7. Binary Options (Medium, 15 points) One particular type of exotic option is the binary option....

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7. Binary Options (Medium, 15 points) One particular type of exotic option is the binary option. These are also known as all-or-nothing options or digital options. Broadly speaking, there are two subtypes: cash-or-nothing options and asset-or-nothing options. In this question, we will focus on cash-or-nothing options. A European cash-or-nothing call option with expiration T and strike price K pays out $100 to the holder if ST > K, $50 if ST = K, and pays out nothing otherwise. A European cash-or- nothing put option with expiration T and strike price K pays out $100 to the holder if ST < K, $50 if ST = K, and pays out nothing otherwise. Note that this means that if ST = K, both the put and the call pay out "half." For this problem, let c denote the price of the European cash-or-nothing call, and p denote the price of the European cash-or-nothing put. Let r denote the risk-free interest rate at horizon T (continuously compounded). (a) (5 points) Unlike regular options, both of these European cash-or-nothing options pay out a maximum of $100. Use this observation to find an upper bound on the prices. That is, find the tightest upper bounds and p in terms of (some of) r, T, K, and $100, but not So- (b) (5 points) Consider a European cash-or-nothing call. Suppose, for simplicity, that K = $100, the same as the payout. Can you give a bound on the option price in terms of So? Hint: compare the payoffs from holding the stock to holding the option. (c) (5 points) Find an equation that relates c to p ("put-call parity" for your binary options). Hint: think about the payoff of holding both options. Your "put-call parity" equation will be about cB+pB (instead of the c - p as with normal options). 7. Binary Options (Medium, 15 points) One particular type of exotic option is the binary option. These are also known as all-or-nothing options or digital options. Broadly speaking, there are two subtypes: cash-or-nothing options and asset-or-nothing options. In this question, we will focus on cash-or-nothing options. A European cash-or-nothing call option with expiration T and strike price K pays out $100 to the holder if ST > K, $50 if ST = K, and pays out nothing otherwise. A European cash-or- nothing put option with expiration T and strike price K pays out $100 to the holder if ST < K, $50 if ST = K, and pays out nothing otherwise. Note that this means that if ST = K, both the put and the call pay out "half." For this problem, let c denote the price of the European cash-or-nothing call, and p denote the price of the European cash-or-nothing put. Let r denote the risk-free interest rate at horizon T (continuously compounded). (a) (5 points) Unlike regular options, both of these European cash-or-nothing options pay out a maximum of $100. Use this observation to find an upper bound on the prices. That is, find the tightest upper bounds and p in terms of (some of) r, T, K, and $100, but not So- (b) (5 points) Consider a European cash-or-nothing call. Suppose, for simplicity, that K = $100, the same as the payout. Can you give a bound on the option price in terms of So? Hint: compare the payoffs from holding the stock to holding the option. (c) (5 points) Find an equation that relates c to p ("put-call parity" for your binary options). Hint: think about the payoff of holding both options. Your "put-call parity" equation will be about cB+pB (instead of the c - p as with normal options).