Journal of Applied Probability

On the monitoring error of the supremum of a normal jump diffusion process

Ao Chen, Liming Feng, and Renming Song

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We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the form a0 / N1/2 + a1 / N3/2 + · · · + b1 / N + b2 / N2 + b4 / N4 + · · ·, where N is the number of monitoring intervals. We obtain explicit expressions for the coefficients {a0, a1, . . . , b1, b2, . . .}. In particular, a0 is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

Article information

J. Appl. Probab., Volume 48, Number 4 (2011), 1021-1034.

First available in Project Euclid: 16 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes 91G60: Numerical methods (including Monte Carlo methods)
Secondary: 65C20: Models, numerical methods [See also 68U20]

Normal jump diffusion process supremum discrete monitoring Spitzer's identity Euler-Maclaurin formula Riemann zeta function Lerch transcendent


Chen, Ao; Feng, Liming; Song, Renming. On the monitoring error of the supremum of a normal jump diffusion process. J. Appl. Probab. 48 (2011), no. 4, 1021--1034. doi:10.1239/jap/1324046016.

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  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Government Printing Office, Washington, DC.
  • Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Prob. 5, 875–896.
  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Econom. 81, 637–654.
  • Boyarchenko, S. I. and Levendorskiǐ, S. Z. (2002). Non-Gaussian Merton–Black–Scholes Theory. World Scientific, River Edge, NJ.
  • Broadie, M., Glasserman, P. and Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance Stoch. 3, 55–82.
  • Calvin, J. M. (1995). Average performance of passive algorithms for global optimization. J. Math. Anal. Appl. 191, 608–617.
  • Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.
  • Dahlquist, G. and Björck, Å. (2008). Numerical Methods in Scientific Computing, Vol. I. Society for Industrial and Applied Mathemetics, Philadelphia, PA.
  • Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. (1953). Higher Transcendental Functions, Vol. I. McGraw Hill, New York.
  • Feng, L. and Linetsky, V. (2008). Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach. Math. Finance 18, 337–384.
  • Feng, L. and Linetsky, V. (2009). Computing exponential moments of the discrete maximum of a Lévy process and lookback options. Finance Stoch. 13, 501–529.
  • Gould, H. W. (1972). Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, 44–51.
  • Janssen, A. J. E. M. and van Leeuwaarden, J. S. H. (2007). On Lerch's transcendent and the Gaussian random walk. Ann. Appl. Prob. 17, 421–439.
  • Janssen, A. J. E. M. and van Leeuwaarden, J. S. H. (2009). Equidistant sampling for the maximum of a Brownian motion with drift on a finite horizon. Electron. Commun. Prob. 14, 143–150.
  • Jeannin, M. and Pistorius, M. (2010). A transform approach to compute prices and greeks of barrier options driven by a class of Lévy processes. Quant. Finance 10, 629–644.
  • Kou, S. G. (2008). Discrete barrier and lookback options. In Handbooks in Operations Research and Management Science, Vol. 15, eds J. Birge and V. Linetsky, pp. 343–373.
  • Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Manag. Sci. 50, 1178–1192.
  • Kudryavtsev, O. and Levendorskiĭ, S. (2009). Fast and accurate pricing of barrier options under Lévy processes. Finance Stoch. 13, 531–562.
  • Kuznetsov, A., Kyprianou, A., Pardo, J. and van Schaik, K. (2010). A Wiener-Hopf Monte-Carlo simulation technique for Lévy processes. Working paper.
  • Merton, R. C. (1973). Theory of rational option pricing. Bell J. Econom. Manag. Sci. 4, 141–183.
  • Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3, 125–144.
  • Petrella, G. and Kou, S. (2004). Numerical pricing of discrete barrier and lookback options via Laplace transforms. J. Comput. Finance 8, 1–37.
  • Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. John Wiley, Hoboken, NJ.
  • Spitzer, F. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.