We propose an efficient lattice method for valuation of options with barrier in a regime switching model. Specifically, we extend the trinomial tree method of Yuen and Yang (2010) by calculating the local average of prices near a node of the lattice. The proposed method reduces oscillations of the lattice method for pricing barrier options and improves the convergence speed. Finally, computational results for the valuation of options with barrier show that the proposed method with interpolation is more efficient than the other tree methods.
Barrier options are most popular options among the exotic options. The barrier options are the contingent claims whose payoffs depend on the relationship between the specified barriers and the path of the underlying asset. If the underlying asset crosses a specified barrier or barriers before the maturity, a barrier option of a knock-out type becomes worthless. And a barrier option of a knock-in type is activated when the underlying asset crosses a specified barrier. Barrier options are important in the financial market because barrier options are cheaper than standard European options and provide flexibility. Thus, they have been studied by many researchers.
The pricing formula of a down-and-out barrier option is first presented by Merton [
The various types of barrier option are developed by many researchers. Most studies are carried out using the Black-Scholes model. However, it is well known that the Black-Scholes model is not adequate to explain the market behavior of option prices. In other words, the Black-Scholes model does not explain well the volatility smile phenomenon in the real market. For the more realistic model, many extensions to the Black-Scholes model have been introduced for valuing options. Among many extensions, we focus on the regime switching model for valuing options with barrier in this paper.
The regime switching model is one of the popular alternative models to overcome the limitations of Black-Sholes model. Since the regime switching model was first introduced by Hamilton [
The lattice methods for valuation of options with regime switching have received much attention by many researchers in recent year. Bollen [
In this paper, we propose the efficient lattice methods for pricing options including American type options in a regime switching model. More concretely, we develop the trinomial tree methods for valuing options with barriers. In order to construct these lattice methods, we adopt the local average method and the interpolation method. As expected, we can find that our lattice methods provide efficiently the prices of options with barriers.
The remainder of the paper is organized as follows. In Section
In this section, we describe the tree method for the regime switching model to price the options. For this, we review the trinomial tree method of Yuen and Yang [
In order to describe the evolution of
We now introduce the tree method of Yuen and Yang for valuing options with regime switching. Let
We choose
Let
In this section, we propose the efficient lattice methods for pricing options with barrier including American type options. We construct the trinomial tree method using local averages with regime switching (LARS) in Section
The tree method using local averages was introduced by Moon and Kim [
Let
We denote that the value of underlying asset is
We consider the average option prices at time
From relation (
For the American option, which allows early exercise of the option before the maturity, we find the optimal boundary
If
When the lattice methods such as binomial or trinomial trees are used for valuing of the barrier options, it is well known that a large number of time steps are required to obtain reasonably accurate results. Therefore, the convergence speed of the lattice methods becomes very slow. This phenomenon occurs since barrier being assumed by the lattice is different from the true barrier. In order to overcome this phenomenon, the interpolation method has been used when options with barrier are priced by the lattice methods. We propose the LARSI method by combining the interpolation method into LARS method, which provides efficiently accurate prices of options with barrier in a regime switching model.
To describe the LARSI method, we define the inner barrier as the barrier formed by the nodes just on the inside of the true barrier and the outer barrier as the barrier formed by nodes just outside the true barrier. Then the LARSI method is as follows.
Compute the price Compute the price Compute the price of the barrier option with regime
Based on the model described in the previous section, we calculate the prices of various options with regime. In this section, we study all types of barrier options including the European type and the American type. Specifically, prices of these options with two-state regime (it was shown in [
First, we assume that the initial underlying asset price
Comparison of different methods in pricing the European call option in LARS method.
