance sampling is a promising variance reduction technique for Monte Carlo simulation-based derivative pricing. Existing importance sampling methods are based on a parametric choice of the proposal. This article proposes an algorithm that estimates the optimal proposal non-parametrically using a multivariate frequency polygon estimator. In contrast to parametric methods, non-parametric estimation allows for close approximation of the optimal proposal. Standard non-parametric importance sampling is inefficient for high-dimensional problems. We solve this issue by applying the procedure to a low-dimensional subspace, which is identified through principal component analysis and the concept of the effective dimension. The mean square error properties of the algorithm are investigated and its asymptotic optimality is shown. Quasi-Monte Carlo is used for further improvement of the method. It is easy to implement, particularly it does not require any analytical computation, and it is computationally very efficient. We demonstrate through path-dependent and multi-asset option pricing problems that the algorithm leads to significant efficiency gains compared with other algorithms in the literature.

MCMC 방법을 이용한 ARMA-GARCH 모형에서의 예측 방법 연구

The volatility is one of most important parameters in the areas of pricing of financial derivatives an measuring risks arising from a sudden change of economic circumstance. We propose a Bayesian approach to estimate the volatility varying with time under a linear model with ARMA(p, q)-GARCH(r, s) errors. This Bayesian estimate of the volatility is compared with the ML estimate. We also present the probability of existence of the unit root in the GARCH model.

Quasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction

Quasi-Monte Carlo (QMC) methods are playing an increasingly important role in the pricing of complex financial derivatives. For models in which the prices of the underlying assets are driven by Brownian motions, the performance of QMC methods is known to depend crucially on the construction of Brownian motions. This paper focuses on the impact of various constructions. Although the Brownian bridge (BB) construction often yields very good results, as Papageorgiou pointed out, there are financial derivatives for which the BB construction performs badly [Papageorgiou, A. 2002. The Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. J. Complexity 18(1) 171–186]. In this paper we first extend Papageorgiou's analysis to establish an equivalence principle: if the BB construction (or any other construction) is the preferred construction for a particular financial derivative, then for any other construction, there is another financial derivative for which the latter construction is the preferred one. In this sense, all methods of construction are equivalent and no method is consistently superior to others; it all depends on the particular financial derivative. We then show how to find a good construction for a particular class of financial derivatives. In practice, our strategy is to find a good construction for an “easy” problem and then apply it to more complicated problems related to the easy one. This strategy is applied to the arithmetic Asian options (including Bermudan Asian options) based on the weighted average of the stock prices. We do this by studying a simpler problem, namely, the geometric Asian option, for which the best construction is easily available, and applying it to the arithmetic Asian option. Numerical experiments confirm the success of this strategy: whereas in QMC all the common methods (the standard method, BB, and principal component analysis) may lose their power in some situations, the new method behaves very well in all cases. Further large variance reduction can be achieved in combination with a control variate. The new method can be interpreted as a practical way of reducing the effective dimension for some class of functions.