PS7_Solutions_final_v17

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FIN650/Kim 20 points Page 1 of 7 Problem Set 7 Solutions Q1. Suppose you run a Fama-French three- factor regression analysis for an investment fund’s historical performance and get the following results: What kind of stocks does the fund invest in? (Solution) Small growth stocks. The loading for the size factor is positive, and the loading for the value factor is negative. Q2. What does each of the three forms of market efficiency postulate? (Solution) • The weak form postulates that the market prices fully reflect all the past market data. Therefore, investors will not be able to generate any positive alpha using technical analysis. On the other hand, the investors will be able to generate positive alpha by engaging in insider trading, or using fundamental analysis. • The semi-strong form postulates that the market prices fully reflect all publicly known information. Therefore, investors will not be able to generate any positive alpha using technical analysis or fundamental analysis. On the other hand, the investors will be able to generate positive alpha by engaging in insider trading.

FIN650/Kim 20 points Page 2 of 7 • The strong form argues that prices fully reflect all public and private information. Therefore, investors will not be able to generate any positive alpha using insider trading, technical analysis, or the fundamental analysis. Note that, expect for very special circumstances, insider trading is illegal in most cases. Q3. List all forms of market efficiency that the momentum anomaly violates. (Solution) Momentum anomaly violates all three forms of the EMH because it uses past prices (i.e., the information set which is available to investors in all three forms of the EMH) yet generates stable and positive abnormal returns over the years. Q4. Today, you buy a European put option on Tesla stock with an exercise price of $270. The option expires after one month. Assume that the risk-free interest rate is 0% per month and that the one option contract represents one share of the underlying. Also, assume today’s option premium is $30. a. What is your payoff of this put option at expiration if Tesla’s stock price is trading for $200 at expiration? b. What is your payoff of this put option at expiration if Tesla’s stock price is trading at $310 at expiration? c. What is your profit on this put option at expiration if Tesla’s stock price is trading at $200 at expiration? d. What is your profit on this put option at expiration if Tesla’s stock price is trading at $310 at expiration? (Solution) Part a. The payoff from buying a put option is as follows: +𝑃 𝑇 = max (𝑋 − 𝑆 𝑇 , 0) In this case:

FIN650/Kim 20 points Page 3 of 7 +𝑃 𝑇 = max (270 − 200,0) = 𝑚𝑎𝑥(70,0) = +$70 If the put buyer (the long position holder) chooses to exercise the option at expiration, the put option buyer can sell the underlying asset (one share of Tesla stock) that is worth $200 in the market (on the day of expiration) for $270 (the exercise price). Therefore, the put buyer will exercise the put option at expiration by: [Step 1] Buying one share of Tesla stock for $200 in the market [Step 2] Immediately delivering the Tesla stock to the put option seller [Step 3] Receiving $270 (the exercise price) from the put seller. Through this process, the put buyer will generate a payoff of +$70 from the put option. There are a couple of points to note here: 1. This also shows that, in general, the put option buyer will only exercise the put option if the 𝑆 𝑇 < 𝑋 at expiration. 2. In practice, the difference between 𝑋 and 𝑆 𝑇 (i.e., 𝑋 − 𝑆 𝑇 ), +$70, is settled in cash without any actual exchange of the underlying asset. Therefore, the put seller simply gives $70 in cash to the put option buyer. 3. Also note that long put is a bearish strategy – you are betting that the underlying price will be below the strike price at expiration. Part b. Again, the payoff from buying a put option is as follows: +𝑃 𝑇 = max (𝑋 − 𝑆 𝑇 , 0) In this case: +𝑃 𝑇 = max (270 − 310,0) = 𝑚𝑎𝑥(−40,0) = $0 If the put buyer chooses to exercise the option, the put option buyer can sell the underlying asset (one share of Tesla stock) that is worth $310 in the market (on the day of expiration) for just $270. If the put buyer sells the underlying that is worth $310 for $270, then the put buyer would suffer a loss of $40. Therefore, the put option buyer does not exercise the option. Instead, the put option buyer simply lets the option expire worthless, which results in a payoff of $0 for the put option buyer. Part c. We know from Part (a) that the payoff from the long put position is +$70 at expiration. But the

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FIN650/Kim 20 points Page 4 of 7 buyer has to pay a price, an option premium of $30, in order to buy this put option today. Since this is a cost that the buyer incurred, $30 should be subtracted from the +$70 payoff the put option buyer collects on the expiration day. Therefore, the profit is $70 − $30 = +$40.00 Part d. We know from Part (b) that the payoff from the long put position is +$0 at expiration. But, again, the buyer has to pay a price today, an option premium of $30, in order to buy this put option today, even if it expires worthless at expiration. Since this is a cost that the buyer incurred, $30 should be subtracted from the $0 option payoff the put option buyer collects on the expiration day. Therefore, the profit is $0 − $30 = −$30 . In other words, a loss of $30. Q5. Today, you buy a European call option on Tesla stock with an exercise price of $270. The option expires after one month. Assume that the risk-free interest rate is 0% per month and that the one option contract represents one share of the underlying. Also, assume today’s option premium is $30. a. What is your payoff of this call option at expiration if Tesla’s stock price is trading for $200 at expiration? b. What is your payoff of this call option at expiration if Tesla’s stock price is trading at $310 at expiration? c. What is your profit on this call option at expiration if Tesla’s stock price is trading at $200 at expiration? d. What is your profit on this call option at expiration if Tesla’s stock price is trading at $310 at expiration? (Solution) Part a. The payoff from buying a call option is as follows: +𝐶 𝑇 = max (𝑆 𝑇 − 𝑋, 0) In this case: +𝐶 𝑇 = max (200 − 270, 0) = 𝑚𝑎𝑥(−70,0) = $0