| Regime 1 | Regime 2 | ||||||
---|---|---|---|---|---|---|---|---|
Naik | B&D | Y&Y | LARS | Naik | B&D | Y&Y | LARS | |
| ||||||||
94 | 5.8620 | 5.8579 | 5.8615 | 5.8612 | 8.2292 | 8.2193 | 8.2297 | 8.2296 |
96 | 6.9235 | 6.9178 | 6.9229 | 6.9226 | 9.3175 | 9.3056 | 9.3181 | 9.3180 |
98 | 8.0844 | 8.0775 | 8.0827 | 8.0834 | 10.4775 | 10.4647 | 10.4772 | 10.4779 |
100 | 9.3401 | 9.3324 | 9.3369 | 9.3390 | 11.7063 | 11.6929 | 11.7049 | 11.7066 |
102 | 10.6850 | 10.6769 | 10.6828 | 10.6839 | 13.0008 | 12.9870 | 13.0001 | 13.0010 |
104 | 12.1127 | 12.1045 | 12.1108 | 12.1114 | 14.3575 | 14.3436 | 14.3571 | 14.3576 |
106 | 13.6161 | 13.6082 | 13.6143 | 13.6147 | 15.7729 | 15.7591 | 15.7725 | 15.7729 |
| ||||||||
94 | 6.2748 | 6.2705 | 6.2760 | 6.2758 | 7.8905 | 7.8844 | 7.8943 | 7.8942 |
96 | 7.3408 | 7.3352 | 7.3422 | 7.3420 | 8.9747 | 8.9680 | 8.9789 | 8.9788 |
98 | 8.5001 | 8.4938 | 8.5010 | 8.5017 | 10.1335 | 10.1264 | 10.1374 | 10.1380 |
100 | 9.7489 | 9.7423 | 9.7489 | 9.7509 | 11.3641 | 11.3568 | 11.3673 | 11.3690 |
102 | 11.0820 | 11.0755 | 11.0833 | 11.0844 | 12.6631 | 12.6659 | 12.6674 | 12.6683 |
104 | 12.4937 | 12.4877 | 12.4959 | 12.4965 | 14.0267 | 14.0197 | 14.0317 | 14.0322 |
106 | 13.9777 | 13.9726 | 13.9805 | 13.9810 | 15.4510 | 15.4446 | 15.4565 | 15.4569 |
We now study the values of diverse options by the LARS method. The underlying asset is assumed to be a stock with the initial price 100, following the geometric Brownian motion with no dividend and two-state regime. In regime 1, the risk free interest rate is
The transition probabilities of the offshoot of state up, middle, and down with 20 time steps are 0.1817, 0.6413, and 0.1770 in regime 1 and 0.3512, 0.2970, and 0.3518 in regime 2, respectively. These values depend on the size of time step, but the values with other sizes of times step are not much different from these values because the time step is small in general. We carry out the experiments to see significant properties of the proposed method.
Table
Pricing the European standard option with the LARS method.
| European standard call option | European standard put option | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 12.7776 | −0.0094 | 0.5159 | 15.8842 | −0.0597 | 0.4971 | 8.4479 | 0.0192 | 0.4939 | 10.3216 | −0.0158 | 0.4899 |
40 | 12.7681 | −0.0049 | 0.5072 | 15.8245 | −0.0297 | 0.4985 | 8.4671 | 0.0095 | 0.4972 | 10.3058 | −0.0077 | 0.4948 |
80 | 12.7633 | −0.0025 | 0.5034 | 15.7948 | −0.0148 | 0.4993 | 8.4766 | 0.0047 | 0.4987 | 10.2981 | −0.0038 | 0.4974 |
160 | 12.7608 | −0.0012 | 0.5017 | 15.7801 | −0.0074 | 0.4996 | 8.4814 | 0.0024 | 0.4993 | 10.2942 | −0.0019 | 0.4987 |
320 | 12.7596 | −0.0006 | 0.5008 | 15.7727 | −0.0037 | 0.4998 | 8.4837 | 0.0012 | 0.4997 | 10.2923 | −0.0010 | 0.4993 |
640 | 12.7589 | −0.0003 | 0.5004 | 15.7690 | −0.0018 | 0.4999 | 8.4849 | 0.0006 | 0.4999 | 10.2914 | −0.0005 | 0.4997 |
1280 | 12.7586 | −0.0002 | 0.5001 | 15.7671 | −0.0009 | 0.4999 | 8.4855 | 0.0003 | 0.5000 | 10.2909 | −0.0002 | 0.4998 |
2560 | 12.7585 | −0.0001 | 15.7662 | −0.0004 | 8.4858 | 0.0001 | 10.2907 | −0.0001 | ||||
5120 | 12.7584 | 15.7658 | 8.4859 | 10.2906 |
The result of the American option is similar to that of the CRR model (Table
Pricing the American standard option with the LARS method.
