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Stochastic equity volatility related to the leverage effect

Author

Listed:
  • Alain Bensoussan
  • Michel Crouhy
  • Dan Galai

Abstract

We propose a general framework to model equity volatility for a firm financed by equity and additional non-equity sources of funds. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities. Second, we show for the first time in the option literature, that instantaneous equity volatility is a solution of a partial differential equation similar to Black-Scholes', although it is non-linear and in general does not have any analytical solution. However, analytical approximations for equity volatility are proposed for different capital structures: (1) equity and debt, (2) equity and warrants, and (3) equity, debt and warrants. They are shown to be very accurate.

Suggested Citation

  • Alain Bensoussan & Michel Crouhy & Dan Galai, 1994. "Stochastic equity volatility related to the leverage effect," Applied Mathematical Finance, Taylor & Francis Journals, vol. 1(1), pages 63-85.
  • Handle: RePEc:taf:apmtfi:v:1:y:1994:i:1:p:63-85
    DOI: 10.1080/13504869400000004
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    References listed on IDEAS

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    1. Galai, Dan & Masulis, Ronald W., 1976. "The option pricing model and the risk factor of stock," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 53-81.
    2. Crouhy, Michel & Galai, Dan, 1994. "The interaction between the financial and investment decisions of the firm: the case of issuing warrants in a levered firm," Journal of Banking & Finance, Elsevier, vol. 18(5), pages 861-880, October.
    3. 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..
    4. Galai, Dan & Schneller, Meir I, 1978. "Pricing of Warrants and the Value of the Firm," Journal of Finance, American Finance Association, vol. 33(5), pages 1333-1342, December.
    5. 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.
    6. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    7. 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.
    8. Geske, Robert, 1977. "The Valuation of Corporate Liabilities as Compound Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 541-552, November.
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    Citations

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    Cited by:

    1. Jarrow, Robert A. & Turnbull, Stuart M., 2000. "The intersection of market and credit risk," Journal of Banking & Finance, Elsevier, vol. 24(1-2), pages 271-299, January.
    2. Lai, Wan-Ni, 2016. "Evaluating the sovereign and household credit risk in Singapore: A contingent claims approach," Research in International Business and Finance, Elsevier, vol. 37(C), pages 435-447.
    3. Yu-Fu Chen & Michael Funke & Kadri Männasoo, 2005. "Extracting Leading Indicators of Bank Fragility from Market Prices - Estonia Focus," Dundee Discussion Papers in Economics 185, Economic Studies, University of Dundee.
    4. Hanke, Michael & Potzelberger, Klaus, 2002. "Consistent pricing of warrants and traded options," Review of Financial Economics, Elsevier, vol. 11(1), pages 63-77.
    5. Moraux, Franck, 2004. "Modeling the business risk of financially weakened firms: A new approach for corporate bond pricing," International Review of Financial Analysis, Elsevier, vol. 13(1), pages 47-61.
    6. A. Bensoussan & M. Crouhy & D. Galai, 1995. "Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(1), pages 43-60.

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