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Finance & Accounting 作业代写

1. This is because the control variate estimate of the American put option, which uses the BS price of corresponding European option as an anchor, is more accurate than that of the binomial tree method which assumes stock price only changes discretely.
2. The implied volatility for the three put option chains with different strike prices are summarized in the following table.
Finance & Accounting 作业代写
Maturity  
Strike Price($)  2013/6/22  2014/1/18  2015/1/17 
70  14.00%  21.28%  20.70% 
75  29.15%  18.81%  19.63% 
80  22.16%  16.79%  18.74% 
85  15.86%  15.39%  18.16% 
90  13.72%  14.51%  17.81% 
95  17.17%  14.26%  17.53% 
100  14.60%  17.65% 
Finance & Accounting 作业代写
The volatility smile for JNJ stock option with different maturity is depicted as below.
3. It can be seen that in the real world, low strike price options on a stock would normally have significantly higher implied volatilities and be more volatile than high strike price options on the same stock. In the BlackScholes world, we would expect that the implied volatilities for different strike prices with a given maturity would be the same.
The option chain with the shortest maturity (2013/6/22) exhibits the steepest volatility smile, while the volatility smile for the option chain with the longest maturity (2015/1/17) is the flattest. The smile is more pronounced for shorter expirations because in the longrun, volatility seems to be mean reverting to a constant for every strike price. That is, in the short run, jumps in stock price give rise to steep smile curves but in the longterm, jumps tend to be averaged out.
4. Table:
Maturity  
Strike Price  182 days  365 days  548 days 
80  2.04%  4.23%  6.48% 
90  3.39%  6.25%  9.55% 
100  5.71%  9.33%  12.55% 
110  8.50%  12.40%  15.87% 
120  10.34%  16.16%  20.16% 
Graph:
5. Table:
Interest Rate  
Strike Price  1%  4%  7% 
80  0.74%  4.23%  9.17% 
90  1.07%  6.25%  13.51% 
100  1.57%  9.33%  19.95% 
110  1.94%  12.40%  26.89% 
120  2.53%  16.16%  30.19% 
6. In both graphs, as the strike price increases, the put option becomes more attractive for early exercise. Therefore, the EEP increases. Given the strike price, the EEP also increases with time to maturity (Question 4) and riskfree interest rate (Question 5). In other words, the effect of time to maturity and interest rate depends on the moneyness of the option.
7. In the below figure, European option exhibits a more pronounced oscillation pattern.
Finance & Accounting 作业代写
8. Graph for European option:
Graph for American option:
9. Table:
n  Eur. Bin  Am. Bin  Eur. BBS  Am. BBS  Eur. BBSR  Am. BBSR  BBS error/Bin error  BBSR error/Bin error 
3  $3.720  $3.997  $3.403  $3.766  
4  $3.003  $3.695  $3.410  $3.866  
5  $3.573  $4.001  $3.389  $3.839  
6  $3.115  $3.784  $3.390  $3.892  $3.417  $3.641  31.97%  274.91% 
7  $3.509  $3.966  $3.380  $3.870  
8  $3.172  $3.795  $3.380  $3.869  $3.440  $3.863  4.40%  4.06% 
9  $3.473  $3.928  $3.374  $3.857  
10  $3.207  $3.800  $3.374  $3.865  $3.403  $3.813  1.06%  81.17% 
11  $3.450  $3.926  $3.370  $3.865  
12  $3.230  $3.820  $3.370  $3.872  $3.410  $3.911  14.80%  100.30% 
13  $3.435  $3.922  $3.367  $3.874  
14  $3.247  $3.830  $3.367  $3.873  $3.392  $3.867  21.41%  4.49% 
15  $3.423  $3.912  $3.365  $3.873  
16  $3.260  $3.829  $3.365  $3.868  $3.396  $3.870  5.58%  11.55% 
17  $3.414  $3.901  $3.363  $3.868  
18  $3.270  $3.830  $3.363  $3.867  $3.384  $3.848  4.92%  50.93% 
19  $3.408  $3.899  $3.362  $3.869  
20  $3.277  $3.836  $3.362  $3.870  $3.387  $3.860  15.20%  19.83% 
21  $3.402  $3.899  $3.360  $3.871  
22  $3.284  $3.842  $3.361  $3.872  $3.379  $3.857  28.99%  34.14% 
23  $3.397  $3.898  $3.359  $3.872  
24  $3.289  $3.845  $3.360  $3.873  $3.381  $3.872  33.81%  30.39% 
25  $3.393  $3.895  $3.359  $3.871 
It can be seen from the last two columns in the above table that the BBS error is significantly lower than simple tree error for every tree step. BBS method can reduce the error by 66%99% relative to the simple tree. On the other hand, BBSR error is higher than simple tree error in two cases. In other cases, BBSR method is not better than BBS method in error reduction.
10. In order to implement importance sampling for barrier options, a joint probability of barrier being crossed and final payoff being greater than 0 has to be obtained. This proves much harder than just obtaining the probability of a positive final payoff for the European option.
11. Note that Barrier is $95, which is less than the strike price $105. For the upandout option, if the barrier is never touched, the final stock price must stay under $95 hence the option will not be exercised. If the barrier is touched during the option life, the option ceases to exist and becomes useless. In either case, the option is worthless at maturity; therefore, it is worthless today.
Since a European option is the same as the portfolio of an upandout and an upandin option, the European option price equals the upandin option.
12. For this analysis, nsteps of 10, 15, 20, 25, …, 95,100 are chosen. Simulated and adjusted prices of down&in and down&out options are calculated and graphed.
It can be seen that simulated prices generally show large oscillations around the adjusted prices. The adjusted prices can be seen as the mean value of the stimulated prices which are random for a given nsteps. With the increase of nsteps/observation points, the probability of observing the barrier being touched becomes larger. This is good for down&in option but bad for down&out option. Therefore, the adjusted price gets bigger for down&in and smaller for down&out option.
13. The formula for standard error used in constructing confidence interval for European option price is ‘STDEV (E60:E1059)/SQRT(1000)’, where 1000 is the number of MC trials.
Repeating the simulation for 100 times, the times and frequency of adjusted prices lying outside the 95% confidence interval for the three options are reported in the table below.
European Option  Down& In Option  Down& Out Option  
Times  8  6  6 
Frequency  8%  6%  6% 
14. Using BINOMDIST function in excel, the following table is obtained.
Finance & Accounting 作业代写
Trial  100  
Pr of outside of the CI  0.05  
x  Pr of x  Cumulative Pr of x 
0  0.005920529  0.005921 
1  0.03116068  0.037081 
2  0.081181772  0.118263 
3  0.139575678  0.257839 
4  0.178142642  0.435981 
5  0.180017827  0.615999 
6  0.150014856  0.766014 
7  0.106025537  0.87204 
8  0.064870888  0.93691 
9  0.034901296  0.971812 
10  0.016715884  0.988528 
It can be seen that the probability of 6 and 8 outside of CI events during 100 trials are 15% and 6.5%, respectively. The probability of up to 6 and 8 such events are 76.6% and 93.7%, respectively. These are reasonable values, therefore it can be concluded that previous results are significant at conventional levels.Finance & Accounting 作业代写