| American standard call option | American standard put option | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 12.7776 | −0.0094 | 0.5159 | 15.8842 | −0.0597 | 0.4971 | 8.8606 | 0.0236 | 0.5507 | 10.9320 | −0.0165 | 0.4796 |
40 | 12.7681 | −0.0049 | 0.5072 | 15.8245 | −0.0297 | 0.4985 | 8.8842 | 0.0130 | 0.4353 | 10.9154 | −0.0079 | 0.6365 |
80 | 12.7633 | −0.0025 | 0.5034 | 15.7948 | −0.0148 | 0.4993 | 8.8972 | 0.0057 | 0.4692 | 10.9075 | −0.0050 | 0.5010 |
160 | 12.7608 | −0.0012 | 0.5017 | 15.7801 | −0.0074 | 0.4996 | 8.9028 | 0.0027 | 0.4577 | 10.9024 | −0.0025 | 0.5487 |
320 | 12.7596 | −0.0006 | 0.5008 | 15.7727 | −0.0037 | 0.4998 | 8.9055 | 0.0012 | 0.4671 | 10.8999 | −0.0014 | 0.5362 |
640 | 12.7589 | −0.0003 | 0.5004 | 15.7690 | −0.0018 | 0.4999 | 8.9067 | 0.0006 | 0.4812 | 10.8985 | −0.0007 | 0.5156 |
1280 | 12.7586 | −0.0002 | 0.5001 | 15.7671 | −0.0009 | 0.4999 | 8.9073 | 0.0003 | 0.4756 | 10.8978 | −0.0004 | 0.5194 |
2560 | 12.7585 | −0.0001 | 15.7662 | −0.0004 | 8.9075 | 0.0002 | 10.8974 | −0.0002 | ||||
5120 | 12.7584 | 15.7658 | 8.9077 | 10.8972 |
For the out-type barrier (down-and-out and up-and-out barrier) options, the prices found in both regimes are smaller than those of the European call option due to the presence of the barriers. The barrier level is set as 90 for the down-and-out barrier options and 110 for the up-and-out barrier options. The prices of down-and-out barrier option in the two regimes are closer to each other compared with those of the European option (Tables
Pricing the European down-and-out barrier call option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 11.6335 | −1.4914 | −0.1840 | 13.4120 | −2.1980 | −0.1660 | 8.7988 | −0.0745 | −2.1090 | 9.5653 | −0.0680 | −1.8901 |
40 | 10.1420 | 0.2745 | −1.0490 | 11.2140 | 0.3648 | −1.0776 | 8.7243 | 0.1571 | 0.3370 | 9.4974 | 0.1285 | 0.3463 |
80 | 10.4165 | −0.2879 | 2.1678 | 11.5788 | −0.3931 | 2.0738 | 8.8814 | 0.0529 | 0.0803 | 9.6258 | 0.0445 | 0.0582 |
160 | 10.1286 | −0.6242 | 0.0853 | 11.1857 | −0.8152 | 0.0839 | 8.9343 | 0.0042 | 3.8905 | 9.6703 | 0.0026 | 5.3210 |
320 | 9.5043 | −0.0533 | −0.9695 | 10.3705 | −0.0684 | −0.9553 | 8.9386 | 0.0165 | 0.8524 | 9.6729 | 0.0138 | 0.8811 |
640 | 9.4511 | 0.0516 | −8.5059 | 10.3021 | 0.0654 | −8.4494 | 8.9551 | 0.0141 | −0.1603 | 9.6867 | 0.0121 | −0.1824 |
1280 | 9.5027 | −0.4392 | −0.3926 | 10.3674 | −0.5523 | −0.3892 | 8.9692 | −0.0023 | −1.0424 | 9.6988 | −0.0022 | −0.8923 |
2560 | 9.0635 | 0.1724 | 9.8151 | 0.2150 | 8.9669 | 0.0024 | 9.6966 | 0.0020 | ||||
5120 | 9.2359 | 10.0301 | 8.9693 | 9.6986 |
Pricing the European down-and-out barrier put option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 0.5299 | −0.3710 | −0.1251 | 0.2961 | −0.2107 | −0.1278 | 0.1001 | −0.0005 | 26.3194 | 0.0545 | −0.0011 | 6.1249 |
40 | 0.1589 | 0.0464 | −0.7938 | 0.0854 | 0.0269 | −0.7560 | 0.0996 | −0.0134 | 0.5454 | 0.0534 | −0.0070 | 0.5881 |
80 | 0.2053 | −0.0369 | 1.6459 | 0.1123 | −0.0204 | 1.6562 | 0.0861 | −0.0073 | 0.0927 | 0.0465 | −0.0041 | 0.0881 |
