Binary options are exotic derivative securities. Two main types of binary op tions are the cash-or-nothing option and the asset-or-nothing option. If the option expires in-the-money; the cash-or-nothing binary option pays a fixed amount of cash while the asset or-nothing option pays the amount of cash equal to the price of the underlying security. Binary options are usually European-style options. More precisely; suppose a cash-or-nothing call option on a stock for B dollar cash strikes at X and matures in T years. It pays the option holder B dollars of cash if Sy X or nothing otherwise, where Sy is the stock price at maturity. Suppose an asset-or-nothing call option on a stock strikes at X and matures in T years. It pays the option holder Sy dollars if Sy X or nothing otherwise. For a long time, binary option contracts have existed over-the-counter (OTC), i.e., issuers directly sell contracts to buyers. In addition, they were often embedded in more complex contracts. In recent years, some binary options websites offer platforms to trade and clear binary options, but the Securities and Exchange Commission (SEC) and Commodity Futures Trading Commission (CFTC) issued a joint warning to investors regarding potential fraud of binary options trading. Legitimate binary option contracts can be priced. Suppose the stock of BinEquity.com is currently traded for $39 per share, and its volatility is 30% per annum. The risk-free interest rate is 3%.

1. A cash-or-nothing call option on the stock for $40 cash strikes at 40 and matures in five months. What is the value of the option? (Use the simulation method to value the option. Choose the size of the simulation to be 50,000. Display the first 10 simulations in your answer.)
2. An asset-or-nothing call option on the stock strikes at 40 and matures in five months. What is the value of the option? (Use the simulation method to value the option. Also, choose the size of the simulation to be 50,000. Display the first 10 simulations in your answer.)
3. A plain-vanilla (i.e., standard) European call option with five-month maturity and strike price 40 pays Sy - 40 if Sy > 40 or O if Sy 40 at maturity. How to combine the binary options in (a) and (b) to obtain the equivalent payoff of the plain-vanilla call option? (Show that your combination of binary options gives exactly the same payoff when Sy> 40 and when Sy 40.
4. Based on the combination of binary options in (c) and the values of binary options you obtained in (a) and (b), what should be the value of the plain-vanilla option in (c)? Is this value consistent with the Black-Scholes formula?