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The Implied Distribution for Stocks of Companies with Warrants and/or Executive Stock Options

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  • Darsinos, T.
  • Satchell, S.E.

Abstract

This paper sets out to provide a risk-management tool (namely the distribution of the stock price of a warrant-issuing firm) and at the same time resolves an outstanding issue between the theory and the empirical evidence of the warrant pricing literature. In their seminal article on warrant pricing, Galai and Schneller (1978) make the following statement: “…if the distribution of the firm’s liquidation value is lognormal, the value of its share price is not lognormally distributed”. On the other hand recent empirical studies suggest that assuming lognormality for the stock price distribution of a warrant-issuing firm gives a very good approximation for the value of a warrant (this is the so-called “option-like” warrant valuation approximation). We show that despite of the fact that the (risk-neutral) distribution of a warrant-issuing firm and a non-warrant issuing firm is different, valuation by taking expectations of the discounted payoff of the warrant over the two different risk-neutral distributions produces warrant prices very close to each other for a large number of cases. Exceptions occur for deep-out-of-the-money and close to maturity out-of-the-money warrants in general. In such cases the “option-like” approximation will significantly overprice warrants.

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Bibliographic Info

Paper provided by Faculty of Economics, University of Cambridge in its series Cambridge Working Papers in Economics with number 0217.

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Length: 43
Date of creation: Jun 2002
Date of revision:
Handle: RePEc:cam:camdae:0217

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Keywords: warrants; executive stock options; risk-neutral; distribution;

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References

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  1. Bauwens, Luc & Lubrano, Michel & Richard, Jean-Francois, 2000. "Bayesian Inference in Dynamic Econometric Models," OUP Catalogue, Oxford University Press, number 9780198773139, September.
  2. Veld, C.H., 1994. "Warrant pricing: A review of empirical research," Discussion Paper 1994-34, Tilburg University, Center for Economic Research.
  3. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
  4. Lauterbach, Beni & Schultz, Paul, 1990. " Pricing Warrants: An Empirical Study of the Black-Scholes Model and Its Alternatives," Journal of Finance, American Finance Association, vol. 45(4), pages 1181-209, September.
  5. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
  6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
  7. repec:cup:etheor:v:13:y:1997:i:6:p:791-807 is not listed on IDEAS
  8. Schulz, G. Uwe & Trautmann, Siegfried, 1994. "Robustness of option-like warrant valuation," Journal of Banking & Finance, Elsevier, vol. 18(5), pages 841-859, October.
  9. Carpenter, Jennifer N., 1998. "The exercise and valuation of executive stock options," Journal of Financial Economics, Elsevier, vol. 48(2), pages 127-158, May.
  10. Knight, John L & Satchell, Stephen E., 1997. "Existence of Unbiased Estimators of the Black/Scholes Option Price, Other Derivatives, and Hedge Ratios," Econometric Theory, Cambridge University Press, vol. 13(06), pages 791-807, December.
  11. Jarrow, Robert & Rudd, Andrew, 1982. "Approximate option valuation for arbitrary stochastic processes," Journal of Financial Economics, Elsevier, vol. 10(3), pages 347-369, November.
  12. Geske, Robert, 1979. "The valuation of compound options," Journal of Financial Economics, Elsevier, vol. 7(1), pages 63-81, March.
  13. 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 1333-42, December.
  14. Yacine Aït-Sahalia, 1999. "Transition Densities for Interest Rate and Other Nonlinear Diffusions," Journal of Finance, American Finance Association, vol. 54(4), pages 1361-1395, 08.
  15. 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 637-54, May-June.
  16. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Analysis of the Black-Scholes Option Price," Cambridge Working Papers in Economics 0102, Faculty of Economics, University of Cambridge.
  17. Brian J. Hall & Kevin J. Murphy, 2000. "Optimal Exercise Prices for Executive Stock Options," NBER Working Papers 7548, National Bureau of Economic Research, Inc.
  18. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
  19. Butler, J. S. & Schachter, Barry, 1986. "Unbiased estimation of the Black/Scholes formula," Journal of Financial Economics, Elsevier, vol. 15(3), pages 341-357, March.
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Citations

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Cited by:
  1. Darsinos, T. & Satchell, S.E., 2002. "On the Valuation of Warrants and Executive Stock Options: Pricing Formulae for Firms with Multiple Warrants/Executive Options," Cambridge Working Papers in Economics 0218, Faculty of Economics, University of Cambridge.
  2. Xiao, Wei-Lin & Zhang, Wei-Guo & Yao, Zheng & Wang, Xiao-Hui, 2013. "The impact of issuing warrant and debt on behavior of the firm's stock," Economic Modelling, Elsevier, vol. 31(C), pages 635-641.

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