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Finance & Accounting 作業代寫

    1.      This is because the control variate estimate of the American put option, which uses the B-S 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  作業代寫
     
      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  作業代寫
     
    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 Black-Scholes 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 long-run, 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 long-term, 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%
     
      Graph:

    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 risk-free 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  作業代寫
    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 up-and-out 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 up-and-out and an up-and-in option, the European option price equals the up-and-in 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  作業代寫
     
     
    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  作業代寫
     

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