Theory-Questions 2 (both a&b) and Question 3 (both 1 &2) are to be answered fully. Use the file attached to see the questions under the heading- Theory

Columbia University MATH GR5030 Spring 2017 Numerical Methods in Finance Professor Tat Sang Fung Homework 1 version 1e: Excel practice and Warm up Homework Distribution date Due date Jan 25, 2017 (Wed) Feb 15, 2017 (Wed) 7:30pm Note: please send all homework solution to TA before the due date and time. Spreadsheet should be uploaded from CourseWorks. For theory part, please deposit it into the homework box in MATH building 4/F. If you suspect there are typos in this homework, or some questions are wrong, please feel free to email the instructor QUESTION 1 If you have forgotten the Black-Scholes formula for currency call option1, you may want to review [Hull] 9th edition chapter 17. The Greeks formulas are somehow given in Chapter 19. Make an Excel spreadsheet that does the following: 1. User inputs spot, domestic interest rate, foreign interest rate, volatility, strike, today's date, expiration date 2. The spreadsheet outputs (simultaneously or based on a user choice): Value, Delta, Gamma, Vega, Theta, Phi (rate of change of value with respect to change of foreign interest rate), Rho (rate of change of value with respect to change of domestic interest rate) of the call option2 3. The user can select one Greek from the list of the Greeks above, the starting spot, the end spot and the number of spots between the start and end value, (maybe then press a button), the spreadsheet gives the spot-Greek graph. Save your spreadsheet with the following data while exhibiting the spot-Theta graph (a USD/JPY call option, valued in JPY): Spot is 120, USD zero rate is 4%, JPY zero rate is 0.5%, strike is 119, vol is 10%, time to expiration is 92 calendar days. The user then wants to see the spot-Theta graph from spot = 100 to spot = 150 with 99 points in between. So graph is between from Theta with spot = 100, 100.5, 101, 101.5, ..., 149.5, 150. THEORY QUESTION 1 Since a call option can be thought of, roughly, as insurance, it seems natural to expect that when tomorrow comes, its offers one less day's protection and hence it should lose value. In finance terms, this means its theta should be negative. If you read [Hull] 9th edition chapter 19 page 410, or 8th edition chapter 18 page 388, it says \"Theta is usually negative for an option\". And then the footnote says \"An exception to this could be an in-the-money 1 You should have learned it from the class \"introduction to math finance\" in the fall semester If you are in doubt, in this homework a USD/JPY call with strike 119 is an option such that on expiration the holder has the right but not the obligation to pay JPY 119 and receive USD 1. Sometimes in the market it is called a \"Call USD/Put JPY\" option but we don't need to confuse ourselves here. Professor Fung Page 1 1/26/2017 2 Numerical Methods in Finance European put option on a non-dividend paying stock or an in-the-money European call option on a currency with a very high interest rate.\" a) Use the graph you built in the practice section to see that it is true that for an in-the-money European call option on a currency with an interest rate larger than the domestic currency (e.g. USD zero rate > JPY zero rate when the USD/JPY call is valued in JPY). Print the graph and hand it in. b) Suppose the USD zero rate is less than JPY zero rate (e.g. USD zero rate is 0.5% and JPY zero rate is 4%), do you think the call option theta can be positive? c) A Trader tells you \"for any European call option, if it is sufficiently deep in the money, its theta (given by the BS formula) will become positive.