FIN650/Kim 20 points Page 5 of 7 If the call buyer chooses to exercise the option, the call option buyer can buy the underlying asset (one share of Tesla stock) that is worth $200 in the market (on the day of expiration) for $270 (strike price). If the call buyer buys the underlying that is worth $200 for $270, then the call buyer would suffer a loss of $70. Therefore, the call option buyer does not exercise the option. Instead, the call option buyer simply lets the option expire worthless, which results in a payoff of $0 for the call option buyer. Note that long call is a bullish strategy – you are betting that the underlying price will be above the strike price at expiration. Part b. The payoff from buying a call option is as follows: +𝐶 𝑇 = max (𝑆 𝑇 − 𝑋, 0) In this case: +𝐶 𝑇 = max (310 − 270, 0) = 𝑚𝑎𝑥(40,0) = +$40 If the call buyer (the long position holder) chooses to exercise the option, the call option buyer can buy the underlying asset (one share of Tesla stock) that is worth $310 in the market (on the day of expiration) for just $270 (the exercise price). Therefore, the call buyer will exercise the call option at expiration by: [Step 1] Paying $270 to the call option seller (the short position holder) [Step 2] Collecting one share of Tesla (the underlying) from the call seller [Step 3] Immediately selling it on the stock market for $310. Through this process, the call buyer will generate a payoff of +$40 from the call option. There are a couple of points to note here: 1. This also shows that, in general, thecall option buyer will only exercise the call option if the 𝑆 𝑇 > 𝑋 at expiration. 2. In practice, the difference between X and 𝑆 𝑇 (i.e., 𝑆 𝑇 − 𝑋 ), +$40, is settled in cash without any actual exchange of the underlying asset. Therefore, the call option seller simply gives $40 in cash to the call option buyer. 3. Also note that long call is a bullish strategy – you are betting that the underlying price will be above the strike price at expiration. Part c.

FIN650/Kim 20 points Page 6 of 7 We know from Q3 Part (a) that the payoff from the long call position is +$0 at expiration. But, again, the option buyer has to pay a price today, an option premium of $30, in order to buy the call option today, even if it expires worthless at expiration. Since this is a cost that the buyer incurred, $30 should be subtracted from the $0 option payoff the call option buyer collects on the expiration day. Therefore, the profit is $0 − $30 = −$30 . In other words, a loss of $30 Part d. We know from Part (b) that the payoff from the long call position is +$40 at expiration. But the buyer has to pay a price, an option premium of $30, in order to buy this call option today. Since this is a cost that the buyer incurred, $30 should be subtracted from the +$40 option payoff the call option buyer earns on the expiration day. Therefore, the profit is $40 − $30 = $10 . In other words, the call option buyer makes a profit of $10. Q6. You write (short) a European call option on Tesla stock with an exercise price of $270. Assume that the risk-free interest rate is 0% per month and that the one option contract represents one share of the underlying. The option expires after one month. Also, assume today’s option premium is $20. a. What is the profit on this option if the stock price is $320 at expiration? b. What is the payoff of this option if the stock price is $245 at expiration? (Solution) Part a First, compute the payoff. The payoff at expiration from writing a call option is as follows: Payoff from writing a call option = −𝐶 𝑇 = −max( 𝑆 𝑇 − 𝑋, 0) In our case: Payoff from writing a call option = −𝐶 𝑇 = −max( 320 − 270,0) = −(max (50,0)) = −(50) = −50 Note that you wrote the call option hence you collected a premium of +$20. Therefore, the profit is −50 + 20 = −$?𝟎

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FIN650/Kim 20 points Page 7 of 7 Part b Let us compute the payoff. The payoff at expiration from writing a call option is as follows: −𝐶 𝑇 = −max( 𝑆 𝑇 − 𝑋, 0) −𝐶 𝑇 = −max( 245 − 270,0) = −(max (−25,0)) = −(0) = 𝟎 Q7. You write (short) a European put option on Tesla stock with an exercise price of $270. Assume that the risk-free interest rate is 0% per month and that the one option contract represents one share of the underlying. The option expires after one month. Also, assume today’s option premium is $22. a. What is the profit on this option if the stock price is $300 at expiration? b. What is the payoff of this option if the stock price is $245 at expiration? (Solution) Part a First, compute the payoff. The payoff at expiration from writing a put option is as follows: Payoff from writing a put option = −P T = − max (X − S T , 0) In our case: −P T = − max (270 − 300,0) = −(max(−30,0)) = −(0) = 0 Note that you wrote the put option hence you collected a premium of +$22. Therefore, the profit is 0 + 22 = + $?? Part b Let us compute the payoff. The payoff at expiration from writing a put option is as follows: −P T = − max (X − S T , 0) −P T = − max (270 − 245, 0) = −(max(25,0)) = −(25) = −$?𝟓

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