Option Pricing on Stocks with Known and Path-Dependent Dividends with the Stair Tree

Pricing options on known-dividend-paying stocks has traditionally been solved by assuming that the stock price minus the present value of dividends follows the lognormal diffusion. The same assumption is also popular in tree methods to avoid combinatorial explosion. Unfortunately, this assumption undervalues the option since the volatility of the underlying stock price is underestimated. The biases can be significant especially when the volatility of the stock price or the dividend payout is large. Our paper does away with this assumption with a new recombining tree, the stair tree. The stair tree is guaranteed to converge to the lognormal price process of the known-dividend-paying stock. Thus option pricing can be solved efficiently and without bias. Numerical examples verify the algorithm’s superior performance to existing methods. Besides, the stair tree can be extended so the dividends depend on the stock prices in arbitrary ways. This extension makes our model more realistic and flexible. ∗A Ph.D student in Department of Computer Science & Information Engineering, National Taiwan University, Taipei, Taiwan. E-mail: [email protected]. The author was supported in part by NSC grant 92-2213-E-002-016. †Corresponding author. A professor in Department of Finance and (preferred address) Department of Computer Science & Information Engineering, National Taiwan University, No 1, Sec 4, Roosevelt Rd, Taipei, Taiwan. E-mail: [email protected]. TEL: 886-2-23625336 ext. 429. FAX: 886-2-23628167. The author was supported in part by NSC grant 92-2213-E-002-016.