\" Do you think the statement is true? If yes, prove it. If no, give a counter example Please note that, as we discussed in class, that interest rates may be zero or even negative nowadays when answering this question. QUESTION 2 V (the rate of change of value with respect 2V to the change of volatility). People need to monitor the second derivative . It is known as Volga in the 2 In the market it is no longer sufficient just to monitor Vega case of FX Options. But they cannot seem to find the formula in [Hull]. A risk manager comes to you and says \" given by a) 2V is the same for a (European) call and a put3. The formula is 2 V d 1d 2 \" Do you think what the risk manager says is true? If yes, prove it; if no, give a counter example Seeing that you are so helpful with part (a), another risk manager told you \"I have formulas for Delta and Gamma but not the next one\" and asks you to compute also the third order derivative maybe 3V . He thinks that S 3 3V d1 = + 1 3 S T S b) Do you think what the risk manager says is true? If yes, prove it; if no, give a counter example QUESTION 3 The purpose of this question is to illustrate that in real life we may need to require the interpolated smile be once-differentiable. 1. A call spread is a package where you buy a call and sell a call of different strikes simultaneously. Let c K be the European Call with strike K . Consider the payoff of a call spread, explain why a ( ) digital call4 option with strike K is the limiting portfolio lim K K 1 (c(K ) c(K )) . Let K K 3 Assuming the call and the put has the same strike, days to maturity... etc In [Hull] a digital call is called a \"cash-or-nothing\" call option Professor Fung Page 2 4 Numerical Methods in Finance 1/26/2017 digital (K ) be the price of a digital call option with strike K . We therefore have: c(K ) digital (K ) = K 2. Hence, show that in the presence of a volatility smile, the value of a digital call option is given by e rT N ( d 2 ) Ke rT N ( d 2 ) T ( K ) 2 S ln + r q T 2 K and symbols have their usual standard meaning (e.g Where d 2 = T see [Hull]) QUESTION 4 Basel Committee on Banking Supervision has issued a new standard in Jan 2016 for \"Minimum capital requirements for market risk\". It is related to the now very famous \"fundamental review of trading book\" The 88 pages document can be found at http://www.bis.org/bcbs/publ/d352.pdf In [BIS] Page 26, one finds: One sees that for a standard currency call option valued by the Black-Scholes formula, the sensitive s defined above with a 1% ratio shift is not necessary the same as the theoretical Black-Scholes Delta ... V be the Black-Scholes value of the call option. Let (S ) denote the theoretical Black-Scholes Delta, and FRTB (S ) be the Delta implied by Let S be the underlying FX spot rate for instantaneous exchange. Let the \"Minimum capital requirements for market risk\" sensitivity, so that we Professor Fung Page 3 Numerical Methods in Finance 1/26/2017 have FRTB (S ) = V (1.01S ) V (S ) . The capital markets team is wondering if the difference 0.01S (S ) FRTB(S ) is material... (a) Enhance the spreadsheet done in Practical section Q1 if necessary, provide a plot of the difference (S ) FRTB(S ) against S when S goes from 50 to 200 (while other information being the same as Practical section Q1. i.e. USD zero rate is 4%, JPY zero rate is 0.5%, strike is 119, vol is 10%, time to expiration is 92 calendar days... etc (you don't need to submit the enhanced spreadsheet but you need to submit only the resulting graph print out) (b) A trader took a look at your graph and said \"it seems to me that when the call option is very outof-the-money or very in-the-money, the difference goes to zero\". Do you agree? If yes, can you provide an intuitive explanation5 why you think this should be true for call options of all strikes in general (not just 119)? If no, please provide a counter-example to prove him wrong (c) Upon hearing your converation with the trader, a Quant says to you \"I think mathematically lim+ ( (S ) FRTB (S )) = 0 and also lim ( (S ) FRTB (S )) = S S 0 0 \" Do you agree? If yes please provide a mathematical proof. If no, please provide a counter-example to prove him wrong REFERENCES [Hull] John Hull, Options, Futures and other derivatives, Prentice Hall, 9th Edition [MA] Masaaki Fujii and Akihiko Takahashi, Choice of collateral currency, RISK Jan 2011 [BIS] Basel Committee on Banking Supervision, Minimum capital requirements for market risk, Jan 2016 http://www.bis.org/bcbs/publ/d352.pdf 5 This means to explain with minimal use of formal mathematics... Professor Fung Page 4 Numerical Methods in Finance 1/26/2017