Abstract
This paper compares the American option prices with one known dividend under two alternative specifications of the underlying stock price: displaced log normal and log normal processes. Many option pricing models follow the standard assumption of the Black–Scholes model (Journal of Political Economy 81:637–659, 1973) in which the stock price, follows a log normal process. However, in order to reach a closed form solution for the American option price with one known dividend, Roll (Journal of Financial Economics 5:251–258, 1977), Geske (Journal of Financial Economics 7: 63–81, 1979), and Whaley (Journal of Financial Economics 9:207–211, 1981) assume a displaced lognormal process for the cum-dividend stock price which results in a lognormal process for the ex-dividend stock price. We compare the two alternative pricing results in this paper.
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Notes
- 1.
Note that the call option has a lower bound of S − e − r(T − t) K. As a result, αD > K(1 − e − r(T − t)).
- 2.
Equivalently, according to GRW the ex-dividend price process follows a log normal diffusion:
$$\frac{\mathit{dS}} {S} = \mathit{rdt} + \sigma \mathit{dW}$$that readily implies that f(ln S t , ln S T ) is a bivariate normal with correlation coefficient \(\sqrt{t/T}\).
- 3.
The binomial model is run on 1,000 steps. We verified that the binomial price for the compound option matches up to the third decimal place with the Geske formula (for strikes 10 and 100, both models give 7.996).
References
Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–659.
Geske, R. 1979a. “The valuation of compound options.” Journal of Financial Economics 7, 63–81.
Geske, R. 1979b. “A note on an analytical valuation formula for unprotected American call options on stocks with known dividends.” Journal of Financial Economics 7, 375–380.
Roll, R. 1977. “An analytical valuation formula for unprotected American call options on stocks with known dividends.” Journal of Financial Economics 5, 251–258.
Whaley, R. 1981. “On the valuation of American call options on stocks with known dividends.” Journal of Financial Economics 9, 207–211.
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Chen, RR., Lee, CF. (2010). Displaced Log Normal and Lognormal American Option Pricing: A Comparison. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_29
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DOI: https://doi.org/10.1007/978-0-387-77117-5_29
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