160 | 0.1684 | −0.0607 | 0.0557 | 0.0919 | −0.0337 | 0.0534 | 0.0788 | −0.0007 | 2.7990 | 0.0424 | −0.0004 | 2.8866 |
320 | 0.1078 | −0.0034 | −1.2756 | 0.0582 | −0.0018 | −1.3459 | 0.0782 | −0.0019 | 0.9838 | 0.0420 | −0.0010 | 1.0095 |
640 | 0.1044 | 0.0043 | −6.7772 | 0.0564 | 0.0024 | −6.6444 | 0.0763 | −0.0019 | −0.2083 | 0.0410 | −0.0011 | −0.2195 |
1280 | 0.1087 | −0.0292 | −0.3595 | 0.0588 | −0.0161 | −0.3591 | 0.0744 | 0.0004 | −0.7057 | 0.0399 | 0.0002 | −0.6565 |
2560 | 0.0795 | 0.0105 | 0.0427 | 0.0058 | 0.0748 | −0.0003 | 0.0401 | −0.0001 | ||||
5120 | 0.0900 | 0.0485 | 0.0745 | 0.0400 |
Pricing the American down-and-out barrier call option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 11.6335 | −1.4914 | −0.1840 | 13.4120 | −2.1980 | −0.1660 | 8.7988 | −0.0745 | −2.1090 | 9.5653 | −0.0680 | −1.8901 |
40 | 10.1420 | 0.2745 | −1.0490 | 11.2140 | 0.3648 | −1.0776 | 8.7243 | 0.1571 | 0.3370 | 9.4974 | 0.1285 | 0.3463 |
80 | 10.4165 | −0.2879 | 2.1678 | 11.5788 | −0.3931 | 2.0738 | 8.8814 | 0.0529 | 0.0803 | 9.6258 | 0.0445 | 0.0582 |
160 | 10.1286 | −0.6242 | 0.0853 | 11.1857 | −0.8152 | 0.0839 | 8.9343 | 0.0042 | 3.8905 | 9.6703 | 0.0026 | 5.3210 |
320 | 9.5043 | −0.0533 | −0.9695 | 10.3705 | −0.0684 | −0.9553 | 8.9386 | 0.0165 | 0.8524 | 9.6729 | 0.0138 | 0.8811 |
640 | 9.4511 | 0.0516 | −8.5059 | 10.3021 | 0.0654 | −8.4494 | 8.9551 | 0.0141 | −0.1603 | 9.6867 | 0.0121 | −0.1824 |
1280 | 9.5027 | −0.4392 | −0.3926 | 10.3674 | −0.5523 | −0.3892 | 8.9692 | −0.0023 | −1.0424 | 9.6988 | −0.0022 | −0.8923 |
2560 | 9.0635 | 0.1724 | 9.8151 | 0.2150 | 8.9669 | 0.0024 | 9.6966 | 0.0020 | ||||
5120 | 9.2359 | 10.0301 | 8.9693 | 9.6986 |
Pricing the American down-and-out barrier put option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
| 6.3763 | −1.3074 | −1.0040 | 6.8947 | −1.5517 | −0.9916 | 0.8840 | 2.1826 | 0.6745 | 0.9394 | 2.2911 | 0.6740 |
| 5.0690 | 1.3126 | 0.1710 | 5.3429 | 1.5387 | 0.1778 | 3.0667 | 1.4722 | 0.5649 | 3.2305 | 1.5442 | 0.5954 |
| 6.3816 | 0.2245 | −0.9941 | 6.8817 | 0.2736 | −1.0116 | 4.5389 | 0.8316 | 0.5833 | 4.7747 | 0.9195 | 0.6112 |
| 6.6061 | −0.2232 | −1.0030 | 7.1553 | −0.2768 | −0.9936 | 5.3705 | 0.4851 | 0.6695 | 5.6942 | 0.5620 | 0.6775 |
| 6.3829 | 0.2239 | 0.9866 | 6.8785 | 0.2750 | 1.0062 | 5.8556 | 0.3247 | 0.6666 | 6.2561 | 0.3807 | 0.6768 |
| 6.6067 | 0.2209 | −0.9993 | 7.1535 | 0.2767 | −1.0017 | 6.1803 | 0.2165 | 0.6214 | 6.6369 | 0.2577 | 0.6431 |
| 6.8276 | −0.2207 | −1.0003 | 7.4302 | −0.2772 | −0.9991 | 6.3968 | 0.1345 | 0.7080 | 6.8945 | 0.1657 | 0.7073 |
| 6.6069 | 0.2208 | 7.1530 | 0.2769 | 6.5313 | 0.0952 | 7.0603 | 0.1172 | ||||
| 6.8277 | 7.4299 | 6.6265 | 7.1775 |
Pricing the European up-and-out barrier call option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 0.6005 | −0.4275 | −0.1184 | 0.3355 | −0.2417 | −0.1210 | 0.0478 | 0.0329 | −0.8215 | 0.0255 | 0.0182 | −0.8136 |
40 | 0.1730 | 0.0506 | −3.2236 | 0.0938 | 0.0293 | −3.0933 | 0.0808 | −0.0271 | −0.0449 | 0.0437 | −0.0148 | −0.0481 |
