Valuation of VIX and target volatility options with affine GARCH models

In this paper we propose semiclosed-form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte–Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.     

ON THE CONSISTENCY OF REGRESSION‐BASED MONTE CARLO METHODS FOR PRICING BERMUDAN OPTIONS IN CASE OF ESTIMATED FINANCIAL MODELS

In many applications of regression‐based Monte Carlo methods for pricing, American options in discrete time parameters of the underlying financial model have to be estimated from observed data. In this paper suitably defined nonparametric regression‐based Monte Carlo methods are applied to paths of financial models where the parameters converge toward true values of the parameters. For various Black–Scholes, GARCH, and Levy models it is shown that in this case the price estimated from the approximate model converges to the true price.     

Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999) , as well as Ma and Zhang (2000) , using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.     

LATTICE OPTION PRICING BY MULTIDIMENSIONAL INTERPOLATION

This paper proposes a method for pricing high-dimensional American options based on modern methods of multidimensional interpolation. The method allows using sparse grids and thus mitigates the curse of dimensionality. A framework of the pricing algorithm and the corresponding interpolation methods are discussed, and a theorem is demonstrated, which suggests that the pricing method is less vulnerable to the curse of dimensionality. The method is illustrated by an application to rainbow options and compared to least squares Monte Carlo and other benchmarks.     

Valuation by Simulation of Contingent Claims with Multiple Early Exercise Opportunities

This paper introduces the application of Monte Carlo simulation technology to the valuation of securities that contain many (buying or selling) rights, but for which a limited number can be exercised per period, and penalties if a minimum quantity is not exercised before maturity. These securities combine the characteristics of American options, with the additional constraint that only a few rights can be exercised per period and therefore their price depends also on the number of living rights (i.e., American-Asian-style payoffs), and forward securities. These securities give flexibility-of-delivery options and are common in energy markets (e.g., take-or-pay or swing options) and as real options (e.g., the development of a mine). First, we derive a series of properties for the price and the optimal exercise frontier of these securities. Second, we price them by simulation, extending the Ibáñez and Zapatero (2004) method to this problem.     

The optimal method for pricing Bermudan options by simulation

Least‐squares methods enable us to price Bermudan‐style options by Monte Carlo simulation. They are based on estimating the option continuation value by least‐squares. We show that the Bermudan price is maximized when this continuation value is estimated near the exercise boundary, which is equivalent to implicitly estimating the optimal exercise boundary by using the value‐matching condition. Localization is the key difference with respect to global regression methods, but is fundamental for optimal exercise decisions and requires estimation of the continuation value by iterating local least‐squares (because we estimate and localize the exercise boundary at the same time). In the numerical example, in agreement with this optimality, the new prices or lower bounds (i) improve upon the prices reported by other methods and (ii) are very close to the associated dual upper bounds. We also study the method's convergence.     

Pricing General Barrier Options: A Numerical Approach Using Sharp Large Deviations

In this paper we develop simulation techniques in order to evaluate single and double barrier options with general features. Our method is based on Sharp Large Deviation estimates, which allow one to improve the usual Monte Carlo procedure. Numerical results are provided and show the validity of the proposed simulation algorithm.     

ON THE ERROR IN THE MONTE CARLO PRICING OF SOME FAMILIAR EUROPEAN PATH-DEPENDENT OPTIONS

This paper studies the relative error in the crude Monte Carlo pricing of some familiar European path-dependent multiasset options. For the crude Monte Carlo method it is well known that the convergence rate   O ( n ^−1/2)  , where n is the number of simulations, is independent of the dimension of the integral. This paper also shows that for a large class of pricing problems in the multiasset Black-Scholes market the constant in   O ( n ^−1/2)  is independent of the dimension. To be more specific, the constant is only dependent on the highest volatility among the underlying assets, time to maturity, and degree of confidence interval.     

DUAL REPRESENTATIONS FOR GENERAL MULTIPLE STOPPING PROBLEMS

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cash flows which are subject to volume constraints modeled by integer‐valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers ( 2012 ), Bender ( 2011a ), Bender ( 2011b ), Aleksandrov and Hambly ( 2010 ), and Meinshausen and Hambly ( 2004 ) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cash flow structures than the additive structure in the above references. For example, some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for prices of multiple exercise options and illustrate it with a numerical study on the pricing of a swing option in an electricity market.     

PRICING OF HIGH‐DIMENSIONAL AMERICAN OPTIONS BY NEURAL NETWORKS

Pricing of American options in discrete time is considered, where the option is allowed to be based on several underlyings. It is assumed that the price processes of the underlyings are given Markov processes. We use the Monte Carlo approach to generate artificial sample paths of these price processes, and then we use the least squares neural networks regression estimates to estimate from this data the so‐called continuation values, which are defined as mean values of the American options for given values of the underlyings at time t subject to the constraint that the options are not exercised at time t. Results concerning consistency and rate of convergence of the estimates are presented, and the pricing of American options is illustrated by simulated data.     

Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options

In this paper, we consider the problem of the numerical computation of Greeks for a multidimensional barrier and lookback style options: the payoff function depends in a rather general way on the minima and maxima of the coordinates of the d -dimensional underlying asset process. Using Malliavin calculus techniques, we derive additional weights that enable computation of the Greeks using Monte Carlo simulations. Numerical experiments confirm the efficiency of the method. This work is a multidimensional extension of previous results (see Gobet and Kohatsu-Higa 2001 ).     

