Extremal measures and hedging in American options

We establish existence conditions for extremal probability measures, study their properties, and consider applications of such measures for solving the perfect hedging problem for American options on incomplete “frictionless” markets with finite horizon. We develop an algorithm for computing an American option and solve a corresponding new example with this algorithm.     

A new radial basis functions method for pricing American options under Merton's jump-diffusion model

A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.     

A new numerical method for pricing fixed-rate mortgages with prepayment and default options

In this paper we consider the valuation of fixed-rate mortgages including prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The mathematical model to obtain the value of the contract is posed as a free boundary problem associated to a partial differential equation (PDE) model. The equilibrium contract rate is determined by using an iterative process. Moreover, appropriate numerical methods based on a Lagrange–Galerkin discretization of the PDE, an augmented Lagrangian active set method and a Newton iteration scheme are proposed. Finally, some numerical results to illustrate the performance of the numerical schemes, as well as the qualitative and quantitative behaviour of solution and the optimal prepayment boundary are presented.     

Valuation of American options via basis functions

After a brief review of recent developments in the pricing and hedging of American options, this paper modifies the basis function approach to adaptive control and neuro-dynamic programming, and applies it to develop: 1) nonparametric pricing formulas for actively traded American options and 2) simulation-based optimization strategies for complex over-the-counter options, whose optimal stopping problems are prohibitively difficult to solve numerically by standard backward induction algorithms because of the curse of dimensionality. An important issue in this approach is the choice of basis functions, for which some guidelines and their underlying theory are provided.     

Asynchronous grid computation for American options derivatives

In relation with the mathematics of financial applications, the present study deals with the solution of the time dependent obstacle problem defined in a three-dimensional domain; this problem arises in the pricing of American options derivatives. In order to solve very quickly large scale algebraic systems derived from the discretization of the obstacle problem, the parallelization of the numerical algorithm is necessary. So, we present parallel synchronous, and more generally asynchronous, iterative algorithms to solve this problem. For the considered problem, arguments implying the convergence of parallel synchronous and asynchronous algorithms are given in a general framework. Finally, computational experiments on GRID’5000, the French national grid, are presented and analyzed. They allow us to compare both synchronous and asynchronous versions with local and distributed clusters and to show the interest of such methods in the context of grid computing.     

Evolutionary computation and the vega risk of American put options

While European style options and American call options can be priced using analytical exact valuation models, closed-form solutions for the valuation of American puts have not yet been derived. The American put price as well as the corresponding greeks (e.g., delta, gamma, vega) can be calculated using numerical procedures or analytical approximations. We use a parallel implementation of the genetic programming approach and derive analytical approximations for determining the vega of an American put option because calculating vegas numerically requires even more computational effort than determining deltas or gammas. Applying our approximations to experimental data sets we can show that the genetically derived approximations outperform other approximations based on frequently used American put pricing formulas.     

Pricing American bond options using a penalty method

We develop a novel numerical method to price American options on a discount bond under the Cox–Ingrosll–Ross (CIR) model which is governed by a partial differential complementarity problem. We first propose a penalty approach to this complementarity problem, resulting in a nonlinear partial differential equation (PDE). To numerically solve this nonlinear PDE, we develop a novel fitted finite volume method for the spatial discretization, coupled with a fully implicit time-stepping scheme. We show that this full discretization scheme is consistent, stable and monotone, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the discretized nonlinear system, we design an iterative method and prove that the method is convergent. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of our methods.     

Multiscale methods for the valuation of American options with stochastic volatility

This paper deals with the efficient valuation of American options. We adopt Heston's approach for a model of stochastic volatility, leading to a generalized Black–Scholes equation called Heston's equation. Together with appropriate boundary conditions, this can be formulated as a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid method based on linear finite elements in space and display corresponding numerical experiments.     

Valuing credit default swap in a non-homogeneous semi-Markovian rating based model

In this paper, we use a discrete time non-homogeneous semi-Markov model for the rating evolution of the credit quality of a firm C (see D’Amico, Janssen, and Manca Proceedings of the II international workshop in applied probablity, 2004a) and we determine the credit default swap spread for a contract between two parties, A and B that, respectively, sell and buy a protection about the failure of the firm C. We work both in the case of deterministic and stochastic recovery rate. We also highlight the link between credit risk and reliability theory.     

The valuation of multidimensional American real options using the LSM simulation method

In this paper we show how a multidimensional American real option may be solved using the LSM simulation method originally proposed by Longstaff and Schwartz [2001, The Review of the Financial Studies 14(1): 113–147] for valuing a financial option and how this method can be used in a complex setting. We extend a well-known natural resource real option model, initially solved using finite difference methods, to include a more realistic three-factor stochastic process for commodity prices, more in line with current research. Numerical results show that the procedure may be successfully used for multidimensional models, expanding the applicability of the real options approach.     

