An adjusted binomial model for pricing Asian options

We propose a model for pricing both European and American Asian options based on the arithmetic average of the underlying asset prices. Our approach relies on a binomial tree describing the underlying asset evolution. At each node of the tree we associate a set of representative averages chosen among all the effective averages realized at that node. Then, we use backward recursion and linear interpolation to compute the option price.     

American-style options in jump-diffusion models: estimation and evaluation

We propose dynamic programming coupled with finite elements for valuing American-style options under Gaussian and double exponential jumps à la Merton [J. Financ. Econ., 1976, 3, 125–144] and Kou [Manage. Sci., 2002, 48, 1086–1101], and we provide a proof of uniform convergence. Our numerical experiments confirm this convergence result and show the efficiency of the proposed methodology. We also address the estimation problem and report an empirical investigation based on Home Depot. Jump-diffusion models outperform their pure-diffusion counterparts.     

An efficient algorithm for pricing barrier options in arbitrage-free binomial models with calibrated drift terms

The interrelation between the drift coefficient of price processes on arbitrage-free financial markets and the corresponding transition probabilities induced by a martingale measure is analysed in a discrete setup. As a result, we obtain a flexible setting that encompasses most arbitrage-free binomial models. It is argued that knowledge of the link between drift and transition probabilities may be useful for pricing derivatives such as barrier options. The idea is illustrated in a simple example and later extended to a general numerical procedure. The results indicate that the option values in our fitted drift model converge much faster to closed-form solutions of continuous models for a wider range of contract specifications than those of conventional binomial models.     

Efficient willow tree method for European-style and American-style moving average barrier options pricing

Moving average options are widely traded in financial markets, but exiting methods for pricing this type of option are too slow. This paper proposes two efficient willow tree methods for pricing European-style and American-style moving average barrier options (MABOs). We first solve the finite-dimensional partial differential equation model for discretely monitored MABOs by willow tree methods, and then compute the value of continuously monitored MABOs by Richardson’s two-point extrapolation. Our new willow tree method employs the interpolation error minimization technique to reduce complexity. The corresponding convergence rate and error bounds are also analyzed. It shows that our proposed methods can provide the same accuracy as the binomial tree approach and Monte Carlo simulation, but require much less computing time. The numerical experiments support our claims.     

Accounting for variable stock options

A survey of nine accounting textbooks reveals that only three include the correct procedures for accounting for Variable Stock Option Plans as set forth in FASB Interpretation 28. Two books incorrectly teach the method used for changes in accounting estimates required by APB Opinion 20, and four do not address this unique exception to APB Opinion 20. This last group functionally allows the user to believe that the general rule outlined in APB Opinion 20 applies.     

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.     

Achieving smooth asymptotics for the prices of European options in binomial trees

A new binomial approximation to the Black–Scholes model is introduced. It is shown that, for digital options and vanilla European call and put options, a complete asymptotic expansion of the error in powers of n ^?1 exists. This is the first binomial tree for which an asymptotic expansion has been shown to exist.     

An improved convolution algorithm for discretely sampled Asian options

We suggest an improved FFT pricing algorithm for discretely sampled Asian options with general independently distributed returns in the underlying. Our work complements the studies of Carverhill and Clewlow [Risk, 1990, 3(4), 25–29], Benhamou [J. Comput. Finance, 2002, 6(1), 49–68], and Fusai and Meucci [J. Bank. Finance, 2008, 32(10), 2076–2088], and, if we restrict our attention only to log-normally distributed returns, also Ve?e? [Risk, 2002, 15(6), 113–116]. While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably defined state space grid, our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the first time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the benefits of the new scheme and benchmark its performance against existing finite difference, Monte Carlo, and forward density convolution algorithms.     

Free boundary and optimal stopping problems for American Asian options

We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.      

Efficient and accurate quadratic approximation methods for pricing Asian strike options

This study is on valuing Asian strike options and presents efficient and accurate quadratic approximation methods that work extremely well, both with regard to the volatility of a wide range of underlying assets, and longer average time windows. We demonstrate that most of the well-known quadratic approximation methods used in the literature for pricing Asian strike options are special cases of our model, with the numerical results demonstrating that our method significantly outperforms the other quadratic approximation methods examined here. Using our method for the calculation of hundreds of Asian strike options, the pricing errors (in terms of the root mean square errors) are reasonably small. Compared with the Monte Carlo benchmark method, our method is shown to be rapid and accurate. We further extend our method to the valuing of quanto forward-starting Asian strike options, with the pricing accuracy of these options being largely the same as the pricing of plain vanilla Asian strike options.     

采用信用风险模型计提监管资本:问题和对策

实施基于商业银行内部信用风险模型的资本监管制度对监管当局和商业银行都提出了严峻的挑战。信用风险计量模型不仅在理论上应经得起推敲,其计量结果在实证上需经得起检验,并且在不同银行间应具有可比性。本文为美联储的五位高级监管人员(Beverly J.Hirtle,Mark Levonian,Marc Saidenberg,Stefan Walter,David Wright)发表在《纽约联储经济政策评论》2001年3月号上的研究报告。该报告对采用风险计量模型计提信用风险资本所涉及的许多重大问题进行了深入讨论,揭示了内部评级法的技术原理,业界关于内部评级法的许多争论很大程度上都围绕这些问题展开,部分问题在新资本协议最终稿以及巴塞尔委员会发表的监管文件中可以找到答案,如时间跨度、使用测试、贷款损失准备处理、报告频率等;但有些问题仍未解决,讨论还在继续,如计量模型的具体形式、模型验证技术。本报告对于读者深入理解内部评级技术和监管要求具有很强指导作用。     

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black–Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed form formula for integration. Moreover, these two methods solve the backward dynamic programing problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of the binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite that this model is only bidimensional, the whole history of the process impacts on the price, and how to handle all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.     

Empirical performance of multifactor term structure models for pricing and hedging Eurodollar futures options

This article compares two one-factor, two two-factor, two three-factor models in the HJM class and Black's [Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167-179.] implied volatility function in terms of their pricing and hedging performance for Eurodollar futures options across strikes and maturities from 1 Jan 2000 to 31 Dec 2002. We find that three-factor models perform the best for 1-day and 1-week prediction, as well as for 5-day and 20-day hedging. The moneyness bias and the maturity bias appear for all models, but the three-factor models produce lower bias. Three-factor models also outperform other models in hedging, in particular for away-from-the-money and long-dated options. Making Black's volatility a square root or exponential function performs similar to one-factor HJM models in pricing, but not in hedging. Correctly specified and calibrated multifactor models are thus important and cannot be replaced by one-factor models in pricing or hedging interest rate contingent claims.     

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

We present regression-based Monte Carlo simulation algorithm for solving the stochastic control models associated with pricing and hedging of the guaranteed lifelong withdrawal benefit (GLWB) in variable annuities, where the dynamics of the underlying fund value is assumed to evolve according to the stochastic volatility model. The GLWB offers a lifelong withdrawal benefit, even when the policy account value becomes zero, while the policyholder remains alive. Upon death, the remaining account value will be paid to the beneficiary as a death benefit. The bang-bang control strategy analysed under the assumption of maximization of the policyholder’s expected cash flow reduces the strategy space of optimal withdrawal policies to three choices: zero withdrawal, withdrawal at the contractual amount or complete surrender. The impact on the GLWB value under various withdrawal behaviours of the policyholder is examined. We also analyse the pricing properties of GLWB subject to different model parameter values and structural features.