Assessing the Least Squares Monte-Carlo Approach to American Option Valuation

A detailed analysis of the Least Squares Monte-Carlo (LSM) approach to American option valuation suggested in Longstaff and Schwartz (2001) is performed. We compare the specification of the cross-sectional regressions with Laguerre polynomials used in Longstaff and Schwartz (2001) with alternative specifications and show that some of these have numerically better properties. Furthermore, each of these specifications leads to a trade-off between the time used to calculate a price and the precision of that price. Comparing the method-specific trade-offs reveals that a modified specification using ordinary monomials is preferred over the specification based on Laguerre polynomials. Next, we generalize the pricing problem by considering options on multiple assets and we show that the LSM method can be implemented easily for dimensions as high as ten or more. Furthermore, we show that the LSM method is computationally more efficient than existing numerical methods. In particular, when the number of assets is high, say five, Finite Difference methods are infeasible, and we show that our modified LSM method is superior to the Binomial Model.     

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

A computational definition of financial randomness

The aim of this study is to present an efficient and easy framework for the application of the Least Squares Monte Carlo methodology to the pricing of gas or power facilities as detailed in Boogert and de Jong [J. Derivatives, 2008, 15, 81–91]. As mentioned in the seminal paper by Longstaff and Schwartz [Rev. Financ. Stud. 2001, 113–147], the convergence of the Least Squares Monte Carlo algorithm depends on the convergence of the optimization combined with the convergence of the pure Monte Carlo method. In the context of the energy facilities, the optimization is more complex and its convergence is of fundamental importance in particular for the computation of sensitivities and optimal dispatched quantities. To our knowledge, an extensive study of the convergence, and hence of the reliability of the algorithm, has not been performed yet, in our opinion this is because the apparent infeasibility and complexity uses a very high number of simulations. We present then an easy way to simulate random trajectories by means of diffusion bridges in contrast to Dutt and Welke [J. Derivatives, 2008, 15 (4), 29–47] that considers time-reversal Itô diffusions and subordinated processes. In particular, we show that in the case of Cox-Ingersoll-Ross and Heston models, the bridge approach has the advantage to produce exact simulations even for non-Gaussian processes, in contrast to the time-reversal approach. Our methodology permits performing a backward dynamic programming strategy based on a huge number of simulations without storing the whole simulated trajectory. Generally, in the valuation of energy facilities, one is also interested in the forward recursion. We then design backward and forward recursion algorithms such that one can produce the same random trajectories by the use of multiple independent random streams without storing data at intermediate time steps. Finally, we show the advantages of our methodology for the valuation of virtual hydro power plants and gas storages.     

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.     

Backward simulation methods for pricing American options under the CIR process

In this paper, we focus on backward simulation of the CIR process. The purpose is to solve the memory requirement issue of the Least Squares Monte Carlo method when pricing American options by simulation. The concept of backward simulation is presented and it is classified into two types. Under the framework of the second type backward simulation, we seek the solutions for the existing CIR schemes. Specifically, we propose forward–backward simulation approaches for Alfonsi’s two implicit schemes, the fixed Euler schemes and the exact scheme. The proposed schemes are numerically tested and compared in pricing American options under the Heston model and the stochastic interest rate model. Some numerical properties such as the convergence order of the explicit–implicit Euler schemes, the storage requirement estimation of the forward–backward exact scheme and its computing time comparison with the squared Bessel bridge are also tested. Finally, the pros and cons of the related backward simulation schemes are summarized.     

On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives

This paper analyses the robustness of Least-Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Decision-making in incomplete markets with ambiguity—a case study of a gas field acquisition

We apply utility indifference pricing to solve a contingent claim problem, valuing a connected pair of gas fields where the underlying process is not standard Geometric Brownian Motion and the assumption of complete markets is not fulfilled. First, empirical data are often characterized by time-varying volatility and fat tails; therefore, we use Gaussian generalized autoregressive score (GAS) and GARCH models, extending them to Student’s t-GARCH and t-GAS. Second, an important risk (reservoir size) is not hedgeable. As a result, markets are incomplete which makes preference free pricing impossible and thus standard option pricing methodology inapplicable. Therefore, we parametrize the investor’s risk preference and use utility indifference pricing techniques. We use Least Squares Monte Carlo simulations as a dimension reduction technique in solving the resulting stochastic dynamic programming problems. Moreover, an investor often only has an approximate idea of the true probabilistic model underlying variables, making model ambiguity a relevant problem. We show empirically how model ambiguity affects project values, and importantly, how option values change as model ambiguity gets resolved in later phases of the projects. We show that traditional valuation approaches will consistently underestimate the value of project flexibility and in general lead to overly conservative investment decisions in the presence of time-dependent stochastic structures.     

Pricing via recursive quantization in stochastic volatility models

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.     

American options under stochastic volatility: control variates,maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of an American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow-scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price an American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where the scaling parameter value is equal to unity, fast and slow-scale approximations are equally accurate.     

亚式期权定价的模拟方法研究

由于算术平均价格亚式期权的定价没有解析公式，所以文章用Monte Carlo模拟方法通过Matlab软件编写程序对亚式期权进行了定价。发现在某些情况下，亚式期权的价值并不是国内外一些研究者所认为的低于相应的欧式期权的价值。     

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 35.     

Evaluating stochastic discount factors from term structure models

This paper examines the feasibility of applying the stochastic discount factor methodology to fixed-income data using modern term structure models. Using this approach the researcher can examine returns on bond portfolios whose exact composition is unknown, as is often the case. This paper proposes an observable proxy for the SDF from continuous-time models and documents via Monte Carlo methods the properties of the GMM estimator based on using this proxy.     

美式期权的Monte Carlo模拟法定价理论及应用

近年来随着计算机技术的飞速发展,美式期权的Monte Carlo模拟法定价取得了实质性的突破。本文分析介绍了美式期权的Monte Carlo模拟法定价理论及在此基础上推导出的线性回归MonteCarlo模拟法定价公式及其在实际的应用。     

Fast Ninomiya–Victoir calibration of the double-mean-reverting model

Abstract

We consider the three-factor double mean reverting (DMR) option pricing model of Gatheral [Consistent Modelling of SPX and VIX Options, 2008], a model which can be successfully calibrated to both VIX options and SPX options simultaneously. One drawback of this model is that calibration may be slow because no closed form solution for European options exists. In this paper, we apply modified versions of the second-order Monte Carlo scheme of Ninomiya and Victoir [Appl. Math. Finance, 2008, 15, 107–121], and compare these to the Euler–Maruyama scheme with full truncation of Lord et al. [Quant. Finance, 2010, 10(2), 177–194], demonstrating on the one hand that fast calibration of the DMR model is practical, and on the other that suitably modified Ninomiya–Victoir schemes are applicable to the simulation of much more complicated time-homogeneous models than may have been thought previously.     

互联网企业价值评估模型比较研究

基于互联网企业轻资产、高估值、迭代快以及风险大等特点,比较传统价值评估模型与Schwar-tz-Moon等实物期权价值评估模型,分别运用于评估案例企业泛微网络价值.结果发现,相较于传统现金流贴现模型,实物期权价值评估模型评估结果更接近于公司实际价值.三种实物期权模型敏感性分析表明:Schwartz-Moon模型评估误差最小,且模型稳健性最强,适用于不确定性高的互联网企业估值.