Financial innovation: Credit default hybrid model for SME lending

Credit risk evaluation is an integral part of any lending process, and even more so for financial institutions involved in lending to SMEs. The importance of credit scoring has increased recently because of the financial crisis and increased capital requirements for banks. There are, however, only few studies that develop credit coring models for SME lending. The objective of this study is to introduce a novel, more accurate credit risk estimation approach for SMEs business lending. Based on traditional statistical methods and recent artificial intelligence (AI) techniques, we proposed a hybrid model which combines the logistic regression approach and artificial neural networks (ANN). In order to test the effectiveness and feasibility of the proposed hybrid model, we use the data of Finnish SMEs from the fiscal years 2004 to 2012. Our results suggest that the proposed ANN/logistic hybrid model is more accurate than either of the initial models ANN or logistic regression. This improvement in the accuracy of the credit scoring model decreases evaluation errors and has thereby many potential practical implications. First of all, a more accurate credit scoring model can result in better performance of the whole SME loan portfolio. Second, it can also result in lower capital requirements from the banks perspective and lower interest rates from the individual firm's perspective. Combined, these effects will enhance the banks competitiveness in the market for SME loans.     

Radial-basis-function-based finite difference operator splitting method for pricing American options

We present a new radial-basis-function (RBF)-based numerical method for pricing European and American option problems. The governing equation is time semi-discretized by a linear-implicit backward difference method. The spatial discretization is done by using the RBF-based finite difference method. The numerical scheme first derived for an European option is extended for American options by using an operator splitting method. Numerical experiments with multiquadric RBF for one- and two-asset option problems are carried out, and the results obtained are compared with the existing ones.     

Pricing American options using a space-time adaptive finite difference method

American options are priced numerically using a space- and time-adaptive finite difference method. The generalized Black–Scholes operator is discretized on a Cartesian structured but non-equidistant grid in space. The space- and time-discretizations are adjusted such that a predefined tolerance level on the local discretization error is met. An operator splitting technique is used to separately handle the early exercise constraint and the solution of linear systems of equations from the finite difference discretization of the linear complementarity problem. In numerical experiments three variants of the adaptive time-stepping algorithm with and without local time-stepping are compared.     

基于TF-IDF算法的P2P贷款违约预测模型

针对目前P2P贷款违约预测模型受限于借贷双方信息不对称性,未考虑投资人之间差异性的问题,提出了基于信息检索词频-逆文本频率（TF-IDF）算法的P2P贷款违约预测模型。首先以投资效用理论为基础,利用投资人历史投资收益率、贷款利率出价等信息,建立基于投资人效用的贷款违约预测模型;然后,借鉴信息检索TF-IDF算法,构造投资人逆向投资比例因子,对投资人差异性进行量化度量,优化模型中投资人权重计算因子。实验结果表明,该模型预测准确度与其他模型相比平均提高了6%左右,并在不同的测试数据集上都保持最优。     

Modeling default risk via a hidden Markov model of multiple sequences

Default risk in commercial lending is one of the major concerns of the creditors. In this article, we introduce a new hidden Markov model (HMM) with multiple observable sequences (MHMM), assuming that all the observable sequences are driven by a common hidden sequence, and utilize it to analyze default data in a network of sectors. Efficient estimation method is then adopted to estimate the model parameters. To further illustrate the advantages of MHMM, we compare the hidden risk state process obtained by MHMM with that from the traditional HMMs using credit default data. We then consider two applications of our MHMM. The calculation of two important risk measures: Value-at-risk (VaR) and expected shortfall (ES) and the prediction of global risk state. We first compare the performance of MHMM and HMM in the calculation of VaR and ES in a portfolio of default-prone bonds. A logistic regression model is then considered for the prediction of global economic risk using our MHMM with default data. Numerical results indicate our model is effective for both applications.     

A Monte Carlo approach to American options pricing including counterparty risk

ABSTRACT

In this work, we propose a numerical technique to compute the total value adjustment for the pricing of American options when considering counterparty risk. Several linear and nonlinear mathematical models, associated to different choices of the mark-to-market value at default, are deduced and numerically solved, thus leading to approximations of the option price with counterparty risk. The methodology is based on Monte Carlo simulations combined with a dynamic programming strategy. At each time step, an optimal stopping criterion is applied and the decision on either exercising or not the option is taken. We present some numerical tests to illustrate the behaviour of the proposed method.     

Application of the local radial basis function-based finite difference method for pricing American options

In this paper we discuss a local radial basis function-based finite difference (RBF-FD) scheme for numerical solution of multi-asset American option problems. The governing equation is discretized by the θ-method and the option price is approximated by the RBF-FD method. Numerical experiments are performed with the multiquadratic radial basis function for single and double asset problem and results obtained are compared with existing ones. We show numerically that the scheme is second-order accurate. Stability of the scheme is also discussed.     

A front-fixing finite element method for the valuation of American options with regime switching

American option problems under regime-switching model are considered in this paper. The conjectures in [H. Yang, A numerical analysis of American options with regime switching, J. Sci. Comput. 44 (2010), pp. 69–91] about the position of early exercise prices are proved, which generalize the results in [F. Yi, American put option with regime-switching volatility (finite time horizon) – Variational inequality approach, Math. Methods. Appl. Sci. 31 (2008), pp. 1461–1477] by allowing the interest rates to be different in two states. A front-fixing finite element method for the free boundary problems is proposed and implemented. Its stability is established under reasonable assumptions. Numerical results are given to examine the rate of convergence of our method and compare it with the usual finite element method.