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Research Papers

Speed and duration of drawdown under general Markov models

ORCID Icon, ORCID Icon & ORCID Icon
Pages 367-386 | Received 14 Oct 2022, Accepted 21 Mar 2024, Published online: 23 Apr 2024

References

  • Abate, J. and Whitt, W., The Fourier-series method for inverting transforms of probability distributions. Queueing. Syst., 1992, 10(1), 5–87.
  • Cai, N., Song, Y. and Kou, S., A general framework for pricing Asian options under Markov processes. Oper. Res., 2015, 63(3), 540–554.
  • Cai, N., Kou, S. and Song, Y., A unified framework for option pricing under regime-switching models. Working Paper, 2019.
  • Carr, P., Geman, H., Madan, D.B. and Yor, M., The fine structure of asset returns: An empirical investigation. J. Bus., 2002, 75(2), 305–332.
  • Carr, P., Zhang, H. and Hadjiliadis, O., Maximum drawdown insurance. Int. J. Theor. Appl. Finance, 2011, 14(8), 1195–1230.
  • Cázares, J.I.G. and Mijatović, A., Monte Carlo algorithm for the extrema of tempered stable processes. arXiv preprint arXiv:2103.15310, 2021.
  • Cázares, J.G. and Mijatović, A., Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation. Financ. Stoch., 2022, 26(4), 671–732.
  • Chen, J., Fan, L., Li, L. and Zhang, G., A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation. Rev. Deriv. Res., 2022, 25, 189–232.
  • Clark, D.S., Short proof of a discrete Gronwall inequality. Discrete Appl. Math., 1987, 16(3), 279–281.
  • Cui, Z. and Nguyen, D., Omega diffusion risk model with surplus-dependent tax and capital injections. Insur. Math. Econ., 2016, 68, 150–161.
  • Cui, Z. and Nguyen, D., Magnitude and speed of consecutive market crashes in a diffusion model. Methodol. Comput. Appl. Probab., 2018, 20(1), 117–135.
  • Cui, Z., Kirkby, J.L. and Nguyen, D., A general valuation framework for SABR and stochastic local volatility models. SIAM J. Financ. Math., 2018a, 9(2), 520–563.
  • Cui, Z., Lee, C. and Liu, Y., Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes. Eur. J. Oper. Res., 2018b, 266(3), 1134–1139.
  • Davydov, D. and Linetsky, V., Pricing and hedging path-dependent options under the CEV process. Manage. Sci., 2001, 47(7), 949–965.
  • Eriksson, B. and Pistorius, M.R., American option valuation under continuous-time Markov chains. Adv. Appl. Probab., 2015, 47(2), 378–401.
  • Gapeev, P.V. and Rodosthenous, N., On the drawdowns and drawups in diffusion-type models with running maxima and minima. J. Math. Anal. Appl., 2016, 434(1), 413–431.
  • Hadjiliadis, O. and Vecer, J., Drawdowns preceding rallies in the Brownian motion model. Quant. Finance, 2006, 6(5), 403–409.
  • Hirsa, A. and Madan, D.B., Pricing American options under variance gamma. J. Comput. Finance, 2004, 7(2), 63–80.
  • Jacod, J. and Shiryaev, A., Limit Theorems for Stochastic Processes, Vol. 288, 2nd ed., 2013 (Springer Science & Business Media).
  • Kou, S.G. and Wang, H., Option pricing under a double exponential jump diffusion model. Manage. Sci., 2004, 50(9), 1178–1192.
  • Landriault, D., Li, B. and Zhang, H., On magnitude, asymptotics and duration of drawdowns for Lévy models. Bernoulli, 2017a, 23(1), 432–458.
  • Landriault, D., Li, B. and Zhang, H., A unified approach for drawdown (drawup) of time-homogeneous Markov processes. J. Appl. Probab., 2017b, 54(2), 603–626.
  • Lehoczky, J.P, Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probab., 1977, 5, 601–607.
  • Li, L. and Linetsky, V., Optimal stopping and early exercise: An eigenfunction expansion approach. Oper. Res., 2013, 61(3), 625–643.
