The data listed in Table 5.1 show how the the number of reset dates influence the option value. Suppose that it is a put option, the initial stock price is 100, the initial strike price is 95, the interest rate 5%, the volatility is 30%, the length of the monitoring intervals is 5 periods, and the number of periods in the lattice is 50.
Assume the lifespan of the option is one year. Since the options are more likely being reset when the number of reset dates increase, the option value also increases as the reset dates increase. Monte Carlo simulations results are also listed in the same table to show the accuracy of our algorithms.
The data listed in Table 5.2 show how the different lengths of monitoring intervals influence the option value. Suppose that the initial stock price is 100, the initial strike price is 90, the interest rate 6%, the volatility is 30%, and the number of periods in the lattice is 65. The reset dates are at period 10,20,30,40,50,60. Assume the lifespan of the option is one year. We can see from the result that the option value decreases as the length of the intervals increases.
Table 5.3, Table 5.4, and Table 5.5 show that the relative difference between the value of geometric- and arithmetic-average-trigger reset options is insignificant. Fig 5.1 demonstrates the oscillation of option value with respect to n, the number of periods in the lattice. We can see that the option value converges very quickly when n ≥ 50. Table 5.6 compares the running time of the two algorithms. Obviously the combinatorial approach is far more efficient then the lattice approach.
Reset Dates (Year) Lattice Lattice(Euro) MC 1 8.73217 8.3018 8.3020 1,0.8 10.8541 10.4507 10.3667 1,0.8,0.6 12.4521 11.9824 11.9054 1,0.8,0.6,0.4 13.7323 13.1883 13.1036 1,0.8,0.6,0.4,0.2 14.735 14.1174 14.0317
Table 5.1: Option Values with Respect to Different Reset Dates. The option value increases with the number of reset dates. After 1,000,000 sample paths with Monte Carlo simulation with 1,000 time steps, the option values are relatively close to the values calculated by the lattice approach.
Length(year) Lattice MC 1/65 22.8105 22.795 2/65 22.7031 22.699 3/65 22.6586 22.625 4/65 22.5909 22.571 5/65 22.5191 22.490
Table 5.2: Option Values w.r.t. Different Lengths of Monitoring Intervals Option values increase as the length of monitoring intervals decrease. The 1,000,000 times sampling from Monte Carlo Simulation agrees with the values calculated by the lattice approach.
# of periods vol=0.5 vol=0.8 vol=1.0 vol=1.2 vol=1.5
10 26.04028647 37.70778614 44.85940238 51.51328682 60.57598176 100 26.1147983 37.73757635 44.95541639 51.63071751 60.76180327 1000 26.13326108 37.76117174 45.18583957 52.01828236 61.19064968 10000 26.16468427 37.78172752 44.97135585 51.56229747 60.4602232
Table 5.3: Evaluate Geometric-Average-Trigger Reset Calls with Monte Carlo simulations. T = 1, S0=100, K=95, r=0.05, h=0.2, and reset date=0.5.
# of periods vol=0.5 vol=0.8 vol=1.0 vol=1.2 vol=1.5
10 26.00424102 37.46460782 44.73355475 51.56247684 60.40588585 100 26.06763879 37.68977758 44.79836599 51.49789968 60.31990169 1000 26.10137742 37.69141599 44.71671987 51.51986505 61.07059802 10000 26.10477572 37.65250356 44.78363601 51.42352387 60.16191354
Table 5.4: Evaluate Arithmetic-Average-Trigger Reset Calls with Monte Carlo simulations. T = 1, S0=100, K=95, r=0.05, h=0.2, and reset date=0.5.
# of periods vol=0.5 vol=0.8 vol=1.0 vol=1.2 vol=1.5
10 0.138421878 0.644902152 0.280537915 -0.095489974 0.280797615 100 0.180585399 0.126660946 0.349346999 0.257245769 0.72726871 1000 0.122004154 0.184728774 1.038200693 0.958157949 0.196192818 10000 0.228967233 0.342027665 0.417420891 0.269137735 0.493398217
Table 5.5: Difference Percentage. The Relative difference is insignificant.
Behavioral Analysis 26
Oscillation
21.15 21.2 21.25 21.3 21.35 21.4 21.45
0 100 200 300 400 500 600 700 800 900 1000
price
Figure 5.1: Oscillation of Option Value. This figure demonstrate the oscillation of option value with respect to n, the number of periods in the lattice. The option value converges very quickly when n ≥ 50.
n = 50 n = 100 n = 200 n = 400 Lattice 103300 N.A. N.A. N.A.
Combinatorial 0.1400 0.4800 21.400 2201800
Table 5.6: Running-Time Comparison. The combinatorial approach is far more efficient then the lattice approach. Assume T = 1, the number of monitoring interval is 2, the interval length is 0.2, and the reset dates are at 0.4 and 0.8 . n denotes the number of periods. The experiment is done on an IBM PC with an Intel Pentium III 866 MHz processor and 1 GB DRAM.
Conclusion and Future Work
The geometric-average-trigger reset option resets the strike price based on the geo-metric average of the underlying asset’s prices over a monitoring interval. Similar con-tracts have been traded on exchanges in Asia. An O(n4h2)-time algorithm for general American-style reset options is presented in the thesis. A more efficient O(n3 hm)-time algorithms is derived to price European-style options. Besides, it is proved that an American reset call option won’t be exercised early if the underlying assets won’t pay the dividends. Numerical results are given to suggest the correctness of these two approaches. Besides, numerical evidence suggests that our pricing approaches give very tight lower (upper) bounds on arithmetic-average-trigger reset calls (puts, respectively). Research on arithmetic-average-trigger reset options is under way.
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