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Exact Simulation of the SABR Model This article was downloaded by: [137.189.59.117] On: 28 August 2017, At: 23:13 Publisher: Institute for Operations Research and the Management Sciences (INFORMS) INFORMS is located in Maryland, USA Operations Research Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org Exact Simulation of the SABR Model http://orcid.org/0000-0002-3235-3340Ning Cai, Yingda Song, Nan Chen To cite this article: http://orcid.org/0000-0002-3235-3340Ning Cai, Yingda Song, Nan Chen (2017) Exact Simulation of the SABR Model. Operations Research 65(4):931-951. https://doi.org/10.1287/opre.2017.1617 Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions This article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval, unless otherwise noted. For more information, contact [email protected]. The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. Copyright © 2017, INFORMS Please scroll down for article—it is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org OPERATIONS RESEARCH Vol. 65, No. 4, July–August 2017, pp. 931–951 http://pubsonline.informs.org/journal/opre/ ISSN 0030-364X (print), ISSN 1526-5463 (online) CONTEXTUAL AREAS Exact Simulation of the SABR Model Ning Cai,a Yingda Song,b Nan Chenc a Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology, Kowloon, Hong Kong; b Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China 200030; c Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong Contact: [email protected], http://orcid.org/0000-0002-3235-3340 (NiC); [email protected] (YS); [email protected] (NaC) Received: July 16, 2011 Abstract. The stochastic alpha-beta-rho (SABR) model becomes popular in the financial Revised: April 13, 2013; December 17, 2014 industry because it is capable of providing good fits to various types of implied volatility Accepted: January 25, 2017 curves observed in the marketplace. However, no analytical solution to the SABR model Published Online in Articles in Advance: exists that can be simulated directly. This paper explores the possibility of exact simulation June 30, 2017 for the SABR model. Our contribution is threefold. (i) We propose an exact simulation Subject Classifications: finance: asset pricing; method for the forward price and its volatility in two special but practically interesting simulation: efficiency, application cases, i.e., when the elasticity β 1, or when β < 1 and the price and volatility processes Area of Review: Financial Engineering are instantaneously uncorrelated. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson https://doi.org/10.1287/opre.2017.1617 and the noncentral chi-squared distribution functions, respectively. Two novel simulation Copyright: © 2017 INFORMS schemes are proposed to achieve numerical accuracy, efficiency, and stability. One stems from numerical Laplace inversion and Asian option literature, and the other is based on recent developments in evaluating the noncentral chi-squared distribution functions in a robust way. Numerical examples demonstrate that our method is fast and accurate under various market environments. (ii) When β < 1 but the price and volatility processes are correlated, our simulation method becomes a semi-exact one. Numerical results suggest that it is still quite accurate when the time horizon is not long, e.g., no greater than one year. For long time horizons, a piecewise semi-exact simulation scheme is developed that reduces the biases substantially. (iii) For European option pricing under the SABR model, we propose a conditional simulation method, which reduces the variance of the plain simulation significantly, e.g., by more than 99%. Funding: The research of the first author is partially supported by the General Research Fund of the Hong Kong Research Grants Council (Project No. 16206415) and University Grants Committee grant (Project DAG12EG07-11). The research of the second author is supported by Natural Science Foundation of China [Grant 71571129]. Part of the research of the third author is supported by Hong Kong Research Grant Council via the General Research Fund Scheme (Project No. 14201114). Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2017.1617. Keywords: exact simulation • SABR model • semi-exact simulation • piecewise semi-exact simulation • time change • Asian options • noncentral chi-squared distributions • Hartman-Watson distributions 1. Introduction implied volatilities indicates that we have to model It has long been observed in the derivatives markets these smiles accurately to achieve an accurate and sta- that the implied volatility is not constant across options ble hedging. contingent on the same underlying but with different Stimulated by practitioners’ need to deal with the strike prices. The implied volatility is often relatively smile risk, there is a growing interest in stochastic lower for near-the-money foreign currency options, volatility models in the area of financial engineering; and tends to rise in the in-the-money and the out-of- see, for example, Hull and White (1987, 1988), Scott the-money directions. Traders refer to this stylized pat- (1987), Wiggins (1987), Johnson and Shanno (1987), tern as the volatility smile. In the equity and interest Stein and Stein (1991), Heston (1993), Duffie et al. Downloaded from informs.org by [137.189.59.117] on 28 August 2017, at 23:13 . For personal use only, all rights reserved. rate markets, a typical shape of the implied volatil- (2000), Fouque et al. (2000), Barndorff-Nielsen and ity known as the volatility skew is that it decreases as Shephard (2001), and so forth. The stochastic alpha- the strike price increases. The ubiquitous existence of beta-rho (SABR) model introduced by Hagan et al. volatility smiles and skews poses a great challenge to (2002) has gained popularity in the financial indus- the practice of risk management in fixed income and try to model foreign exchange and interest rate mar- foreign exchange trading desks. These desks usually kets. On the one hand, the SABR model can fit implied have large exposure across a wide range of strikes. volatility smiles very well with only a few parameters; The fact that different options correspond to different on the other hand, it is capable of generating correct 931 Cai et al.: Exact Simulation of the SABR Model 932 Operations Research, 2017, vol. 65, no. 4, pp. 931–951, © 2017 INFORMS comovements between the underlying and the smile the same time, to increase the number of time steps curves, which overcomes a salient drawback of con- in each path to reduce the discretization bias. This ventional local volatility models proposed by Dupire may incur a huge computational burden. More seri- (1994, 1997) and Derman and Kani (1994, 1998). ously, the discretization methods potentially lead to Despite the above attractive features of the SABR an extra distortion of the price distribution for the model, its complicated structure prevents us from SABR model besides the conventional one caused by obtaining closed-form solutions to related pricing the discretization. Under the SABR model, the price problems of financial derivatives, including European process is governed by a constant elasticity of variance options, swaptions, and other path-dependent options (CEV)-type diffusion whose value ranges in the inter- such as forward starting options. The literature so far val 0; + . However, the discretization methods may » 1º mainly relies on partial differential equation (PDE) generate negative values in the intermediate steps with based approaches to find asymptotic expansions for a significant probability. Therefore we have to restrict option prices under the SABR model. Hagan et al. the values within the domain by, e.g., truncating the (2002) apply the singular perturbation technique in the negative values to zero. This would then bring a new PDE to obtain an explicit asymptotic expansion for the distortion to the original price distribution. implied volatility, and then substitute it into Black’s for- In light of the absence of closed-form option pric- mula to yield an analytical approximation to the Euro- ing formulas and the drawbacks of PDE-based asymp- pean option price. Obłój (2008) points out an inconsis- totic expansions and conventional discretization simu- tency between the results in Hagan et al. (2002) and lation, the aim of this paper is to explore the possibility Berestycki et al. (2004) under a more general framework of exact simulation for the SABR model. Apparently, this and obtains a corrected version of the implied volatility relies heavily on the knowledge about the exact joint dis- formula. Wu (2012) constructs a series expansion for tribution of the forward price and the volatility. Unfor- the joint density of the price and the volatility for the tunately, in general cases, it seems very difficult to SABR model through a hierarchy of parabolic equa- derive it, if not impossible. To the authors’ best knowl- tions and a near-Gaussian coordinate transformation. edge, there exists no “complete” answer to this ques- However, the aforementioned approaches may have tion in the literature. A “partial” solution is given as three limitations. First, to the best knowledge of follows.
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