A Refined Binomial Lattice for Pricing American Asian Options

We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the Cox-Ross-Rubinstein (1979) binomial lattice of stock prices. Each node in the lattice is partitioned into nodelets, each of which represents all paths arriving at the node with a specific geometric stock price average. The upper bound uses an interpolation idea similar to the Hull-White (1993) method. From the backward-recursive upper-bound computation, we estimate a good exercise rule that is consistent with the refined lattice. This exercise rule is used to obtain a lower bound on the option price using a modification of a conditional-expectation based idea from Rogers-Shi (1995) and Chalasani-Jha-Varikooty (1998). Our algorithms run in time proportional to the number of nodelets in the refined lattice, which is smaller than n^4/20 for n > 14 periods.     

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.     

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.     

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.     

Analytical pricing of discrete arithmetic Asian options with mean reversion and jumps

Empirical evidence indicates that commodity prices are mean reverting and exhibit jumps. As some commodity option payoffs involve the arithmetic average of historical commodity prices, we derive an analytical solution to arithmetic Asian options under a mean reverting jump diffusion process. The analytical solution is implemented with the fast Fourier transform based on the joint characteristic function of the terminal asset price and the realized average value. We also examine the accuracy and computational efficiency of the proposed method through numerical studies.     

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.     

Pricing European-type,early-exercise and discrete barrier options using an algorithm for the convolution of Legendre series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.     

Static versus dynamic hedges: an empirical comparison for barrier options

We conduct an empirical comparison of static versus dynamic hedges of barrier options. Using more than five years of data, we compare a number of static hedges from the literature with dynamic hedges based on the local volatility model. The main result is that the variability of profit-and-loss distributions from certain static hedges is significantly smaller than that of dynamic hedges and robust to changing market scenarios. Furthermore, these static hedges are able to provide a robust tracking of barrier options’ sensitivities. This article reflects the authors’ personal opinion and not necessarily the opinion of their employers.     

Pricing options with Green's functions when volatility,interest rate and barriers depend on time

We derive the Green's function for the Black–Scholes partial differential equation with time-varying coefficients and time-dependent boundary conditions. We provide a thorough discussion of its implementation within a pricing algorithm that also accommodates American style options. Greeks can be computed as derivatives of the Green's function. Generic handling of arbitrary time-dependent boundary conditions suggests our approach to be used with the pricing of (American) barrier options, although options without barriers can be priced equally well. Numerical results indicate that knowledge of the structure of the Green's function together with the well-developed tools of numerical integration make our approach fast and numerically stable.     

Static hedges for reverse barrier options with robustness against skew risk: an empirical analysis

We conduct an empirical evaluation of a static super-replicating hedge of barrier options. The hedge is robust to uncertainty about the future skew. Using almost seven years of current data on the DAX, we evaluate the performance of the hedge and compare it with those of both a dynamic and a static replicating hedge. The main result is that the robustness of the static super-replicating portfolio is also empirically confirmed in practice such that the hedge sets an upper bound for the price of skew risk for barrier options.     

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.     

美国政策再宽松、美元陷阱与我国中期政策选择

文章在深入分析美国政策再宽松可能带来的美元贬值和通胀风险的基础上,指出了我国经济所面临的增长放缓与美元贬值的内外挑战,并从主动调节经济增速与加快经济结构调整等方面对化解上述困境提出了针对性的政策建议。     

Binomial basis for linear information dynamics: real options, dividends and the valuation of equity

Analytical research has confirmed that real options give rise to the kind of nonlinearities observed in practice between equity prices and the figures appearing on corporate financial statements. We develop these real option values in terms of a quasi 'supply-side' model of linear information dynamics based on simple discrete time binomial filtration processes. Our analysis shows that the linear models that pervade the empirical (and analytical) work of the area, will almost certainly suffer from an omitted variables problem. Parameter estimation will then be inconsistent and inefficient.     

A Markov-switching generalized additive model for compound Poisson processes,with applications to operational loss models

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.