80 | 0.2236 | −0.1631 | −0.3359 | 0.1230 | −0.0905 | −0.3348 | 0.0537 | 0.0012 | −0.5066 | 0.0289 | 0.0007 | −0.3764 |
160 | 0.0604 | 0.0548 | −0.9761 | 0.0326 | 0.0303 | −0.9728 | 0.0549 | −0.0006 | −0.5856 | 0.0296 | −0.0003 | −0.8087 |
320 | 0.1152 | −0.0535 | −0.2984 | 0.0629 | −0.0295 | −0.2982 | 0.0543 | 0.0004 | −1.5676 | 0.0293 | 0.0002 | −1.3799 |
640 | 0.0617 | 0.0160 | −0.9796 | 0.0334 | 0.0088 | −0.9766 | 0.0547 | −0.0006 | −0.0076 | 0.0295 | −0.0003 | −0.0253 |
1280 | 0.0777 | −0.0156 | −0.0301 | 0.0422 | −0.0086 | −0.0306 | 0.0541 | 0.0000 | −52.8157 | 0.0292 | 0.0000 | −16.3714 |
2560 | 0.0621 | 0.0004 | 0.0336 | 0.0002 | 0.0541 | −0.0002 | 0.0292 | −0.0001 | ||||
5120 | 0.0625 | 0.0338 | 0.0539 | 0.0291 |
Pricing the European up-and-out barrier put option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 7.5614 | −1.0591 | −0.1951 | 8.5033 | −1.4915 | −0.1726 | 5.2933 | −0.1666 | −1.1509 | 5.5685 | −0.1550 | −1.0705 |
40 | 6.5023 | 0.2066 | −5.9780 | 7.0118 | 0.2575 | −5.8889 | 5.1267 | 0.1918 | −0.0217 | 5.4135 | 0.1659 | −0.0306 |
80 | 6.7089 | −1.2352 | −0.4839 | 7.2693 | −1.5164 | −0.4677 | 5.3184 | −0.0042 | −3.5735 | 5.5794 | −0.0051 | −2.3325 |
160 | 5.4737 | 0.5977 | −0.9975 | 5.7530 | 0.7092 | −1.0004 | 5.3143 | 0.0149 | −0.0632 | 5.5744 | 0.0118 | −0.1022 |
320 | 6.0715 | −0.5963 | −0.3598 | 6.4622 | −0.7095 | −0.3538 | 5.3291 | −0.0009 | −6.2294 | 5.5862 | −0.0012 | −4.0389 |
640 | 5.4752 | 0.2145 | −0.9983 | 5.7527 | 0.2510 | −1.0003 | 5.3282 | 0.0059 | 0.0862 | 5.5850 | 0.0049 | 0.0695 |
1280 | 5.6897 | −0.2142 | −0.0297 | 6.0037 | −0.2511 | −0.0291 | 5.3340 | 0.0005 | 3.6497 | 5.5899 | 0.0003 | 4.6204 |
2560 | 5.4756 | 0.0064 | 5.7526 | 0.0073 | 5.3346 | 0.0018 | 5.5902 | 0.0016 | ||||
5120 | 5.4819 | 5.7599 | 5.3364 | 5.5918 |
Pricing the American up-and-out barrier call option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 7.1590 | −1.6580 | −0.9986 | 7.5712 | −1.8630 | −0.9921 | 0.1801 | 2.2234 | 0.8266 | 0.1714 | 2.3199 | 0.8042 |
40 | 5.5009 | 1.6556 | −1.0003 | 5.7082 | 1.8482 | −1.0019 | 2.4035 | 1.8378 | 0.5322 | 2.4913 | 1.8656 | 0.5625 |
80 | 7.1565 | −1.6561 | −0.9997 | 7.5564 | −1.8517 | −0.9981 | 4.2413 | 0.9781 | 0.6761 | 4.3569 | 1.0495 | 0.6833 |
160 | 5.5504 | 1.6556 | −0.3718 | 5.7047 | 1.8482 | −0.3781 | 5.2195 | 0.6613 | 0.6434 | 5.4064 | 0.7171 | 0.6578 |
320 | 7.1560 | −0.6155 | −0.9998 | 7.5528 | −0.6988 | −0.9987 | 5.8808 | 0.4255 | 0.6971 | 6.1235 | 0.4717 | 0.6999 |
640 | 6.5405 | 0.6154 | −0.2311 | 6.8541 | 0.6979 | −0.2333 | 6.3063 | 0.2966 | 0.6693 | 6.5952 | 0.3301 | 0.6781 |
1280 | 7.1559 | −0.1422 | −0.9998 | 7.5520 | −0.1628 | −0.9986 | 6.6030 | 0.1985 | 0.7010 | 6.9253 | 0.2238 | 0.7028 |
2560 | 7.0136 | 0.1423 | 7.3892 | 0.1625 | 6.8015 | 0.1391 | 7.1491 | 0.1573 | ||||
5120 | 7.1559 | 7.5517 | 6.9406 | 7.3064 |
Pricing the American up-and-out barrier put option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 7.9544 | −1.0884 | −0.1976 | 9.0574 | −1.5517 | −0.1724 | 5.6154 | −0.1700 | −1.1779 | 5.9979 | −0.1575 | −1.1179 |
