December 8th, 2014, 3:34 pm
A Binomial Tree to Price European Options ? Model parameter valuesAthos BrogiThe model presented in the short paper ?A Binomial Tree to Price European Options? was applied to price (European) put options written on the S&P500 index, and model prices were compared with market prices on pages 15 and 16 of version 1.1 of the ?OptionCity Calculator: Tutorial and Help? of Alan Lewis. In what follows, the notation used is the same as in the abovementioned short paper.S&P500 index options expiring in September 2001, December 2001, March 2002, June 2002 and December 2002 were priced on Monday 27/8/2001 when the index closed at 1179.21.First of all, considering that apart from mu and sigma_(t_0 ) model volatility process has only one parameter, alpha, once appropriate values of m, number of Monte Carlo simulation tree paths, and n, number of time steps for each simulated path, were selected and fixed, the model was calibrated by adjusting by trial and error alpha and sigma_(t_0 ), until what seemed to be the best fit with respect to market bid and ask prices was obtained for different strikes starting from the September 2001 expiration and progressing to the December 2001, December 2002, and intermediate expirations. For calibration m = 500,000 and n = 100, S_(t_0 )=1179.21 (Monday 27/8/2001 close), S_(t_(-1) )=1184.93 (Friday 24/8/2001 close), mu=r, where r is the continuous compounding interest corresponding to the Eurodollar futures interest for each expiration which is simple interest, T was set equal to the ratio of number of days till expiration and 365, and k was the strike. After calibration alpha = 0.045 and sigma_(t_0 )=20% and put options across all strikes and expirations were priced with these parameter values.Using Matlab on an Intel Core i5 processor each Monte Carlo simulation to calculate one option price took between about 15 and 17.5 minutes. The following tables show market bid and ask prices and model prices. Standard errors are in parentheses.SPX Index Options: Market Quotes and Model Prices, August 27, 2001Option expiration: September 21, 2001 Eurodollar rate: 3.48%Type Strike Volume Bid Ask Model pricePut 900 0 0.15 0.20 0.1249 (0.0048)Put 1050 2183 1.65 1.90 2.3585 (0.0218)Put 1150 4708 11.50 11.80 14.7730 (0.0527)Put 1175 1885 18.70 20.20 22.7126 (0.0637)Put 1200 212 31.50 32.50 34.0102 (0.0741)Option expiration: December 21, 2001 Eurodollar rate: 3.54%Type Strike Volume Bid Ask Model pricePut 900 51 2.90 3.60 3.4253 (0.0338)Put 1050 351 14.10 15.60 15.4219 (0.0740)Put 1150 2971 35.90 37.90 37.8977 (0.1143)Put 1175 174 45.40 47.40 46.6715 (0.1260)Put 1200 102 56.60 58.60 56.8401 (0.1364)Option expiration: March 15, 2002 Eurodollar rate: 3.67%Type Strike Volume Bid Ask Model pricePut 900 0 6.70 7.70 7.6051 (0.0556)Put 1050 0 23.60 25.60 24.7555 (0.1034)Put 1150 100 49.40 51.40 50.1608 (0.1455)Put 1175 0 58.70 60.70 58.8495 (0.1574)Put 1200 40 69.30 71.30 68.7481 (0.1678)Option expiration: June 21, 2002 Eurodollar rate: 3.99%Type Strike Volume Bid Ask Model pricePut 900 0 11.80 13.30 12.5417 (0.0755)Put 1050 0 33.00 35.00 33.4084 (0.1273)Put 1150 101 60.90 62.90 59.8000 (0.1703)Put 1200 20 80.90 82.90 78.1089 (0.1936)Option expiration: December 20, 2002 Eurodollar rate: 4.77%Type Strike Volume Bid Ask Model pricePut 900 0 20.10 23.10 19.8145 (0.1010)Put 1050 0 46.50 49.50 43.5733 (0.1556)Put 1150 0 75.70 78.70 69.5913 (0.1986)Put 1200 0 95.00 98.00 86.3251 (0.2218)