Pricing two-asset alternating barrier options with icicles and their variations. (English) Zbl 1485.62147

J. Korean Stat. Soc. 49, No. 2, 626-672 (2020); correction ibid. 49, No. 2, 625 (2020).
Summary: This paper introduces a new class of barrier options and its variations. We call the new class of options as two-asset alternating barrier options, since we consider alternating barrier levels for two underlying assets. The alternating barrier levels are placed in the sub-periods of the option’s lifetime; each being applied to one of the two underlying assets. We also consider vertical branches of the barrier, which are termed as icicles. The alternating barrier with icicles can be often seen as an embedded form in various equity-linked financial products. To price such new options, we obtain the joint distribution of two underlying asset prices at an intermediate time point and the maturity, along with their partial maximums under the Black-Scholes model. This joint distribution plays a critical role in the derivation of the pricing formulas for alternating barrier options and their variants. As in ordinary barrier options, we consider eight types of alternating barrier options and derive their explicit option pricing formulas. To our knowledge, the pricing formulas for these options have never been obtained explicitly in the literature even under the Black-Scholes model. We also examine an autocallable equity-linked investment product to derive its explicit pricing formula. Our results are illustrated with numerical examples, showing the effect of different barrier levels and different values of correlation coefficient between two underlying asset prices.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91G20 Derivative securities (option pricing, hedging, etc.)
Full Text: DOI


[1] Buchen, P., An introduction to exotic option pricing (2012), Boca Raton: CRC Press, Boca Raton · Zbl 1242.91183
[2] Gerber, HU; Shiu, ESW, Option pricing by Esscher transform, Transactions of Society of Actuaries, 46, 99-140 (1994)
[3] Gerber, HU; Shiu, ESW, Actuarial bridges to dynamic hedging and option pricing, Insurance: Mathematics and Economics, 18, 183-218 (1996) · Zbl 0896.62112
[4] Gerber, HU; Shiu, ESW; Yang, H., Valuing equity-linked benefits and other contingent options: A discounted density approach, Insurance: Mathematics and Economics, 51, 73-92 (2012) · Zbl 1284.91233
[5] Guillaume, T., Step double barrier options, Journal of derivatives, 18, 59-80 (2010)
[6] Harrison, JM, Brownian motion and stochastic flow systems (1990), Florida Malabar: Krieger Publishing Company, Florida Malabar · Zbl 0716.60092
[7] Heynen, RC; Kat, HM, Partial barrier options, The Journal of Financial Engineering, 3, 4, 253-274 (1994) · Zbl 1153.91507
[8] Huang, Y-C; Shiu, ESW, Discussion of “Pricing Dynamic Investment Fund Protection”, North American Actuarial Journal, 4, 1, 153-157 (2001) · Zbl 1083.91542
[9] Hwang, Y-W; Chang, S-C; Wu, Y-C, Capital forbearance, ex ante life insurance guaranty schemes, and interest rate uncertainty, North American Actuarial Journal, 19, 2, 94-115 (2015) · Zbl 1414.91204
[10] Lee, H., Pricing equity-indexed annuities with path-dependent options, Insurance: Mathematics and Economics, 33, 677-690 (2003) · Zbl 1103.91368
[11] Lee, H.; Ko, B., Valuing equity-indexed annuities with icicled barrier options, Journal of the Korean Statistical Society, 47, 330-346 (2018) · Zbl 1410.91457
[12] Ng, AC-Y; Li, JS-H, Valuing variable annuity guarantees with the multivariate Esscher transform, Insurance: Mathematics and Economics, 49, 393-400 (2011) · Zbl 1228.91044
[13] Tiong, S., Valuing equity-indexed annuities, North American Actuarial Journal, 4, 4, 149-163 (2000) · Zbl 1083.62545
[14] Wang, X., Discussion of “Capital forbearance, ex ante life insurance guaranty schemes, and interest rate uncertainty”, North American Actuarial Journal, 20, 1, 88-93 (2016) · Zbl 1414.91242
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