Data: we will price a call option on the S&P 500 index. The index level at the close of Monday September 21 was equal to 1,065. Assume an annualized long term volatility for the index of 20% per year. The 1-year LIBOR rate is at 1.25%. For the stochastic volatility process take a = 0.05 and c = 0.4. Assume a correlation coecient between 1 and 2 of -50%. Objective: we want to price an European call option on the S&P 500 index with maturity equal to 1 year (250 trading days) and strike price equal to 1,100. Volatility is stochastic.

Platform: you can use any software package you know how to use. You will be able to do this exercise in Excel, just be careful in constructing your spreadsheet. In Matlab, if you know the language, it should not take you more than 30 lines of code. Suggestion: I would set up the excel le so that you have 4 dierent excel sheets: one for 1, one for 2, one for V and one for S. If you do that you can x the draws by sampling all of them at once and keep them xed (which literally means you copy and paste the values.) In that way excel will not re-draw every time you press enter.

1. Here a simple way to implement a Monte Carlo method: (a) rst of all you need to simulate 1000 paths of the underlying and the volatility/variance under the risk neutral probability. I will guide you on how to simulate one path. Essentially you will use a recursive formula S(t+t) = Ste(r0.5Vt)t+t Vt 1 (t+t) (1) V(t+t) = Vt + a(VL Vt)t + cpt |Vt|h1 (t+t) +p(12)2 (t+t)i (2) NOTE: it is possible that the volatility might still go negative. If that is the case, just force it to be positive by considering the absolute value of V(t+t). 1 Start with S(0) = 1,065 and V0 = VL = (0.2)2. At each step draw 1 1 and 2 1 from a standard normal random variable. Now go ahead and compute V1 and S1: V1 = V0 + cpt |V0|h1 1 +p(12)2 1iS 1 = S0e(r0.5V0)t+t V0 1 1

Now go ahead and compute V2 and S2: V2 = V1 + a(VL V1)t + cpt |V1|h1 2 +p(12)2 2iS 2 = S1e(r0.5V1)t+t V1 1 2 Lets say that we are going to simulate the paths at daily frequency, so that t = 1/250. Now keep going until you reach S250. (b) After simulating one path, go back and repeat the procedure so that you have N=1000 paths. (c) Once you have got the paths for the stock price, compute the payo of the option at maturity Pi(T) = max(0,Si (T) K) where Pi(T) is the payo of the option at maturity of the i-th path, with i = 1...N, and Si(T) is the stock price at maturity for the i-th simulation (d) The price of the option at time t will be just Ct = er(Tt) E[Pi(T)] where P(T) denotes the expiration payo of the option. The expected value can be approximated by taking the average payo over the 100 simulated cases E[P(T)] = 1 NPN i=1 Pi(T) where N is the number of simulation and Pi(T) is the option payo for the i-th simulation.

2. What is the price of the call option today?

3. How does the price of the call change if the number of paths is increased to 5000?

4. How does the price of the call option compare to the price of a call option without stochastic volatility (say that the volatility of the stock is 20%)? You should use the same set of shocks (1) and lets say we are going to compare the price with 1,000 simulations. 5. Lets now do something fun: we are going to compute the implied volatility of the Monte Carlo pricing model with CONSTANT volatility. You just computed the price for the option with = 20%. So the question is, how much do you have to change so that you obtain the same price you got from the model with stochastic volatility (point 2. above)?

6. Deliverable: READ CAREFULLY! Please print the top left corner of each worksheet only (i.e., the rst ten rows and ten columns are enough). Print both the numbers and a version that displays the formulas. For the formulas make sure you enlarge the cell enough so that the entire formula gets printed. If you are coding this, please print the code. Other than that, please prepare a neatly organized report (in words) that contains the answers to all questions and a DETAILED explanation of how you approached the problem. In other words, I should be able to understand all you have done by just reading the description, as opposed to looking at the excel le. Also, prepare a zipped version of the excel le. I will give you instructions on how to deliver it soon.