Two-State Option Pricing allowing the number of time differencing intervals to become infinite over the fixed life of the option Before deriving the continuous time version of the model we will develop a valuation equation for the discrete time case under the assumptions that the distribution of returns of the stock is stationary over time and the stock pays no dividends. The Discrete Time model When the option matures, there will be a one-to-one correspondence between the value of the option and the value of its underlying stock. The value of a call option at maturity, wr, is max(o, ST-X), where Sr is the value of the underlying stock at the maturity date, T, and X is the exercise price of the option. At the period prior to the option's maturity date, the value of the option is given by u(1+R一H)+(H-(1+R)) WUT-Y (A.1 H-H)(1+R) Similarly, the value of the option two periods prior to maturity is u-(1+R一H)+lr-1(H-(1+R) LUT-2 (H-H)(1+R) By substituting equation (A 1)into (A 2) and noting that the term w is the option value at maturity, given that the price of the underlying stock advances in period T-1 and falls in period T, the value of the option at period T-2 (1+R-H)+u(Ht-(1+R))(1+R-H) (H+-H)2(1+R) (ur(1+R-H)+lr(H+-(1+R))(H-(1+).(A3) H)(1+R)2 Equation(.3)can be simplified by noting that wt wT, since the value of the underlying stock at maturity will be the same whether or not it advances first and then declines, or declines first and then advances. With this substitution equation(A 3) can be restated as u(1+R-H)2+2u(H-(1+R)(1+R-H) +ur(H+-(1+R)2 (H'-H)2(1+R)2 (A.4) If this same type of procedure is repeated for a total of T periods, there always be T 1 terms in the numerator of the option valuation equation. A T periods, there are exactly ( T a ways that a sequence of T pluses can occur, there are[ ways that T-1 pluses can occur along with one minus, there are 2/ways for T-2 pluses and 2 minuses, and so on. In addition, the power to