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athos20145
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Option pricing model

December 6th, 2014, 9:30 pm

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: athos20145QuoteOriginally posted by: Cuchulainn6. I cannot seem to find the reference in Hull. BTW "Brute force" sounds a bit scary.5. The article does no mention how many thousands (MC is slow in general).3. In the case rho = -1 in any case the corresponding is not elliptic anymore but parabolic, that Alan also alludes to. 1. IMHO, the article is still incomplete. // Clewlow and Strickland also discuss time-varying volatility and they suggest using trinomial method instead.6. In Hull, 4th Edition, Chapter 16 "Numerical Procedures", Section 16.6 "Monte Carlo Simulation", there's a paragraph called "Sampling through a tree", and it goes: "Instead of implementing Monte Carlo simulation by randomly sampling from the stochastic process for an underlying variable, we can sample paths for the underlying variable using a binomial tree. Suppose we have a binomial tree where the probability of an "up" movement is 0.6. The procedure for sampling a random path through the tree is as follows. At each node, we sample a random number between zero and one. If the number is less than 0.4, we take the down path. If it is greater than 0.4, we take the up path. Once we have a complete path from the initial node to the end of the tree we can calculate a payoff. This completes the first trial. A similar procedure is used to complete more trials. The mean of the payoffs is discounted at the risk-free rate to get an estimate of the value of the derivative."5. Looking at the formula of the standard error in step 10 of the option pricing algorithm (second to last page), you see that the sample standard deviation is divided by the square root of the number of simulated payoffs, so that, for example, to double the accuracy of the simulation you need to quadruple the number of simulated payoffs, to increase the accuracy by a factor of 10 you need to increase the number of simulated payoffs by a factor of 100.Athos6.I have Hull 6th edition, chapter 17. 17.6 is "MC Simulation" but I don't see sampling from a binomial tree. Maybe I'm missing something.IMO have a strategy for computing the binomial parameters??I went to check that paragraph I mentioned in the latest Hull edition in the university library, and it's there. You should find it under "MC Simulation".I've also priced S&P500 index options with the model. If you are interested I can email you results. You can leave me your email at [email protected] if you want.Ciao.
Last edited by athos20145 on December 5th, 2014, 11:00 pm, edited 1 time in total.
 
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athos20145
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Option pricing model

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)
 
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athos20145
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Re: Option pricing model

May 3rd, 2018, 7:53 pm

I looked briefly. I suspect it does have a continuous-time limit, perhaps something like[$]dx_t = (r - \frac{1}{2} \sigma^2_t) \, dt + \sigma_t dW_t[$],[$]d \sigma_t = -\alpha \, \sigma_t \,  dW_t[$],where [$]x_t = \log S_t[$].If my guess is correct, this would be equivalent to[$]dS_t = r  S_t dt + \sigma_t S_t dW_t[$],[$]d \sigma_t = \alpha \, \sigma_t \, dB_t[$],[$]dW_t dB_t = \rho \, dt[$], with [$]\rho = -1[$].This system  is a degenerate case of the lognormal SABR model [with stock price drift], perfect negative correlation, and vol-of-vol = [$]\alpha[$]. There may also be a drift on the [$]d \sigma_t[$] SDE (not sure). If so, and it is also proportional to [$]\sigma_t[$],this limit would be more properly described as a degenerate case of the GBM-vol model, which is a running example in my stoch. vol. book.       In addition to Nelson & Ramaswamy, whch Daniel mentioned, take a look at  "ARCH models as diffusion approximations" by Nelson Your model will gain more interest if it can be tied to a continuous-time limit.
HI Alan. Thanks for the post. I had a closer look. I think that the model you write differs from the tree, because the volatility process of the tree doesn't have a Brownian motion in the limit. I think that's the reason why the tree doesn't have a continuous-time limit.