Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 2474305, 14 pages
http://dx.doi.org/10.1155/2016/2474305

Research Article

Efficient Lattice Method for Valuing of Options with Barrier in a Regime Switching Model

Youngchul Han¹ and Geonwoo Kim²

¹Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea
²Department of Mathematical Science, Seoul National University, Seoul 151-747, Republic of Korea

Received 25 July 2016; Revised 7 September 2016; Accepted 14 September 2016

Academic Editor: Francisco R. Villatoro

Copyright © 2016 Youngchul Han and Geonwoo Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1] under the Black-Scholes model. Rubinstein and Reiner [2] proposed pricing formulas of various kinds of options with single barrier. For the pricing of double barrier options, Kunitomo and Ikeda [3] proposed a pricing formula of options with exponentially curved barriers. The results are expressed as an infinite sum of normal distribution functions. Alternatively, Geman and Yor [4] provided the Laplace transform of the double barrier option price by a probabilistic approach. Other researchers also presented the double barrier option price formula using different methods including the path counting and the method of images (for the details, see Sidenius [5], Lin [6], and Buchen and Konstandatos [7]).

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 [8], there have been many studies for the regime switching model in finance area. In particular, many researchers have used the regime switching model which leads to transition of volatilities of the underlying assets for valuation of various options. Naik [9] derived the price of the European option using the conditional probability density functions of occupation time of the volatility in high state. Buffington and Elliott [10] provided a method for valuation of the American options by partial differential equation approach. Guo and Zhang [11] derived a closed-form pricing solution for the perpetual American options. In addition, Boyle and Draviam [12] presented the general numerical methods for valuation of exotic options. Elliott et al. [13] also dealt with pricing of barrier options with regime switching.

The lattice methods for valuation of options with regime switching have received much attention by many researchers in recent year. Bollen [14] first introduced a lattice method for valuation of options with a single underlying asset in regime switching model. Lin [6] suggested the new recombining tree method for efficient valuation of the European and the American options with regime switching. Liu and Zhao [15] extended the method of Lin [6] to options with two underlying assets which follow the regime switching model. In addition, Yuen and Yang [16] developed a trinomial tree method for valuation of options in a regime switching model. Costabile et al. [17] presented a multinomial approach for pricing options under the regime switching jump-diffusion model. Costabile [18] proposed a trinomial lattice model for approximating the evolution of the investment fund value with regime switching.

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 2, we review the trinomial method for option pricing in a regime switching model. In Section 3, we propose efficient lattice method based on the local averages and interpolation. Section 4 presents the numerical results of diverse options with barrier. Finally, we give the concluding remarks in Section 5.

2. Trinomial Tree Method for the Regime Switching Model

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 [16].

In order to describe the evolution of -state of the economy with regime switching, we first define that is a continuous-time Markov chain with finite -state space , which represents general market trends and economic conditions. And we assume that is observable. Then, under the risk neutral measure, the dynamics are the underlying asset with regime switching are

where is a standard Brownian motion. For observed state at time , the interest rate and the volatility are constants. In addition, we assume that is the generator matrix of to be state dependent. Then functions of elements are continuous and bounded and are constants satisfying for and for each

We now introduce the tree method of Yuen and Yang for valuing options with regime switching. Let be the risk neutral probabilities of the underlying asset price up, middle, and down, respectively. We put uniform time grids between 0 and maturity with time mesh . Then we have

where is the risk free interest rate and should be greater than 1, so that the risk neutral probability measure exists.

We choose as the size of the up and down move, where . For each ,, let be the risk neutral probabilities of the underlying asset price up, middle, and down, respectively. Then, for the implementation of the regime switching model, we can obtain the following equations:

If is defined as for each , we have and can be expressed in terms of as

Here, by the suggestion of Yuen and Yang [16], we choose as

where is the mean of .

Let be the option value with strike at node and time step when the state at time step is . Then the option values at each node can be calculated by backward induction algorithm with the terminal condition (call option) or (put option) for all , where underlying stock price is given by . Since the Markov chain is independent of the Brownian motions, the transition probabilities are not affected by changing the probability measure. Therefore, the option values under the regime switching model can be calculated by employing the following equation recursively:

where is given by

with the identity matrix and the generator matrix .

