oBIN( ) - Constant Step BOPM Derivation

 Absence of arbitrage opportunities requires that any two portfolios that have the same terminal payoffs must have the same value. Thus, an option can be valued using the known values of securities used to form a replicating portfolio. Determination of potential payoffs to an option requires a specification for the process of the underlying asset. Here, we assume that S follows a binomial random walk. Initially, assume that there is only one period, and that S can take one of two values at the end of the period, If S = 50, u = 15, d = 0.5, then Assume a European call option is written on S with X = 50. Then at the end of the period, Now consider a replicating portfolio of shares of stock and \$B in riskless bonds. The potential values are then in general terms given by: where, r is defined here to be one plus the riskless rate over a fixed time period. If and B are chosen to equate payoffs, then This gives two equations in two unknowns, with the following solutions To prevent arbitrage, we require that Substituting using the previous definitions of and B gives: (1) where, Note that the option value is not dependent on the expected stock return or q. Thus C will be agreed on by all investors, irrespective of their degree of risk aversion or subjective opinions regarding the probabilities of upward or downward moves. Thus, if risk attitudes are not important we can assume that everyone is risk neutral and the expected return for all assets is the riskless rate. In fact, this is implied by the definition of p. If we set the expected return equal to the riskless rate, then: (2) Also, as 0 < p < 1 and p + (1 - p) = 1, we can interpret p as a probability  a probability in a risk neutral world. With this interpretation, C can be regarded as the discounted expected value in a risk neutral world. Now, how can we extend the analysis to a two-period world? (Assuming that the outcome at the second step is independent from that at the first step  the probability of an up-jump at time 2 is the same irrespective of whether there was an up- or down-jump at the previous step). Given the stock price process, we know Cuu, Cud, and Cdd. We can determine Cu and Cd in the same manner as that used to determine C above  this reduces to applying equation (1). And, working back to the present we have, Note how this reflects that for the two step process, we have one path to both Cuu and Cdd (with probabilities p2 and (1 - p)2 and two paths to Cud (with probability p(1 - p). Thus C is simply the expected value of the option at maturity, discounted back to today. With that observation, we can generalize the process to n periods. After n binomial steps, there are n + 1 potential outcomes and 2n potential paths. For n = 3, the outcomes are d3, ud2, u2d and u3, and there are 23 = 8 potential paths. In general, the number of possible paths to a particular terminal outcome (with j up-jumps from a total of n jumps) are computed as: The total set of possible outcomes after n steps is for j = 0, 1, 2, .... , n. Recall that call value can be determined as the discounted expected payoff at maturity. That is: If a is the minimum number of up-jumps such that , then Note that the second term in square brackets is the complementary (cumulative) binomial distribution function, . That is, the cumulative probability that there will be at least a successes from n trials. We can also interpret the first expression in square brackets as a complementary binomial distribution, if we define Thus where, a is the smallest non-negative integer such that , or the smallest integer greater than: For realistic applications of the Binomial model, we need to divide the fixed calendar interval (T) into a 'large enough' number of intervals (n) such that the binomial step in prices over each interval of length is a reasonable approximation to reality. As and in the limit we have a continuous stochastic process. But, we still need to choose appropriate estimates for u, d, (and therefore p). It makes sense that these be chosen such that the mean and variance of stock price changes over the interval h match those implied by the risk-neutral version of the Geometric Brownian Motion process, i.e . Cox, Ross and Rubinstein (1979) show that this requires: where r is now defined as the continuously compounded riskless rate. With these choices, the binomial process for stock prices approaches the GBM process when . 