See attached the assignment
McGill Desautels Daniel Andrei, Financial Derivatives FINE 448, Winter 2022 Assignment #1 Due date: Thursday, Feb. 17, 5pm This is an open book group assignment. While you are not allowed to copy from different groups, you are encouraged to discuss conceptual problems with all your colleagues. No late assignments will be accepted, no exceptions. Please send the assignment by email (PDF file) to
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[email protected]. Make sure your answers are clear and legible. Write down clearly the names of all group members on the assignment. No need to send your codes. Any appeal to your grade must be submitted in writing no later than one week after the grades have been announced. A request for a re-grade will result in a full second evaluation of all questions. The new outcome may be higher, the same, or lower than the initial grade. For clarification questions about the assignment, please consult with Nan Ma, the T.A. of the class. Alternatively, you can consult with me during office hours. No more questions will be answered 36 hours before the submission deadline. Good luck! 1 mailto:
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[email protected] 1 Equity Index Futures a. Find information online about the NYSE FANG+ Index and answer. How is it calculated? What are the current constituents of the index? What does FANG represent? What is the difference between FANG and FANG+? Note that there are two common versions of the index: price index (or price return index) and total return index. What is the difference between these two versions? b. Check the contract specifications of NYSE FANG+ Index futures on the web- sites of the Intercontinental Exchange (ICE) and answer. What is the contract size? How is the price quoted? What is the tick size? When was the last trad- ing day for the September 2018 contract? Is there a daily price limit for this contract? How is the contract settled? What is the final settlement price? Is it based on the price index or the total return index? c. Table 1 shows the daily settlement price, trading volume, and open interest of September 2018 NYSE FANG+ Index futures between 23 July 2018 and 1 August 2018. What is the relation between daily trading volume and open interest? Is it possible that the daily trading volume exceeds open interest? What does this imply about the liquidity of the futures contract? Table 1: Daily market data of September 2018 FANG+ Index futures and your ac- count. Date Settlement Volume Open interest P&L ($) Margin account ($) Margin call ($) 2018-07-23 2,936.9 1,873 1,274 - 2018-07-24 2,942.4 4,774 1,433 825 2018-07-25 3,012.6 2,879 1,424 2018-07-26 2,925.5 2,307 1,456 2018-07-27 2,822.3 1,911 1,435 2018-07-30 2,741.8 1,687 1,468 2018-07-31 2,755.4 1,851 1,468 2018-08-01 2,757.2 1,184 1,301 Source: Bloomberg Alicia, an expert in computer science, is optimistic about the future development of the high-tech industry. After some investigations about the financial market, she decides to trade the NYSE FANG+ Index futures. d. On 23 July 2018, Alicia takes a position in the September 2018 contract at the settlement price and her profit and loss (P&L) on the next day is shown in the table. How many contracts does she purchase or sell short? Does the clearing house have a long or a short position corresponding to her trade? Assume that the current maintenance margin of the September 2018 NYSE FANG+ Index futures contract is $7, 574 per contract and that the initial margin for her is 2 10% more than her maintenance margin. What are the initial and maintenance margin requirements for her position? e. Alicia initially deposits an amount equal to her initial margin requirement in her margin account when she opens the position and she will not add extra cash into her margin account unless she has to. Complete the three columns of Table 1 in blank: daily P&L, margin account balance right after daily settlement (prior to margin call), and the variation margin if a margin call is received. How many margin calls does Alicia receive? f. Note that there was a large drop in the futures price (as well as the index level) starting from July 25. What event triggered this plunge in FANG+ stocks? g. Calculate the return on capital (ROC) of Alicia’s position between July 24 and August 1. ROC is defined as the P&L divided by the amount of total investment (ignoring the time value of money and opportunity cost for simplicity). 3 2 Exotic Options: The Log Contract A log contract is an important building block in volatility derivatives and in the calculation of the VIX. As you will see later during this course, the log contract allows you to achieve pure exposure to fluctuations in volatility. The payoff from a log contract at maturity T is simply the natural logarithm of the underlying asset divided by the strike price, ln(ST/K). The payoff is thus nonlinear and has many similarities with options. The value of this contract at time t = 0 for a non-dividend paying asset is L0 = e −rT [ ln ( S0 K ) + ( r − 1 2 σ2 ) T ] . (1) In this exercise, you will find the price of a log contract using a binomial model. We will consider a log contract that pays LT = 1000 ln(ST/K) (which is equivalent with 1000 individual log contracts). Report answers with 4 decimal digits. Consider the following inputs for a non-dividend paying stock • The initial stock price S0 • The payoff function LT = 1000 ln(ST/K) • The interest rate r • The length of the period h • The up and down factors u and d • The number of periods n a. Compute the initial value of a log contract with n = 4, r = 0.05, h = 1, u = erh+σ √ h, d = erh−σ √ h, S0 = 100, σ = 0.3, and strike K = 90. Show in a table the values of the log contract on all nodes of the tree. What is the replicating portfolio at time 0? What is the difference between the closed-form price (use Eq. (1)) and the initial value of the log contract that you found above? b. Compute the initial value of a log contract with n = 48, r = 0.05, h = 1/12, u = erh+σ √ h, d = erh−σ √ h, S0 = 100, σ = 0.3, and strike K = 90. (I have underlined the parameters that are different from the previous point.) What is the replicating portfolio at time 0? What is the difference between the closed-form price (use Eq. (1)) and the initial value of the log contract that you found above? 4 3 Exotic Options: Asian Options An Asian option is an option on the average price of the underlying asset. One particular application of Asian options is hedging exchange rate risk. Asian options are also popular in the energy OTC market and many commodity markets. Further- more, because an Asian option is based on a price average, an attempt to manipulate the asset price just before expiration will normally have little effect or no effect on the option’s value. Asian options should therefore be of particular interest in markets for thinly traded assets. Consider the case of an Asian call option with discrete arithmetic averaging. An option with maturity of T years and strike K has the payoff max { 1 n n∑ i=1 S(ti)−K, 0 } (2) where ti = i× h and h = T/n. Notice that we do not include in the summation the price of the asset today, i.e. S(t0). Furthermore, assume S(t0) = 200, r = 0.02, σ = 0.20, K = 220, T = 1, n = 365. a. Use Monte Carlo simulation to obtain one price path starting from S(t0) = 200 to S(tn). To simulate the process, consider its values at each step: S(ti+1) = S(ti) exp [( r − 1 2 σ2 ) ∆t+ σ √ ∆t�t ] (3) where ∆t = T/n is the time interval, r and σ are given above, and �t is a random value drawn from a standard normal distribution. Plot the resulting price path. b. Build a second plot that has two lines: (i) The price path from the previous point (ii) The arithmetic average of the price path at each point in time ti, starting from S(t0). That is, your second line on the plot should be: S(t0), S(t1) 1 , S(t1) + S(t2) 2 , S(t1) + S(t2) + S(t3) 3 , · · · ∑n i=1 S(ti) n (4) What do you observe? Do you think the price of the Asian option should be lower, equal, or higher compared with a similar standard option? Why? c. Price the option by Monte Carlo simulation using 100 paths. Compute the price of the option on each path with the formula CA0 = e −rT ×max { 1 n n∑ i=1 S(ti)−K, 0 } (5) 5 What is the average of CA0 over the 100 simulated prices? State the 95 percent confidence interval of this average. (For this last question, you might need to revisit statistical notions like “standard error of the mean” and “confidence interval”). d. Redo the previous point, but now with 100,000 paths. Compare the confidence interval obtained in this case with the confidence interval from the previous point. 6