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 u as “e raised to sigma square root delta t” and d as the inverse of u. A model originally for these things makes 4 assumptions about the market, including no arbitrage and no transaction costs. (-5[1])The (15[1])(*) American type of these things can be exercised at any time, in contrast 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 that give the holder (-5[1])the sell and purchase rights for an underlying asset. ■END■