实物期权评价方法及波动率全周期估算研究

在不确定性投资中,实物期权评价方法充分考虑了项目投资中的管理灵活性、不确定性和不可逆性,因而更能准确地评估项目投资的价值。项目建设期的现金流出和经营期可能出现的亏损使得现行波动率估算方法难以应用于项目评价中。在分析波动率性质的基础上,以净现值法为基础,应用蒙特卡洛原理,提出了在现金流随机变动条件下实物期权模型中基于全周期的波动率参数估算方法,该方法适用于项目投资且易于操作。     

Local Vega Index and Variance Reduction Methods

In this article we discuss a generalization of the Greek called vega which is used to study the stability of option prices and hedging portfolios with respect to the volatility in various models. We call this generalization the local vega index. We compute through Monte Carlo simulations this index in the cases of Asian options under the classical Black-Scholes setup. Simulation methods using Malliavin calculus and kernel density estimation are compared. Variance reduction methods are discussed.     

国际碳贸易价格波动对可再生能源投资的影响机制——基于实物期权理论的分析

有效利用可再生能源是促进节能减排实现绿色循环经济的重要手段。目前,可再生能源技术成本仍高于传统能源技术,因而需要获得额外的经济激励以增加其投资。《京都议定书》所建立的国际碳贸易体系是支持发展中国家实现碳减排的重要机制,但该贸易体系发展前景不明确,这将深刻影响我国可再生能源投资。本文分析和揭示了国际碳贸易体系的不确定性对可再生能源投资决策的影响,在此基础上提出了分别存在于可再生能源项目前期规划阶段和项目建设阶段的增长期权和延迟期权;通过构建两阶段期权模型研究国际碳价格波动下企业延迟投资的灵活性,并量化确定可再生能源项目投资期权价值,采用Monte-Carlo仿真分析法进一步验证模型,推导得出国际碳价格波动对可再生能源项目投资的作用机制。     

Canonical Distribution,Implied Binomial Tree,and the Pricing of American Options

In this study, a new approach to pricing American options is proposed and termed the canonical implied binomial (CIB) tree method. CIB takes advantage of both canonical valuation (Stutzer, 1996) and the implied binomial tree method (Rubinstein, 1994). Using simulated returns from geometric Brownian motions (GBM), CIB produced very similar prices for calls and European puts as those of Black–Scholes (BS). Applied to a set of over 15,000 American‐style S&P 100 Index puts, CIB outperformed BS with historic volatility in pricing out‐of‐the‐money options; in addition, it outperformed the canonical least‐squares Monte Carlo (Liu, 2010) in the dynamic hedging of in‐the‐money options. Furthermore, CIB suggests that regular GBM‐based Monte Carlo can be extended to American options pricing by also utilizing the implied binomial tree. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

专利实物期权定价的蒙特卡罗模拟方法及其改进技术

专利作为现代企业重要的无形资产,对企业募集发展资金、扩大收益回流、形成竞争优势发挥着巨大的作用,因此专利定价问题也成为现代企业管理决策中的重要内容.本文基于专利的实物期权特征,运用蒙特卡罗模拟方法对专利的实物期权定价问题进行研究与探讨.首先,分析探讨专利投资项目的决策过程及其实物期权特征;在此基础上,建立专利定价的实物期权蒙特卡罗模拟模型,并引入对偶变量技术用以提高蒙特卡罗模拟的效率;最后,以生物医药企业专利定价为例进行实证模拟.研究结论认为,引入适当方差减少技术的蒙特卡罗模拟则成为专利实物期权定价的一种有效的分析方法.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

TRUE UPPER BOUNDS FOR BERMUDAN PRODUCTS VIA NON-NESTED MONTE CARLO

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic "true" stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale can be employed for computing tight dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator. Numerical experiments indicate the efficiency of the proposed algorithms.     

A Forward Monte Carlo Method for American Options Pricing

This study proposes a forward Monte Carlo method for the pricing of American options. The main advantage of this method is that it does not use backward induction as required by other methods. Instead, the proposed approach relies on a wise determination about whether a simulated stock price has entered the exercise region. The validity of the proposed method is supported by the mathematical proofs for the vanilla cases. With some adaption, it is shown that this forward method can be extended to price other American style options such as chooser and exchange options. This study demonstrates the effectiveness of the proposed approach using a series of numerical examples, revealing significant improvements in numerical efficiency and accuracy in contrast with the standard regression‐based method of Longstaff and Schwartz (2001). © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:369‐395, 2013     

A NEW METHOD OF PRICING LOOKBACK OPTIONS

A new method for pricing lookback options (a.k.a. hindsight options) is presented, which simplifies the derivation of analytical formulas for this class of exotics in the Black-Scholes framework. Underlying the method is the observation that a lookback option can be considered as an integrated form of a related barrier option. The integrations with respect to the barrier price are evaluated at the expiry date to derive the payoff of an equivalent portfolio of European-type binary options. The arbitrage-free price of the lookback option can then be evaluated by static replication as the present value of this portfolio. We illustrate the method by deriving expressions for generic, standard floating-, fixed-, and reverse-strike lookbacks, and then show how the method can be used to price the more complex partial-price and partial-time lookback options. The method is in principle applicable to frameworks with alternative asset-price dynamics to the Black-Scholes world.