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A delay financial model with stochastic volatility; martingale method

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  • Lee, Min-Ku
  • Kim, Jeong-Hoon
  • Kim, Joocheol

Abstract

In this paper, we extend a delayed geometric Brownian model by adding a stochastic volatility term, which is driven by a hidden process of fast mean reverting diffusion, to the delayed model. Combining a martingale approach and an asymptotic method, we develop a theory for option pricing under this hybrid model. The core result obtained by our work is a proof that a discounted approximate option price can be decomposed as a martingale part plus a small term. Subsequently, a correction effect on the European option price is demonstrated both theoretically and numerically for a good agreement with practical results.

Suggested Citation

  • Lee, Min-Ku & Kim, Jeong-Hoon & Kim, Joocheol, 2011. "A delay financial model with stochastic volatility; martingale method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(16), pages 2909-2919.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:16:p:2909-2919
    DOI: 10.1016/j.physa.2011.03.032
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    References listed on IDEAS

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    1. Grech, D & Mazur, Z, 2004. "Can one make any crash prediction in finance using the local Hurst exponent idea?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(1), pages 133-145.
    2. Dibeh, Ghassan, 2005. "Speculative dynamics in a time-delay model of asset prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(1), pages 199-208.
    3. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    4. Scheinkman, Jose A & LeBaron, Blake, 1989. "Nonlinear Dynamics and Stock Returns," The Journal of Business, University of Chicago Press, vol. 62(3), pages 311-337, July.
    5. Kim, Jeong-Hoon, 2004. "Asymptotic theory of noncentered mixing stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 161-174, November.
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    Cited by:

    1. Chendur Kumaran, R. & Venkatesh, T.G. & Swarup, K.S., 2022. "Stochastic delay differential equations: Analysis and simulation studies," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    2. A Chunxiang & Shao Yi, 2018. "Worst-Case Investment Strategy with Delay," Journal of Systems Science and Information, De Gruyter, vol. 6(1), pages 35-57, February.
    3. Wei, Yu, 2012. "Forecasting volatility of fuel oil futures in China: GARCH-type, SV or realized volatility models?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5546-5556.

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