40 | 6.8660 | 0.2151 | −5.9265 | 7.5057 | 0.2674 | −5.8956 | 5.4454 | 0.2002 | −0.0200 | 5.8404 | 0.1760 | −0.0338 |
80 | 7.0811 | −1.2748 | −0.4846 | 7.7732 | −1.5768 | −0.4676 | 5.6456 | −0.0040 | −3.9377 | 6.0165 | −0.0059 | −2.0446 |
160 | 5.8064 | 0.6177 | −0.9968 | 6.1964 | 0.7373 | −1.0009 | 5.6416 | 0.0158 | −0.0564 | 6.0105 | 0.0121 | −0.1183 |
320 | 6.4241 | −0.6157 | −0.3600 | 6.9337 | −0.7380 | −0.3537 | 5.6574 | −0.0009 | −6.8705 | 6.0227 | −0.0014 | −3.5386 |
640 | 5.8084 | 0.2216 | −0.9980 | 6.1957 | 0.2611 | −1.0004 | 5.6565 | 0.0061 | 0.0889 | 6.0212 | 0.0051 | 0.0687 |
1280 | 6.0300 | −0.2212 | −0.0297 | 6.4568 | −0.2612 | −0.0291 | 5.6626 | 0.0005 | 3.5307 | 6.0263 | 0.0003 | 4.6524 |
2560 | 5.8088 | 0.0066 | 6.1956 | 0.0076 | 5.6631 | 0.0019 | 6.0267 | 0.0016 | ||||
5120 | 5.8154 | 6.2032 | 5.6650 | 6.0283 |
For further improvement of the convergence speed and stability of LARS method in valuing options with barrier, we apply the LARSI method. In down-and-out barrier option case, the convergence patterns of LARSI method are more stable and faster than LARS method, though step size is small. In particular, in up-and-out barrier option, the prices of LARSI method converge uniformly, while those in LARS method oscillate. Therefore, we can think that LARSI method is efficient in valuing the up-and-out barrier option and compensate the defect of LARS method.
Tables
Pricing the European double barrier call option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 7.4788 | −0.0969 | 12.0395 | 5.9400 | −0.0507 | 24.3064 | 5.6723 | 0.0724 | 0.3143 | 4.1254 | 0.0460 | 0.8110 |
40 | 7.3819 | −1.1669 | −0.2170 | 5.8893 | −1.2334 | −0.1959 | 5.7447 | 0.0228 | 0.6148 | 4.1714 | 0.0373 | 0.3820 |
80 | 6.2150 | 0.2532 | −0.9646 | 4.6559 | 0.2416 | −0.9230 | 5.7675 | 0.0140 | 0.4757 | 4.2086 | 0.0142 | 0.4819 |
160 | 6.4682 | −0.2442 | 0.9075 | 4.8975 | −0.2230 | 1.0357 | 7.7815 | 0.0067 | 0.5439 | 4.2229 | 0.0069 | 0.4879 |
320 | 6.2240 | −0.2216 | 0.5402 | 4.6745 | −0.2310 | 0.5033 | 5.7881 | 0.0036 | 0.5865 | 4.2297 | 0.0033 | 0.4995 |
640 | 6.0024 | −0.1197 | −1.0097 | 4.4435 | −0.1163 | −1.0204 | 5.7918 | 0.0021 | 0.4641 | 4.2331 | 0.0017 | 0.5644 |
1280 | 5.8827 | 0.1209 | −0.9951 | 4.3273 | 0.1186 | −0.9899 | 5.7939 | 0.0010 | 0.2764 | 4.2348 | 0.0009 | 0.5660 |
2560 | 6.0035 | −0.1202 | 4.4459 | −0.1174 | 5.7949 | 0.0002 | 4.2357 | 0.0005 | ||||
5120 | 5.8833 | 4.3285 | 5.7951 | 4.2362 |
Pricing the European double barrier put option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
| 12.6678 | 0.9570 | −0.9571 | 9.2339 | 1.0463 | −0.9747 | 11.8281 | 0.0825 | 0.5818 | 8.2848 | 0.1559 | 0.3274 |
| 13.6248 | −0.9160 | 0.4357 | 10.2802 | −1.0198 | 0.3518 | 11.9106 | 0.0480 | 0.6612 | 8.4407 | 0.0511 | 0.6639 |
| 12.7088 | −0.3991 | −1.0572 | 9.2604 | −0.3587 | −1.1282 | 11.9586 | 0.0317 | 0.3365 | 8.4918 | 0.0339 | 0.4967 |
| 12.3098 | 0.4219 | −1.0389 | 8.9017 | 0.4047 | −1.1260 | 11.9903 | 0.0107 | 0.6280 | 8.5257 | 0.0168 | 0.5402 |
