A 1977 paper by Phelim Boyle pioneered Monte Carlo methods for valuing these things which are optimal to use when there are multiple sources of uncertainty. William Sharpe proposed a binomial tree model for the price of these things which under the CRR method sets (15[1])u as “e raised to sigma square root delta t” (-5[1])and d as the (15[1])inverse of u. A model originally for these things makes 4 assumptions about the market, including (15[1])no arbitrage and no (15[1])transaction costs. (15[1]-5[2])The (15[1])(*) American type of these things can (10[1])be exercised (10[1])at any time, (-5[1])in contrast (10[1])to the European type of these things, whose price can be calculated using the Black-Scholes formula. For 10 points, (10[1])name these derivatives whose main varieties are called puts and calls (10[1])that give the holder (-5[1])the sell and purchase rights for an underlying asset. ■END■ (10[1]0[2])