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 four assumptions about the market, including no arbitrage and no transaction costs. The (*) 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, name these derivatives whose main varieties are (10[1])called puts and calls (10[1])and give the holder the sell and purchase rights (-5[1])for an underlying asset. ■END■ (10[1]0[2])