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Optimal Hedging of Basket Barrier Options with Additive Models and Its Application to Equity Value Separation Problem

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  • Yuji Yamada

    (University of Tsukuba)

Abstract

At the heart of optimal hedging with additive models in Yamada (Recent advances in financial engineering: proceedings of the KIER-TMU international workshop on financial engineering, World Scientific, pp 225–245, 2010; Proceedings of the 2011 American control conference, pp 3856–3861, 2011; Asia-Pac Financ Mark 19(2):149–179, 2012) is to replicate the payoff of European basket options using separate options as close as possible. In this paper, we extend their technique for the case of path-dependent barrier options, where the mean square error of the payoffs between the basket barrier option and the sum of options on the individual assets is minimized over any smooth payoff functions. To this end, we propose to represent the underlying assets using the Brownian bride decomposition and show that computations involving conditional expectations of basket barrier options boil down to those of unconditional expectations. This procedure enables us to provide an algorithm to compute the necessary and sufficient condition for the optimal hedging problem based on the Monte Carlo method. Then, we consider to apply our methodology to the Black–Cox type first passage time structural model, where a defaultable company possesses/runs multiple assets/projects and the default may occur the first time the asset value hits a certain lower threshold before the maturity. We formulate the equity value separation problem using additive models, in which individual equity values are introduced so that their sum approximates the total equity value as close as possible. It is also shown that any portion of total equity value may be assigned as an initial value of each individual equity when using the optimal smooth functions. Finally, we examine the contributions of individual equity values to default or survival by applying a certain normalization for conditional expectations via numerical experiments to illustrate our proposed methodology.

Suggested Citation

  • Yuji Yamada, 2017. "Optimal Hedging of Basket Barrier Options with Additive Models and Its Application to Equity Value Separation Problem," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 24(1), pages 1-18, March.
    • Handle: RePEc:kap:apfinm:v:24:y:2017:i:1:d:10.1007_s10690-016-9221-y
    DOI: 10.1007/s10690-016-9221-y
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    References listed on IDEAS

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    2. 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..
    3. David Hobson & Peter Laurence & Tai-Ho Wang, 2005. "Static-arbitrage upper bounds for the prices of basket options," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 329-342.
    4. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    5. Yuji Yamada, 2012. "Properties of Optimal Smooth Functions in Additive Models for Hedging Multivariate Derivatives," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 19(2), pages 149-179, May.
    6. repec:bla:jfinan:v:53:y:1998:i:3:p:1165-1190 is not listed on IDEAS
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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