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, if 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 ST X or nothing otherwise, where ST is the stock price at maturity. If an asset-or-nothing call option on a stock strikes at X and matures in T years, it pays the option holder ST if ST X or nothing otherwise.

For a long time, binary options 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 BinaryBets.com is currently traded for $48 per share, and its volatility is 30% per annum. The risk-free interest rate is 6%.

(a) A cash-or-nothing call option on the stock for $40 cash strikes at 50 and matures in six months. What is the value of the option? (Use the simulation method to value the option. Choose the size of the simulation to be 10,000. Display the first 10 simulations in your answer.)

(b) Use pricing formula to calculate the value of the above cash-or-nothing call option and compare it with the value you obtained from simulation.

(c) An asset-or-nothing call option on the stock strikes at 50 and matures in six 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 10,000. Display the first 10 simulations in your answer.)

(d) Use the pricing formula to calculate the value of the above asset-or-nothing call option and compare it with the value you obtained from simulation.