| 12.7317 | −0.4383 | 0.1409 | 9.3064 | −0.4557 | 0.1572 | 12.0010 | 0.0067 | 0.5624 | 8.5425 | 0.0091 | 0.5195 |
| 12.2934 | −0.0617 | −1.0474 | 8.8507 | −0.0716 | −1.0811 | 12.0077 | 0.00368 | 0.5285 | 8.5516 | 0.0047 | 0.4499 |
| 12.2316 | 0.0647 | −0.9775 | 8.7791 | 0.0774 | −0.9625 | 12.0115 | 0.0020 | 0.3737 | 8.5563 | 0.0021 | 0.3817 |
| 12.2963 | −0.0632 | 8.8565 | −0.0745 | 12.0135 | 0.0007 | 8.5585 | 0.0008 | ||||
| 12.2331 | 8.7820 | 12.0142 | 8.5593 |
Pricing the American double barrier call option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 12.4324 | 0.0648 | −1.2446 | 14.9452 | 0.1245 | −1.7594 | 12.0743 | 0.1873 | 0.4084 | 14.2092 | 0.3145 | 0.4477 |
40 | 12.4972 | −0.0807 | −0.6282 | 15.0697 | −0.2190 | −0.3987 | 12.2616 | 0.0765 | 0.6891 | 14.5237 | 0.1408 | 0.7128 |
80 | 12.4165 | 0.0507 | 0.1797 | 14.8506 | 0.0873 | 0.3136 | 12.3381 | 0.0527 | 0.5628 | 14.6645 | 0.1003 | 0.5815 |
160 | 12.4672 | 0.0091 | −1.2180 | 14.9380 | 0.0274 | −1.4265 | 12.3908 | 0.0297 | 0.6462 | 14.7648 | 0.0584 | 0.6694 |
320 | 12.4763 | −0.0111 | −0.0077 | 14.9654 | −0.0391 | 0.0678 | 12.4204 | 0.0192 | 0.6734 | 14.8232 | 0.0391 | 0.6892 |
640 | 12.4653 | 0.0001 | 218.2816 | 14.9263 | −0.0026 | −15.9678 | 12.4396 | 0.0129 | 0.6694 | 14.8622 | 0.0269 | 0.6824 |
1280 | 12.4653 | 0.0187 | −0.2806 | 14.9236 | 0.0423 | −0.3198 | 12.4525 | 0.0086 | 0.6627 | 14.8891 | 0.0184 | 0.6721 |
2560 | 12.4840 | −0.0052 | 14.9659 | −0.0135 | 12.4611 | 0.0058 | 14.9075 | 0.0124 | ||||
5120 | 12.4788 | 14.9524 | 12.4669 | 14.9199 |
Pricing the American double barrier put option with the LARS-type method.
| LARS method | LARSI method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regime 1 | Regime 2 | Regime 1 | Regime 2 | |||||||||
Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 21.8204 | 0.0549 | −0.6749 | 23.4097 | −0.0197 | 8.4024 | 21.7067 | 0.0826 | 0.1183 | 23.0915 | 0.0456 | −0.2495 |
40 | 21.8754 | −0.0371 | −0.5947 | 23.3900 | −0.1657 | −0.2468 | 21.7893 | 0.0098 | 1.4698 | 23.1372 | −0.0114 | −0.9766 |
80 | 21.8383 | 0.0221 | −0.4942 | 23.2243 | 0.0409 | −1.0437 | 21.7991 | 0.0144 | 0.3350 | 23.1258 | 0.0111 | 0.1763 |
160 | 21.8603 | −0.0109 | 1.1220 | 23.2652 | −0.0427 | 0.9272 | 21.8135 | 0.0048 | 0.5809 | 23.1369 | 0.0020 | 0.7872 |
320 | 21.8494 | −0.0122 | 0.6107 | 23.2225 | −0.0396 | 0.5892 | 21.8183 | 0.0028 | 0.6598 | 23.1389 | 0.0015 | 0.8910 |
640 | 21.8372 | −0.0075 | −1.1900 | 23.1829 | −0.0233 | −0.9767 | 21.8211 | 0.0018 | 0.4232 | 23.1404 | 0.0014 | 0.3589 |
1280 | 21.8297 | 0.0089 | −0.9289 | 23.1596 | 0.0228 | −1.0140 | 21.8229 | 0.0008 | 0.2702 | 23.1418 | 0.0005 | −0.2092 |
2560 | 21.8386 | −0.0082 | 23.1824 | −0.0231 | 21.8237 | 0.0002 | 23.1423 | −0.0001 | ||||
5120 | 21.8304 | 23.1593 | 21.8239 | 23.1422 |
We now consider a few more examples. First, we compare the prices between different barrier levels in double barrier option. Table
Price of the European double barrier call options with different barrier levels: LARS-type method.
