Relative forecasting performance of volatility models: Monte Carlo evidence

Abstract

A Monte Carlo (MC) experiment is conducted to study the forecasting performance of a variety of volatility models under alternative data-generating processes (DGPs). The models included in the MC study are the (Fractionally Integrated) Generalized Autoregressive Conditional Heteroskedasticity models ((FI)GARCH), the Stochastic Volatility model (SV), the Long Memory Stochastic Volatility model (LMSV) and the Markov-switching Multifractal model (MSM). The MC study enables us to compare the relative forecasting performance of the models accounting for different characterizations of the latent volatility process: specifications that incorporate short/long memory, autoregressive components, stochastic shocks, Markov-switching and multifractality. Forecasts are evaluated by means of mean squared errors (MSE), mean absolute errors (MAE) and value-at-risk (VaR) diagnostics. Furthermore, complementarities between models are explored via forecast combinations. The results show that (i) the MSM model best forecasts volatility under any other alternative characterization of the latent volatility process and (ii) forecast combinations provide systematic improvements upon most single misspecified models, but are typically inferior to the MSM model even if the latter is applied to data governed by other processes.     

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Portfolio credit derivatives are contracts that are tied to an underlying portfolio of defaultable reference assets and have payoffs that depend on the default times of these assets. The hedging of credit derivatives involves the calculation of the sensitivity of the contract value with respect to changes in the credit spreads of the underlying assets, or, more generally, with respect to parameters of the default-time distributions. We derive and analyze Monte Carlo estimators of these sensitivities. The payoff of a credit derivative is often discontinuous in the underlying default times, and this complicates the accurate estimation of sensitivities. Discontinuities introduced by changes in one default time can be smoothed by taking conditional expectations given all other default times. We use this to derive estimators and to give conditions under which they are unbiased. We also give conditions under which an alternative likelihood ratio method estimator is unbiased. We illustrate the application and verification of these conditions and estimators in the particular case of the multifactor Gaussian copula model, but the methods are more generally applicable.      

基于蒙特卡罗模拟的多标的资产期权定价研究

在单标的资产价格随机模型的基础上,推导了具相关性的多标的资产价格的随机过程公式,以此构造蒙特卡罗模拟高维欧式期权定价的随机模型,给出模拟算法,并分析了影响蒙特卡罗模拟效果的几个关键因素。模拟算例的结果显示模拟效果较好。     

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。     

蒙特卡罗方差减小技术以及在金融中的应用

本文主要介绍MonteCarlo方差减小技术及其在金融中衍生证券定价的一些的应用。对近几年来国际上取得的主要研究成果作了简要的介绍和比较。并提出一些目前待解决的问题。     

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。     

On the Malliavin approach to Monte Carlo approximation of conditional expectations

Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X ₂)|X₁]. The different representations are determined by some localizing functions. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated mean-square-error minimizer among the class of separable localizing functions. For general localizing functions, we prove existence and uniqueness of the optimal localizing function in a suitable Sobolev space. We also provide a PDE characterization of the optimal solution which allows to draw the following observation : the separable exponential function does not minimize the integrated mean square error, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle.Mathematics Subject Classification: 60H07, 65C05, 49-00JEL Classification: G10, C10The authors gratefully acknowledge for the comments raised by an anonymous referee, which helped understanding the existence result of Sect. [4.2] of this paper.     

Least-squares Importance Sampling for Monte Carlo security pricing

We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least-squares optimization procedure. With several numerical examples, we show that such Least-squares Importance Sampling (LSIS) provides efficiency gains comparable to the state-of-the-art techniques, for problems that can be formulated in terms of the determination of the optimal mean of a multivariate Gaussian distribution. In addition, LSIS can be naturally applied to more general Importance Sampling densities and is particularly effective when the ability to adjust higher moments of the sampling distribution, or to deal with non-Gaussian or multi-modal densities, is critical to achieve variance reductions.     

A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.     

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.     

Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.     

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.     

Testing Conditional Asset Pricing Models Using a Markov Chain Monte Carlo Approach

We use Markov Chain Monte Carlo (MCMC) methods for the parameter estimation and the testing of conditional asset pricing models. In contrast to traditional approaches, it is truly conditional because the assumption that time variation in betas is driven by a set of conditioning variables is not necessary. Moreover, the approach has exact finite sample properties and accounts for errors‐in‐variables. Using S&P 500 panel data, we analyse the empirical performance of the CAPM and the Fama and French (1993) three‐factor model. We find that time‐variation of betas in the CAPM and the time variation of the coefficients for the size factor (SMB) and the distress factor (HML) in the three‐factor model improve the empirical performance. Therefore, our findings are consistent with time variation of firm‐specific exposure to market risk, systematic credit risk and systematic size effects. However, a Bayesian model comparison trading off goodness of fit and model complexity indicates that the conditional CAPM performs best, followed by the conditional three‐factor model, the unconditional CAPM, and the unconditional three‐factor model.     

Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes

Lévy processes are popular models for stock price behavior since they allow to take into account jump risk and reproduce the implied volatility smile. In this paper, we focus on the tempered stable (also known as CGMY) processes, which form a flexible 6-parameter family of Lévy processes with infinite jump intensity. It is shown that under an appropriate equivalent probability measure a tempered stable process becomes a stable process whose increments can be simulated exactly. This provides a fast Monte Carlo algorithm for computing the expectation of any functional of tempered stable process. We use our method to price European options and compare the results to a recent approximate simulation method for tempered stable process by Madan and Yor (CGMY and Meixner Subordinators are absolutely continuous with respect to one sided stable subordinators, 2005).     

Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model

We introduce the two-dimensional shifted square-root diffusion (SSRD) model for interest-rate and credit derivatives with (positive) stochastic intensity. The SSRD is the unique explicit diffusion model allowing an automatic and separated calibration of the term structure of interest rates and of credit default swaps (CDSs), and retaining free dynamics parameters that can be used to calibrate option data. We propose a new positivity preserving implicit Euler scheme for Monte Carlo simulation. We discuss the impact of interest-rate and default-intensity correlation and develop an analytical approximation to price some basic credit derivatives terms involving correlated CIR processes. We hint at a formula for CDS options under CIR + + CDS-calibrated stochastic intensity.Received: March 2004, Mathematics Subject Classification (2000): 60H10, 60J60, 60J75, 91B70JEL Classification: G13     

On an efficient multiple time step Monte Carlo simulation of the SABR model

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options.     

基于VaR的投资组合保险策略绩效评价研究

本文用引入VaR作为投资组合保险策略绩效评价指标,分析CPPI策略、TIPP策略、OBPI策略的绩效,并与CM策略和B&H策略进行比较。发现基于VaR的绩效评价与基于SHARP比率的绩效评价相悖。