Practice Problems Implied volatility puzzle and stochastic volatility This exercise is intended to show how to obtain the price of an option on an underlying with stochastic volatility through numerical methods, and verify that the stochastic volatility model helps We will apply the Monte Carlo simulation method. 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+18). Start with S(0) = 1, 065 and Vo = V1 = (0.2)). At each step drawe and e? from a standard normal random variable. Now go ahead and compute V1 and 51: h Vi = V0 + c A VOT pe + (1 - ple - (-0.5v.)Ar='4*. Now go ahead and compute V2 and 52: v2 = V1 + a(vz v:)At+c87 bit "pek + = pre - Gel-0.56.)Ar+'*** Let's say that we are going to simulate the paths at daily frequency, so that At = 1/250. Now keep going until you reach S250. After simulating one path, go back and repeat the procedure so that you have N=1000 paths. Data: we will price a call option on the S&P 500 index. The index level at the close of yesterday 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 = 0.95 and c = 0.85. Objective: we want to price several European call options on the S&P 500 index with maturity equal to 1 year (250 trading days) and strike prices between 100 and 2000. We will do this under different correlation scenarios. The purpose is to show how a stochastic volatility process can generate model prices that exhibit a Black-Scholes implied volatility smile. 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. Suggestion: I would set up the excel file so that you have 4 different excel sheets: one for et, one for el, one for Vand one for S. I would also make sure that Excel does not draws new random variables every time you press enter. To do so, please draw all of epsilons at once. Then copy and paste the values. You want to keep the same draws from the normal distribution as a base to price all options. Here a simple way to implement a Monte Carlo method: - first 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 Spl-0.5V.JA:+' V(+44) = V: + a(VL - V.)At + c At TVET Pete+48) + (1 - p%)*7+At) (2) 1. First things first, let's get a benchmark. Price the call options with strikes that range from 100 to 2000 USD (at 100 USD increases), for a total of 20 options using the Black and Scholes formula. Use a volatility parameter of 20% (i.e., the long run volatility that was used for the stochastic volatility model). We are going to call those the BS prices. 2. Using the simulated underlying prices that you obtained from the Monte Carlo simulation, price the same 20 call options with strikes that range from 100 to 2000 USD (at 100 USD increases). Assume a correlation of p = -30% between the shocks. (Note: you do not need to simulate new paths for the price and variance for each option. You can just use the same simulation to price all options. We are going to call those the SVM prices. Next, compute the Black-Sholes implied volatility for each one of the options prices that you have computed. What that means is that you use the implied volatility formula that you set up for the last homework, but you match the BS prices to the ones you have obtained from the SVM model. 3. Now we are going to compare the stochastic volatility model to the log-normal pricing SE+A1 = h 1 Practice Problems Implied volatility puzzle and stochastic volatility This exercise is intended to show how to obtain the price of an option on an underlying with stochastic volatility through numerical methods, and verify that the stochastic volatility model helps We will apply the Monte Carlo simulation method. 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+18). Start with S(0) = 1, 065 and Vo = V1 = (0.2)). At each step drawe and e? from a standard normal random variable. Now go ahead and compute V1 and 51: h Vi = V0 + c A VOT pe + (1 - ple - (-0.5v.)Ar='4*. Now go ahead and compute V2 and 52: v2 = V1 + a(vz v:)At+c87 bit "pek + = pre - Gel-0.56.)Ar+'*** Let's say that we are going to simulate the paths at daily frequency, so that At = 1/250. Now keep going until you reach S250. After simulating one path, go back and repeat the procedure so that you have N=1000 paths. Data: we will price a call option on the S&P 500 index. The index level at the close of yesterday 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 = 0.95 and c = 0.85. Objective: we want to price several European call options on the S&P 500 index with maturity equal to 1 year (250 trading days) and strike prices between 100 and 2000. We will do this under different correlation scenarios. The purpose is to show how a stochastic volatility process can generate model prices that exhibit a Black-Scholes implied volatility smile. 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. Suggestion: I would set up the excel file so that you have 4 different excel sheets: one for et, one for el, one for Vand one for S. I would also make sure that Excel does not draws new random variables every time you press enter. To do so, please draw all of epsilons at once. Then copy and paste the values. You want to keep the same draws from the normal distribution as a base to price all options. Here a simple way to implement a Monte Carlo method: - first 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 Spl-0.5V.JA:+' V(+44) = V: + a(VL - V.)At + c At TVET Pete+48) + (1 - p%)*7+At) (2) 1. First things first, let's get a benchmark. Price the call options with strikes that range from 100 to 2000 USD (at 100 USD increases), for a total of 20 options using the Black and Scholes formula. Use a volatility parameter of 20% (i.e., the long run volatility that was used for the stochastic volatility model). We are going to call those the BS prices. 2. Using the simulated underlying prices that you obtained from the Monte Carlo simulation, price the same 20 call options with strikes that range from 100 to 2000 USD (at 100 USD increases). Assume a correlation of p = -30% between the shocks. (Note: you do not need to simulate new paths for the price and variance for each option. You can just use the same simulation to price all options. We are going to call those the SVM prices. Next, compute the Black-Sholes implied volatility for each one of the options prices that you have computed. What that means is that you use the implied volatility formula that you set up for the last homework, but you match the BS prices to the ones you have obtained from the SVM model. 3. Now we are going to compare the stochastic volatility model to the log-normal pricing SE+A1 = h 1
