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Uncertainty Quantification of Derivative Instruments

Part of: Applications Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  02 May 2017

Xianming Sun  and
Michèle Vanmaele
Xianming Sun*
Affiliation:
School of Finance, Zhongnan University of Economics and Law, Wuhan 430073, China Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Ghent B-9000, Belgium
Michèle Vanmaele*
Affiliation:
Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Ghent B-9000, Belgium
*
*Corresponding author. Email address:Xianming.Sun@hotmail.com (X. Sun), Michele.Vanmaele@ugent.be (M. Vanmaele)
*Corresponding author. Email address:Xianming.Sun@hotmail.com (X. Sun), Michele.Vanmaele@ugent.be (M. Vanmaele)
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Abstract

Model and parameter uncertainties are common whenever some parametric model is selected to value a derivative instrument. Combining the Monte Carlo method with the Smolyak interpolation algorithm, we propose an accurate efficient numerical procedure to quantify the uncertainty embedded in complex derivatives. Except for the value function being sufficiently smooth with respect to the model parameters, there are no requirements on the payoff or candidate models. Numerical tests carried out quantify the uncertainty of Bermudan put options and down-and-out put options under the Heston model, with each model parameter specified in an interval.

Keywords

Parameter uncertaintyDerivative pricingSmolyak algorithmMonte CarloEntropy

MSC classification

Secondary: 62P05: Applications to actuarial sciences and financial mathematics 65C05: Monte Carlo methods 65C50: Other computational problems in probability 68U20: Simulation
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 343 - 362
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Assing, S., Jacka, S. and Ocejo, A., Monotonicity of the value function for a two-dimensional optimal stopping problem, Ann. Appl. Probab. 24, 1554–1584 (2014).CrossRefGoogle Scholar
[2] Avellaneda, M., Levy, A. and Parás, A., Pricing and hedging derivative securities in markets with uncertain volatilities, Appl. Math. Financ. 2, 73–88 (1995).CrossRefGoogle Scholar
[3] Bannör, K. and Scherer, M., Capturing parameter risk with convex risk measures, Euro. Actuar. J. 3, 97–132 (2013).CrossRefGoogle Scholar
[4] Barthelmann, V., Novak, E. and Ritter, K., High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12, 273–288 (2000).CrossRefGoogle Scholar
[5] Beiglböck, M., Henry-Labordère, P. and Penkner, F., Model-independent bounds for option prices – a mass transport approach, Financ. Stoch. 17, 477–501 (2013).CrossRefGoogle Scholar
[6] BIS, Revisions to the Basel II market risk framework – updated 31 December 2010, Number December 2010. Bank for International Settlements, available online at: http://www.bis.org/publ/bcbs193.htm (2011).Google Scholar
[7] Brissaud, J., The meanings of entropy, Entropy 7, 68–96 (2005).CrossRefGoogle Scholar
[8] Chen, N. and Liu, Y., American option sensitivities estimation via a generalised infinitesimal perturbation analysis approach, Oper. Res. 62, 616–632 (2014).CrossRefGoogle Scholar
[9] Chen, X., Deelstra, G., Dhaene, J. and Vanmaele, M., Static super-replicating strategies for a class of exotic options, Insur. Math. Econ. 42, 1067–1085 (2008).CrossRefGoogle Scholar
[10] Chiarella, C., Kang, B. and Meyer, G., The evaluation of barrier option prices under stochastic volatility, Comput. Math. Appl. 64, 2034–2048 (2012).CrossRefGoogle Scholar
[11] Cont, R., Model uncertainty and its impact on the pricing of derivative instruments, Math. Financ. 16, 519–547 (2006).CrossRefGoogle Scholar
[12] Cox, A. and Obłój, J., Robust pricing and hedging of double no-touch options, Financ. Stoch. 15, 573–605 (2011).CrossRefGoogle Scholar
[13] Duffie, D., Garleanu, N. and Pedersen, L.. Valuation in over-the-counter markets, Rev. Financ. Stud. 20, 1865–1900 (2007).CrossRefGoogle Scholar
[14] Duembgen, M. and Rogers, L., Estimate nothing, Quant. Financ. 14, 2065–2072 (2014).CrossRefGoogle Scholar
[15] Eraker, B., Do stock prices and volatility jump? Reconciling evidence from spot and option prices, J. Financ. 59, 1367–1404 (2004).CrossRefGoogle Scholar
[16] Fang, F. and Oosterlee, C., A Fourier-based valuation method for Bermudan and barrier options under Heston's model, SIAM J. Financ. Math. 2, 439–463 (2011).CrossRefGoogle Scholar
[17] Federal Reserve, Supervisory guidance on model risk management, volume 2011, Board of Governors of the Federal Reserve System Office, Office of the Comptroller of the Currency, available online at: http://www.occ.treas.gov/news-issuances/bulletins/2011/bulletin-2011-12.html, (2011).Google Scholar
