Two-State Option Pricing 1099 IV. Applications of the Model European Puts and Calls on Non Dividend Paying Stocks In this section we price European put and call options on non-dividend paying stocks using the two-state model as an approximation procedure for the case in which stock prices are assumed to follow a lognormal distribution. Given the assumptions of no dividends and lognormal returns, the Black-Scholes[3] model provides the exact values for both types of options, thereby serving as a bench mark to assess the accuracy of the two-state model as a numerical procedure In Table l, we present prices of one-year European put and call options with exercise prices of $75, $100, and $125 assuming a current stock price of $100. The riskless interest rate is assumed to be 5% per year. To conform with the black- Scholes model, continuous compounding of interest is assumed. Thus, R=e5/N 1, where N is the number of time intervals per year employed in the analysis The values of h+ and H- are chosen so that the annual standard deviation of the logarithmic return is, 324 as in the previous four-period example. The expected value of the logarithmic return is assumed to take on values of. 5,. 1, 0, -1 and 5 per year, and a value of 8 equal to 5 is assumed. Finally, option pric ces are calculated by partitioning the year into 12, 52, and 100 time periods Consider the panel of Table 1 in which the stock' s growth rate (u)is assumed to be 0%. When the year is divided into 100 time intervals, the two-state prices of the put and call options are quite close to their corresponding Black-Scholes prices. With these two parameters (u =0, T=100), the greatest absolute percentage difference between the black-Scholes and two-state prices is. 6% Even if only 12 time differencing intervals are assumed, the two-state and black Scholes prices are remarkably close For growth rates of 10% and-10%, the two-state prices do not appear to be significantly different from those obtained when a zero growth rate is assumed Thus, within this range of growth rates, the option price does not appear to be significantly affected by the growth rate If extreme growth rates are assumed (u= 5 and 4=-.5), the two-state model does not appear to provide an accurate approximation to the Black-Scholes price for low T values. However for 100 time intervals, the two-state and Black-scholes prices are reasonably close The entries in Table 1 reveal that the option price is slightly dependent upon the stock's growth rate. In addition, if 0 were varied we would also discover a slight dependence on investor probability beliefs. These findings seem to contra dict the earlier observation that two state prices are independent of both investor preferences(which would be revealed through p) and probability beliefs This dependence results from the fact that h* and H- are chosen in the two state model to conform with a given continuous distribution Since the two-stat model is only an approximation, the values of u and e implicit in the continuous distribution may be reflected in the two-state solution. In the limit as T-o, the two distributions will be identical, and therefore, preferences and probabilities 2 See Brennan and Schwartz [4] and Parkinson [7] for descriptions of alternative numerical procedures for solving option pncing equations