Technical Note—On Matrix Exponential Differentiation with Application to Weighted Sum Distributions

Published Online:https://doi.org/10.1287/opre.2021.2257

References

  • Abate J, Choudhury GL, Lucantoni DM, Whitt W (1995) Asymptotic analysis of tail probabilities based on the computation of moments. Ann. Appl. Probab. 5(4):983–1007.CrossrefGoogle Scholar
  • Akhiezer NI (1965) The Classical Moment Problem and Some Related Questions in Analysis. University Mathematical Monographs (Oliver & Boyd, Edinburgh and London). Google Scholar
  • Baumann B, Merkle CW, Leitgeb RA, Augustin M, Wartak A, Pircher M, Hitzenberger CK (2019) Signal averaging improves signal-to-noise in OCT images: But which approach works best, and when? Biomedical Optics Express 10(11):5755–5775.CrossrefGoogle Scholar
  • Bertsimas D, Popescu I (2002) On the relation between option and stock prices: A convex optimization approach. Oper. Res. 50(2):358–374.LinkGoogle Scholar
  • Bertsimas D, Popescu I (2005) Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15(3):780–804.CrossrefGoogle Scholar
  • Bertsimas D, Sethuraman J (2000) Moment problems and semidefinite optimization. Wolkowicz H, Saigal R, Vandenberghe L, eds. Handbook of Semidefinite Programming, vol. 27. International Series in Operations Research & Management Science (Springer, Boston), 469–509.CrossrefGoogle Scholar
  • Brignone R, Kyriakou I, Fusai G (2021) Moment-matching approximations for stochastic sums in non-Gaussian Ornstein–Uhlenbeck models. Insurance Math. Econom. 96:232–247.CrossrefGoogle Scholar
  • Cai N, Kou S (2012) Pricing Asian options under a hyper-exponential jump diffusion model. Oper. Res. 60(1):64–77.LinkGoogle Scholar
  • Cai N, Yang X (2018) International reserve management: A drift-switching reflected jump-diffusion model. Math. Finance 28(1):409–446.CrossrefGoogle Scholar
  • Cai N, Yang X (2021) A computational approach to first passage problems of reflected hyperexponential jump diffusion processes. INFORMS J. Comput. 33(1):216–229.LinkGoogle Scholar
  • Cai N, Kou S, Song Y (2020) A unified framework for regime-switching models. Working paper, Department of Industrial Engineering and Decision Analytics, The Hong Kong University of Science and Technology. Google Scholar
  • Cai N, Li C, Shi C (2014) Closed-form expansions of discretely monitored Asian options in diffusion models. Math. Oper. Res. 39(3):789–822.LinkGoogle Scholar
  • Cai N, Song Y, Kou S (2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3):540–554.LinkGoogle Scholar
  • Chen X, He S, Jiang B, Ryan CT, Zhang T (2021) The discrete moment problem with nonconvex shape constraints. Oper. Res. 69(1):279–296.LinkGoogle Scholar
  • Choudhury GL, Lucantoni DM (1996) Numerical computation of the moments of a probability distribution from its transform. Oper. Res. 44(2):368–381.LinkGoogle Scholar
  • Creemers S (2018) Moments and distribution of the net present value of a serial project. Eur. J. Oper. Res. 267(3):835–848.CrossrefGoogle Scholar
  • Cui Z, Kirkby JL, Nguyen D (2017) A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps. Eur. J. Oper. Res. 262(1):381–400.CrossrefGoogle Scholar
  • Cui Z, Kirkby JL, Nguyen D (2021) Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations. Eur. J. Oper. Res. 290(3):1046–1062.CrossrefGoogle Scholar
  • Cui Z, Lee C, Liu Y (2018) Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes. Eur. J. Oper. Res. 266(3):1134–1139.CrossrefGoogle Scholar
  • Ding K, Cui Z, Wang Y (2021) A Markov chain approximation scheme for option pricing under skew diffusions. Quant. Finance 21(3):461–480.CrossrefGoogle Scholar