[18] Gaß, M., Glau, K., Mahlstedt, M. and Mair, M., Chebyshev interpolation for parametric option pricing, arXiv:1505.04648v2, 1-50 (2016).Google Scholar
[19] Gaß, M., Glau, K. and Mair, M., Magic points in finance: Empirical integration for parametric option pricing, arXiv:1511.00884v3, 1-34 (2016).Google Scholar
[20] Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer, New York (2004).Google Scholar
[21] Guillaume, F. and Schoutens, W., Calibration risk: Illustrating the impact of calibration risk under the Heston model. Rev. Deriv. Res. 15, 57–79 (2012).CrossRefGoogle Scholar
[22] Gupta, A. and Reisinger, C., Robust calibration of financial models using Bayesian estimators, J. Comput. Financ. 17, 3–36 (2014).CrossRefGoogle Scholar
[23] Haentjens, T. and in't Hout, K., ADI schemes for pricing American options under the Heston Model, Appl. Math. Financ. 22, 207–237 (2015).CrossRefGoogle Scholar
[24] Hansen, L., Nobel Lecture: Uncertainty outside and inside economic models, J. Polit. Econ. 122, 945–987 (2014).CrossRefGoogle Scholar
[25] Hénaff, P. and Martini, C., Model validation: Theory, practice and perspectives, J. Risk Model Validat. 5, 3–15 (2011).CrossRefGoogle Scholar
[26] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6, 327–343 (1993).CrossRefGoogle Scholar
[27] Hobson, D., Robust hedging of the lookback option, Financ. Stoch. 2, 329–347 (1998).CrossRefGoogle Scholar
[28] in't Hout, K. and Foulon, S., ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Model. 7, 303–320 (2010).Google Scholar
[29] Klimke, A. and Wohlmuth, B., Algorithm 847: Spinterp: Piecewise multilinear hierarchical sparse grid interpolation in MATLAB, ACM Trans. Math. Softw. 31, 561–579 (2005).CrossRefGoogle Scholar
[30] Knight, F.. Risk, Uncertainty and Profit, Houghton Mifflin, Boston (1921).Google Scholar
[31] Lin, X., Zhang, C. and Siu, T., Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Math. Method Oper. Res. 75, 83–100 (2012).CrossRefGoogle Scholar
[32] Lyons, T., Uncertain volatility and the risk-free synthesis of derivatives, Appl. Math. Financ. 2, 117–133 (1995).CrossRefGoogle Scholar
[33] Miao, J. and Yang, X., Pricing model for convertible bonds: A mixed fractional Brownian motion with jumps, East Asian J. Appl. Math. 5, 222–237 (2015).CrossRefGoogle Scholar
[34] Narayan, A. and Xiu, D., Constructing nested nodal sets for multivariate polynomial interpolation, SIAM J. Sci. Comput. 35, A2293–A2315 (2013).CrossRefGoogle Scholar
[35] Narayan, A. and Zhou, T., Stochastic collocation methods on unstructured meshes, Commun. Comput. Phys. 18, 1–36 (2015).CrossRefGoogle Scholar
[36] Park, S. and Bera, A., Maximum entropy autoregressive conditional heteroskedasticity model, J. Econom. 150, 219–230 (2009).CrossRefGoogle Scholar
[37] Routledge, B. and Zin, S., Model uncertainty and liquidity, Rev. Econom. Dyn. 12, 543–566 (2009).CrossRefGoogle Scholar
[38] Schoutens, W., Simons, E. and Tistaert, J., A perfect calibration! Now what?, Wilmott Magazine, pp. 66–78, March (2004).CrossRefGoogle Scholar
[39] Shannon, C., A mathematical theory of communication, Bell Syst. Tech. J. 27, 379–423 (1948).CrossRefGoogle Scholar
[40] Sun, X., Quantifying Uncertainty in Financial Markets, PhD thesis, Central South University and Ghent University (2016).Google Scholar
[41] Tan, X. and Touzi, N., Optimal transportation under controlled stochastic dynamics, Ann. Probab. 41, 3201–3240 (2013).CrossRefGoogle Scholar
[42] Tang, T. and Zhou, T., On discrete least-squares projection in unbounded domain with random evaluations and its application to parametric uncertainty quantification, SIAM J. Sci. Comput. 36, A2272–A2295 (2014).CrossRefGoogle Scholar
[43] Tang, T. and Zhou, T., Recent developments in high order numerical methods for uncertainty quantification, Scientia Sinica (Mathematica) 7, 891–928 (2015).Google Scholar
[44] von Sydow, L., Höök, L., Larsson, E., Lindström, E., Milovanović, S., Persson, J., Shcherbakov, V., Shpolyanskiy, Y., Sirén, S., Toivanen, J., Waldén, J., Wiktorsson, M., Levesley, J., Li, J., Oosterlee, C., Ruijter, M., Toropov, A. and Zhao, Y., BENCHOP – The BENCHmarking project in option pricing, Int. J. Comput. Math. 92, 2361–2379 (2015).CrossRefGoogle Scholar
[45] Wasilkowski, G. and Wozniakowski, H., Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11, 1–56 (1995).CrossRefGoogle Scholar
[46] Xiu, D. and Hesthaven, J., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput. 27, 1118–1139 (2005).CrossRefGoogle Scholar
[47] Xu, Z. and Zhou, T.. On sparse interpolation and the design of deterministic interpolation points, SIAM J. Sci. Comput. 36, A1752–A1769 (2014).CrossRefGoogle Scholar
[48] Zeng, P. and Kwok, Y.. Pricing barrier and Bermudan style options under time-changed Lévy Processes: Fast Hilbert transform approach, SIAM J. Sci. Comput. 36, B450–B485 (2014).CrossRefGoogle Scholar