LARS method | LARSI method | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
90 | 80 | 70 | 60 | 50 | 90 | 80 | 70 | 60 | 50 | |
| ||||||||||
110 | 0.0016 | 0.0397 | 0.0746 | 0.0791 | 0.0793 | | 0.0248 | 0.0498 | 0.0542 | 0.0544 |
120 | 0.1113 | 0.4554 | 0.6065 | 0.6210 | 0.6218 | 0.1028 | 0.4304 | 0.5759 | 0.5941 | 0.5949 |
130 | 0.7303 | 1.6610 | 1.9502 | 1.9741 | 1.9753 | 0.7090 | 1.6224 | 1.9079 | 1.9387 | 1.9400 |
140 | 2.0142 | 3.5940 | 3.9971 | 4.0276 | 4.0290 | 1.8812 | 3.4030 | 3.7974 | 3.8365 | 3.8381 |
150 | 3.3872 | 5.4423 | 5.9130 | 5.9469 | 5.9485 | 3.2997 | 5.3257 | 5.7937 | 5.8378 | 5.8395 |
| ||||||||||
110 | | 0.0086 | 0.0322 | 0.0416 | 0.0430 | | 0.0050 | 0.0201 | 0.0282 | 0.0294 |
120 | 0.0174 | 0.1552 | 0.3127 | 0.3537 | 0.3584 | 0.0159 | 0.1446 | 0.2899 | 0.3374 | 0.3422 |
130 | 0.2009 | 0.7600 | 1.1536 | 1.2335 | 1.2412 | 0.1933 | 0.7364 | 1.1129 | 1.2087 | 1.2169 |
140 | 0.8040 | 2.0252 | 2.6775 | 2.7907 | 2.8005 | 0.7321 | 1.8843 | 2.4997 | 2.6349 | 2.6454 |
150 | 1.6795 | 3.5227 | 4.3689 | 4.5025 | 4.5133 | 1.6185 | 3.4164 | 4.2349 | 4.3980 | 4.4097 |
Second, we predict that the convergence rate of the proposed model will be harmed if the volatility of different regimes is largely different from each regime to another. All the other conditions are assumed to be the same, but the volatility of the two regimes becomes 0.10 and 0.50. The prices of the European call option are tested (Table
Pricing the European call option with the LARS method: great deviation in volatility.
| Regime 1 | Regime 2 | ||||
---|---|---|---|---|---|---|
Price | Difference | Ratio | Price | Difference | Ratio | |
20 | 9.6172 | 0.0900 | 0.6695 | 20.2019 | −0.1418 | 0.4953 |
40 | 9.7071 | 0.0602 | 0.4450 | 20.0601 | −0.0702 | 0.4968 |
80 | 9.7674 | 0.0268 | 0.4514 | 19.9899 | −0.0349 | 0.4984 |
160 | 9.7942 | 0.0121 | 0.4825 | 19.9550 | −0.0174 | 0.4993 |
320 | 9.8063 | 0.0058 | 0.4925 | 19.9376 | −0.0087 | 0.4996 |
640 | 9.8121 | 0.0029 | 0.4965 | 19.9289 | −0.0043 | 0.4998 |
1280 | 9.8150 | 0.0014 | 0.4983 | 19.9246 | −0.0022 | 0.4999 |
2560 | 9.8164 | 0.0007 | 19.9224 | −0.0011 | ||
5120 | 9.8171 | 19.9214 |
Figures
Convergence of various models for calculation of European down-and-out barrier call option.
European down-and-out barrier call option, regime 1
European down-and-out barrier call option, regime 2
Convergence of various models for calculation of European down-and-out barrier put option.
European down-and-out barrier put option, regime 1
European down-and-out barrier put option, regime 2
Convergence of various models for calculation of European up-and-out barrier call option.
European up-and-out barrier call option, regime 1
European up-and-out barrier call option, regime 2
Convergence of various models for calculation of European up-and-out barrier put option.
European up-and-out barrier put option, regime 1
European up-and-out barrier put option, regime 2
Convergence of various models for calculation of European double barrier call option.
European double barrier call option, regime 1
European double barrier call option, regime 2
Convergence of various models for calculation of European double barrier put option.
European double barrier put option, regime 1
European double barrier put option, regime 2
Regime switching model is one of popular models in finance area. We develop the trinomial tree method based on regime switching method and local averages of the option price and compare its performance with other trinomial tree schemes in terms of their accuracy and efficiency. We also modify the LARS method using interpolation method and show that it works very well for general types of options with barrier including European type and American type. The LARSI method has a smoothing effect to reduce oscillations of the tree method, and it seems to accelerate the convergence rate. Finally, we can find that LARSI method provides good performance for valuing options with barrier in a regime switching model.
We assume that
Let us denote
The authors declare that there are no competing interests regarding the publication of this paper.
This research was supported by the National Research Foundation of Korea grant funded by the Korea government (MSIP) (NRF-2015R1C1A1A02037533).