  • Fusai G, Kyriakou I (2016) General optimized lower and upper bounds for discrete and continuous arithmetic Asian options. Math. Oper. Res. 41(2):531–559.LinkGoogle Scholar
  • Fusai G, Marena M, Roncoroni A (2008) Analytical pricing of discretely monitored Asian-style options: Theory and application to commodity markets. J. Banking Finance 32(10):2033–2045.CrossrefGoogle Scholar
  • Gambaro AM, Kyriakou I, Fusai G (2020) General lattice methods for arithmetic Asian options. Eur. J. Oper. Res. 282(3):1185–1199.CrossrefGoogle Scholar
  • He S, Zhang J, Zhang S (2010) Bounding probability of small deviation: A fourth moment approach. Math. Oper. Res. 35(1):208–232.LinkGoogle Scholar
  • Horn RA, Johnson CR (2012) Matrix Analysis (Cambridge University Press, New York).CrossrefGoogle Scholar
  • Kalbfleisch JD, Lawless JF (1985) The analysis of panel data under a Markov assumption. J. Amer. Statist. Assoc. 80(392):863–871.CrossrefGoogle Scholar
  • Khamis SH (1954) On the reduced moment problem. Ann. Math. Statist. 25(1):113–122.CrossrefGoogle Scholar
  • Kirkby JL, Nguyen D, Cui Z (2017) A unified approach to Bermudan and barrier options under stochastic volatility models with jumps. J. Econom. Dynam. Control 80:75–100.CrossrefGoogle Scholar
  • Kyriakou I, Brignone R, Fusai G (2021) Unified moment-based modelling of integrated stochastic processes. Working paper, Bayes Business School, University of London. Google Scholar
  • Lindsay BG, Basak P (2000) Moments determine the tail of a distribution (but not much else). Amer. Statist. 54(4):248–251.Google Scholar
  • Lo AW (1987) Semi-parametric upper bounds for option prices and expected payoffs. J. Financial Econom. 19(2):373–387.CrossrefGoogle Scholar
  • Magnus JR, Neudecker H (2019) Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd ed. Wiley Series in Probability and Statistics (John Wiley & Sons, Ltd., Chichester, UK).Google Scholar
  • Mijatović A, Pistorius M (2013) Continuously monitored barrier options under Markov processes. Math. Finance 23(1):1–38.CrossrefGoogle Scholar
  • Moler C, Van Loan C (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1):3–49.CrossrefGoogle Scholar
  • Nadarajah S (2008) A review of results on sums of random variables. Acta Appl. Math. 103(2):131–140.CrossrefGoogle Scholar
  • Shampine LF (2008) Vectorized adaptive quadrature in MATLAB. J. Comput. Appl. Math. 211(2):131–140.CrossrefGoogle Scholar
  • Shampine LF, Allen RC Jr, Pruess S (1997) Fundamentals of Numerical Computing (John Wiley & Sons, Inc., New York).Google Scholar
  • Sidje RB (1998) Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Software 24(1):130–156.CrossrefGoogle Scholar
  • Solomon H, Stephens MA (1978) Approximations to density functions using Pearson curves. J. Amer. Statist. Assoc. 73:153–160.CrossrefGoogle Scholar
  • Song Y, Cai N, Kou S (2018) Computable error bounds of Laplace inversion for pricing Asian options. INFORMS J. Comput. 30(4):634–645.LinkGoogle Scholar
  • Tian R, Cox SH, Zuluaga LF (2017) Moment problem and its applications to risk assessment. North Am. Actuarial J. 21(2):242–266.CrossrefGoogle Scholar
  • Tsai H, Chan KS (2003) A note on parameter differentiation of matrix exponentials, with applications to continuous-time modelling. Bernoulli 9(5):895–919.CrossrefGoogle Scholar
  • Yang W, Ma J, Cui Z (2021) Analysis of Markov chain approximation for Asian options and occupation-time derivatives: Greeks and convergence rates. Math. Methods Oper. Res. 93(2):359–412.CrossrefGoogle Scholar
  • Zhang G, Li L (2021) A general approach for Parisian stopping times under Markov processes. Working paper, School of Science and Engineering, The Chinese University of Hong Kong. Google Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.