  • Li, L. and Zhang, G., Option pricing in some non-Lévy jump models. SIAM. J. Sci. Comput., 2016, 38(4), B539–B569.
  • Li, L. and Zhang, G., Error analysis of finite difference and Markov chain approximations for option pricing. Math. Finance, 2018, 28(3), 877–919.
  • Madan, D.B., Carr, P.P. and Chang, E.C., The variance gamma process and option pricing. Rev. Financ., 1998, 2(1), 79–105.
  • Magdon-Ismail, M., Atiya, A.F., Pratap, A. and Abu-Mostafa, Y.S., On the maximum drawdown of a Brownian motion. J. Appl. Probab., 2004, 41(1), 147–161.
  • Meier, C., Li, L. and Zhang, G., Markov chain approximation of one-dimensional sticky diffusions. Adv. Appl. Probab., 2021, 53(2), 335–369.
  • Meier, C., Li, L. and Zhang, G., Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation. Eur. J. Oper. Res., 2023, 305(3), 1292–1308.
  • Mijatović, A. and Pistorius, M.R., On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Their. Appl., 2012, 122(11), 3812–3836.
  • Mijatović, A. and Pistorius, M., Continuously monitored barrier options under Markov processes. Math. Financ., 2013, 23(1), 1–38.
  • Mijatović, A., Vidmar, M. and Jacka, S., Markov chain approximations for transition densities of Lévy processes. Electron. J. Probab., 2014, 19, 1–37.
  • Pedersen, L.H., Efficiently Inefficient: How Smart Money Invests and Market Prices are Determined, 2019 (Princeton University Press).
  • Pospisil, L. and Vecer, J., PDE methods for the maximum drawdown. J. Comput. Finance, 2008, 12(2), 59–76.
  • Pospisil, L. and Vecer, J., Portfolio sensitivity to changes in the maximum and the maximum drawdown. Quant. Finance, 2010, 10(6), 617–627.
  • Pospisil, L., Vecer, J. and Hadjiliadis, O., Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stoch. Process. Their. Appl., 2009, 119(8), 2563–2578.
  • Schuhmacher, F. and Eling, M., Sufficient conditions for expected utility to imply drawdown-based performance rankings. J. Bank. Financ., 2011, 35(9), 2311–2318.
  • Song, Y., Cai, N. and Kou, S., Computable error bounds of Laplace inversion for pricing Asian options. Informs. J. Comput., 2018, 30(4), 634–645.
  • Taylor, H.M., A stopped Brownian motion formula. Ann. Probab., 1975, 3(2), 234–246.
  • Zhang, H., Occupation times, drawdowns, and drawups for one-dimensional regular diffusions. Adv. Appl. Probab., 2015, 47(1), 210–230.
  • Zhang, H. and Hadjiliadis, O., Drawdowns and rallies in a finite time-horizon. Methodol. Comput. Appl. Probab., 2010, 12(2), 293–308.
  • Zhang, H. and Hadjiliadis, O., Drawdowns and the speed of market crash. Methodol. Comput. Appl. Probab., 2012, 14(3), 739–752.
  • Zhang, G. and Li, L., Analysis of Markov chain approximation for option pricing and hedging: Grid design and convergence behavior. Oper. Res., 2019, 67(2), 407–427.
  • Zhang, G. and Li, L., Analysis of Markov chain approximation for diffusion models with nonsmooth coefficients. SIAM J. Financ. Math., 2022, 13(3), 1144–1190.
  • Zhang, G. and Li, L., A general approach for lookback option pricing under Markov models. Quant. Finance, 2023a, 23(9), 1305–1324.
  • Zhang, G. and Li, L., A general approach for Parisian stopping times under Markov processes. Finance Stoch., 2023b, 27, 1–61.
  • Zhang, G. and Li, L., A general method for analysis and valuation of drawdown risk. J. Econom. Dyn. Control, 2023c, 152, 104669.
  • Zhang, H., Leung, T. and Hadjiliadis, O., Stochastic modeling and fair valuation of drawdown insurance. Insur. Math. Econom., 2013, 53(3), 840–850.
  • Zhang, X., Li, L. and Zhang, G., Pricing American drawdown options under Markov models. Eur. J. Oper. Res., 2021, 293(3), 1188–1205.

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