Math 174E Fall 2016 Homework 2 (due: Fr, Oct. 7) The purpose of this homework assignment is to model the dependence of the time value Vn of a call option at the money with a given strike price as a function of the time n to expiration. You will first analyze real market data using (simple) linear regression (data fitting or method of least squares). This will be briefly reviewed in the TA session, but I assume that you know this. The second part of this assignment is to provide a theoretical explanation for your numerical findings. (1) The following table gives the prices on 9/28/16 for call options on Wells Fargo (ticker: WFC) at strike price $45 for various expiration dates. The last price price for WFC was $45.31 and so the options are at the money. The table shows the days n until expiration and the price of the option. The prices shown are averages of the last bid/ask prices. As the spreads were generally small in the range of 10-30 cents, this provides a good approximation of the market value of the options. days n until exp. price of option 15 1.02 22 1.15 29 1.25 36 1.31 50 1.43 78 1.72 113 2.06 204 2.63 260 2.89 477 4.01 841 5.25 (a) Compile a table with the time values of the options. (b) A reasonable assumption seems that the time value Vn for an option in our table with n days until expiration follows an approximate law of the form Vn = An, where A is some coefficient. To justify this linear relation, one could argue that if an option O1 has twice as long until expiration as another option O2 with the same strike, then the time value of O1 should be twice the time value of O2 . Analyze the data by using linear regression (method of least squares) to find the coefficient A that gives the best data fit for our assumed law Vn = An. Hint: You can do the computations by hand (tedious!) or use a spreadsheet like Excel or LibreOffice (free to download from the internet) that has built-in procedures for linear regression of data (recommended!). In the latter case, add a printout with your data. (c) Using the coefficient computed in (b), compile a table of the values An that are supposed to approximate the values Vn and compare them with our real data. What do you think: is our model Vn = An viable? (d) Maybe a more realistic model is a relation of the form Vn = Bn , where B > 0 and > 0 are some coefficients. Is B or more important here for the general characteristics of our model? (e) If we take (natural) logarithms in (d), then we obtain log Vn = b + log n, where b = log B. In other words, in this model there should be a linear relation between log Vn and log n. Use linear regression to find the constants b and here that give the best fit for the data. (f) Based on the numerical evidence in (e), what seems to be a plausible and simple assumption for the \"true\" value of ? (g) Use the best fit value B = eb obtained in (e) and the \"true\" value of from (f), to compile a list of the values Bn . Compare them with the real data for Vn . What is your assessment? (h) Summarize your numerical findings in a simple rule of thumb: if an option O1 at the money with the same strike as another option O2 has a time value that is twice as big as the time value of O1 , then the time to expiration of O1 should be X times the time to expiration of O2 . What is X here? In the next problems you are supposed to develop a theoretical basis for your numerical findings. (2) (a) Suppose X is a random variable that only takes the values X = 1 and X = 1 with equal (= .5) probability. What are the expected value 0 and the standard deviation 0 of X? (b) Suppose X1 , X2 , X3 , . . . are random variables that are independent and have the same distribution as the random variable X in (a). Suppose n is a natural number. What is the mean n and what is the standard deviation n of Zn = X1 + + Xn ? (c) Suppose n in (b) is very large. Then the central limit theorem suggests that Zn has a distribution close to the distribution of an + bn N , where N is a standard normal random variable. What are an and bn here? (d) Suppose we are interested in the expected value En of the random variable (Zn )+ := max{Zn , 0}. Based on the approximation in (c) find an explicit formula for En if n is large. (3) You play a game where you have to pay an upfront cost C to enter the game. You cannot lose any more money and your gain depends on the value of a realvalued random variable Y . Namely, if Y is negative, then you do not gain anything. If Y is positive, then you gain the amount Y . (a) Suppose Y is the random variable X in (2). What is the fair value of C in this case, i.e., the value C where on average (when you play the game repeatedly and one game does not influence the following games) you neither gain nor lose money in total? (b) What is the fair value C of playing the game for a general random variable Y ? Hint: Use expectations and a random variable related to Y ! (4) At a certain date a stock has a price S0 (per share). Let us assume that the stock price has neither an upward nor a downward trend and that it goes \"sideways\". Then a simplistic model for how the stock price develops over time is that it goes up or down in a random way and that the change each day is by , where is a (small) fixed amount and a gain or loss of in the stock price happens with equal probability. We also assume that these day-to-day changes are independent and ignore the effect of interest rates. (a) We denote the stock price at day n by Sn . Based on our model, express Sn in terms of S0 and the random variable Zn is (2). (b) Our model suggests that Sn for large n has a similar distribution as cn + dn N , where N is a standard normal random variable, and cn and dn are suitable constants. What are cn and dn here? (c) Even though cn in (b) is non-negative, the random variable cn + dn N can take negative values, which of course cannot happen for the stock price Sn . We can ignore this problem of our model if cn + dn N is rather sharply centered around its mean value cn > 0. Formulate a corresponding condition in terms of Sn , n, and that is relevant for the validity of our model! (5) We now want to develop a mathematical model for the fair time value of a call stock option at the money. We assume that the stock price develops according to the simplistic random model in (4). Since the the call option is at the money, it has a strike price K = S0 , where S0 is the current stock price of the underlying. We assume that the option has n days before expiration. For simplicity we also assume that we hold on to the option until its expiration at which point we receive the intrinsic value In of the option. We ignore the effect of interest rates. (a) Express In in terms of the stock price at Sn after n days and the strike price K = S0 . (b) Based on the model in (4), what is the expected value En of In for large n? (c) Relate the fair time value Vn of our option to En . (d) Find a relation between Vn , , and n. (6) Use the data in (1) and linear regression to find the value in (5) (d) that gives the best fit with the data. Compile a table of the corresponding values for Vn based on (5) (d) and compare with the real data. What is your assessment