Obtain the price of an option on an underlying with stochastic volatility through numerical methods, and verify that the stochastic volatility model helps . . Apply the Monte Carlo simulation method. Data: 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: 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. 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. Suggestion: Set up the excel file so that you have 4 different excel sheets: one for 1, one for e?, one for V and one for S. I would also make sure that Excel does not draws new random variables every time you press enter. To do so, 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 is a simple way to implement a Monte Carlo method: First simulate 1000 paths of the underlying and the volatility/variance under the risk neutral probability. Essentially you will use a recursive formula . . S(t+At) Ve+at) Sie(-0.5v.) At+ At Vi efe+) Vi+a(V V.)At + c/At |V | [pete+ae ) + V (1 p?)e+e+21] 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+At). Start with S(0) = 1, 065 and Vo = VL = (0.2)2. At each step draw1 and 24 from a standard normal random variable. Now go ahead and compute V1 and 51: Vi Vo+cV At Vol (pe + V (1 p2ei Si Soer-0.5V)At+vAt Voc Now go ahead and compute V2 and 52: V2 Vi + a(V; V;)At + c/ At |Vil (pe: + V (1 -p?) S2 = Sier-0.5V1At+vAt Vie 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. Questions: 1. 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, 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 model (i.e., Black and Scholes): plot the implied volatilities obtained from the SVM prices and those obtained from the BS prices against their respective strike levels. 4. Now construct the implied distributions from the SVM and BS prices (you should have two distributions, one for each). Please use the following formula: c(k3)-c(k2) c(k2)-cki) k3-k2 k2-ki (k2) (k2 ki) (Note: the implied distribution that you get from the BS prices is the proper benchmark to look at. It is basically the better version of trying to overlay the normal distribution as we did in the homework.) Plot both distribution against the strike price. 5. Now we are going to repeat the exercise but compute SVM prices under two alternative scenarios. For the first alternative, change the correlation from -30% to 0. Compute implied volatilities and implied distribution and compare them to the SVM scenario where p = -30% 6. For the second alternative, consider a correlation of -30%, but change c parameter from 0.85 to 0.15. Compute implied volatilities and implied distribution and compare them to the SVM scenario where p = -30% and c = 0.85. strike call price 100 970 200 871 300 772.3 400 674.2 500 576.8 600 481 700 387 800 296 900 209 1000 133 1100 72 1200 34 1300 16.5 1400 7.8 1500 4.3 1600 2.52 1700 1.7 1800 1.25 1900 0.9 2000 0.6 Obtain the price of an option on an underlying with stochastic volatility through numerical methods, and verify that the stochastic volatility model helps . . Apply the Monte Carlo simulation method. Data: 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: 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. 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. Suggestion: Set up the excel file so that you have 4 different excel sheets: one for 1, one for e?, one for V and one for S. I would also make sure that Excel does not draws new random variables every time you press enter. To do so, 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 is a simple way to implement a Monte Carlo method: First simulate 1000 paths of the underlying and the volatility/variance under the risk neutral probability. Essentially you will use a recursive formula . . S(t+At) Ve+at) Sie(-0.5v.) At+ At Vi efe+) Vi+a(V V.)At + c/At |V | [pete+ae ) + V (1 p?)e+e+21] 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+At). Start with S(0) = 1, 065 and Vo = VL = (0.2)2. At each step draw1 and 24 from a standard normal random variable. Now go ahead and compute V1 and 51: Vi Vo+cV At Vol (pe + V (1 p2ei Si Soer-0.5V)At+vAt Voc Now go ahead and compute V2 and 52: V2 Vi + a(V; V;)At + c/ At |Vil (pe: + V (1 -p?) S2 = Sier-0.5V1At+vAt Vie 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. Questions: 1. 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, 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 model (i.e., Black and Scholes): plot the implied volatilities obtained from the SVM prices and those obtained from the BS prices against their respective strike levels. 4. Now construct the implied distributions from the SVM and BS prices (you should have two distributions, one for each). Please use the following formula: c(k3)-c(k2) c(k2)-cki) k3-k2 k2-ki (k2) (k2 ki) (Note: the implied distribution that you get from the BS prices is the proper benchmark to look at. It is basically the better version of trying to overlay the normal distribution as we did in the homework.) Plot both distribution against the strike price. 5. Now we are going to repeat the exercise but compute SVM prices under two alternative scenarios. For the first alternative, change the correlation from -30% to 0. Compute implied volatilities and implied distribution and compare them to the SVM scenario where p = -30% 6. For the second alternative, consider a correlation of -30%, but change c parameter from 0.85 to 0.15. Compute implied volatilities and implied distribution and compare them to the SVM scenario where p = -30% and c = 0.85. strike call price 100 970 200 871 300 772.3 400 674.2 500 576.8 600 481 700 387 800 296 900 209 1000 133 1100 72 1200 34 1300 16.5 1400 7.8 1500 4.3 1600 2.52 1700 1.7 1800 1.25 1900 0.9 2000 0.6