the stock price wiZz rise or fall. Bulls and bears tust agree on the value of the call, relative to its underlying stock price Clearly, our numerical example has been chosen for simplicity, not reals Among other things, it gives no consideration to the existing liquid secondary market, which permits closing transactions any tine priot to expiration, and it posiTs Very tmli't i![ t i,' : aek i t i,',m.\ ments. As it turms out, correcting the former deficiency supplies the key to correcting the latter. II. BINOMIAL OPTION PRICING FORMULA To model the value of a call prior to expiration, we start with the simplest nontrivial situation. Suppose the expiration date is just one "period away. Denoting the current stock price as s, we assume it follows a binomial process so that at the end of the period at the ex- piration of the call, its price is either us or ds, with probability q and 1-q, respectively. Therefore, s with probability q ds with probability I-q Letti r denote one plus the ioterest rate over the period,we require u>r>d. If these inequalities did not hold, there would be profitable riskless arbitrage opportunities involving only the stock and riskless borrowing and lending. Although we could simplify the example