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Pricing formula for a Barrier call option based on stochastic delay differential equation

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  • Kim, Kyong-Hui
  • Kim, Jong-Kuk
  • Sin, Myong Guk

Abstract

We derive new explicit pricing formulae for a type of Barrier call option, down and in call option when underlying asset price processes are represented by a stochastic delay differential equation (hereafter “SDDE”). We note the conditional normality of a stochastic integral with respect to a Wiener process to find the joint distribution of the stochastic integral and their minimum. On the basis of this result, we obtain pricing formulae for the Barrier call option which extends ones in the classical Black-Scholes models without delay. Finally, through Monte-Carlo simulations, we demonstrate that our theoretical prices for a Barrier option are correct.

Suggested Citation

  • Kim, Kyong-Hui & Kim, Jong-Kuk & Sin, Myong Guk, 2024. "Pricing formula for a Barrier call option based on stochastic delay differential equation," Statistics & Probability Letters, Elsevier, vol. 205(C).
    • Handle: RePEc:eee:stapro:v:205:y:2024:i:c:s0167715223001670
    DOI: 10.1016/j.spl.2023.109943
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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