3. Numerical Methods

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 3.1. Finally, we propose the LARSI method by combining LARS method and interpolation method in Section 3.2.

3.1. LARS Method

The tree method using local averages was introduced by Moon and Kim [19]. They modified the standard binomial tree method based on local averages of option prices and showed that their method is more effective than other methods computationally. By combining the local average method into the trinomial tree method mentioned in the previous section, we propose the trinomial tree method for the efficient valuation of options with regime switching.

Let . Then, when the state at time 0 is , the option value can be computed by

where is a payoff function at maturity .

We denote that the value of underlying asset is for , where , and divide the interval at the maturity into nonoverlapping intervals of the uniform length . Then the average option prices on each interval can be calculated by

where at the maturity.

We consider the average option prices at time with regime . Then the average option prices satisfy the following relation:

for .

From relation (10), we can obtain the option price at time with regime . We further use the following scheme to reduce the approximation error of the option price (for more details, see the Appendix)

Then finally becomes the standard option price with regime by the LARS method.

For the American option, which allows early exercise of the option before the maturity, we find the optimal boundary at each regime by

where is the option price with regime from the backward process of the lattice at time and is the exercise price at time . Then we have

If is in the interval , the error between and gets smaller when goes to 0. Let . Then we can update the option price at node with regime by , and the American option price with regime is computed by applying the backward iteration (10).

3.2. LARSI Method

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.

LARSI Method. The LARSI method is as follows:

(1)Compute the price of the barrier option with regime by LARS method on the assumption that the inner barrier is the true barrier .(2)Compute the price of the barrier option with regime by LARS method on the assumption that the outer barrier is the true barrier .(3)Compute the price of the barrier option with regime as

4. Numerical Results

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 [20] that two-state regime switching model is sufficient for accurate option pricing in the real market. Also, it is possible to extend multistate regime switching model easily) are calculated using the LARS and LARSI (LARS-type) methods, and the accuracy and efficiency of LARS-type method for valuing options with barrier are shown by comparing with the results given in [12, 16].

First, we assume that the initial underlying asset price and the strike price are set to be 100. The volatility of the underlying asset in regime 1 and regime 2 is 0.15 and 0.25, respectively. The option has 1-year maturity, and the trinomial tree is set to have 1000 time steps. The generators of the regime switching process of LARS-type are

for the above parameter sets, respectively. We present the numerical results for European call option prices, and the results are compared with benchmarks Naik (Naik) [9], Boyle and Draviam (B&D) [12], and Yuen and Yang (Y&Y) [16] in Table 1. Table 1 shows that numerical results for the option prices obtained by using the LARS method are very close to the value obtained by the analytical solutions derived in [9] and also close to those obtained using partial differential equation in [12]. This verifies that the LARS-type method is applicable (cf. [16]).

Table 1: 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

European call option 1

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

European call option 2

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 and the volatility of stock is 0.25; in regime 2, the risk free interest rate is and the volatility of stock is 0.35. All options expire in one year with a strike price 100. The generator for the regime switching model of LARS-type is taken to be

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 2 shows that the prices of the European call and the European put options converge rapidly. “” is a number of time steps used in the calculation, “Difference” refers to a difference in price calculated using various numbers of time steps, and “Ratio” is a ratio of the difference. We know that the price of the derivative using the Cox-Ross-Rubinstein (CRR) model converges to the corresponding price under the simple geometric Brownian motion model and that the speed of convergence can have order 1; that is, the error of the price is halved if the number of time steps is doubled (for more details, see [21, 22]). We can see that most of the ratios shown in the tables are close to 0.5. The ratios of the European call option for regime 2 are also close to 0.5. Therefore, the convergence patterns in regime 1 and regime 2 are more stable than the method of Y&Y. If we apply put-call parity to each of the regimes, the interest rate implied in two regimes is 4.367 and 5.631, respectively, in the 5120 time steps case. This is reasonable because both of them are between 4 and 6, and the interest rate implied by the numerical results in regime 1 is closer to the rate in regime 1, while the same happens for regime 2. Interestingly, the deviations between the current interest rate and the interest rate implied by the put-call parity in both regimes are close to 0.37. This is because of the symmetry of the two regimes in terms of the transition probabilities.

Table 2: 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 3). The prices of the American call option found by the LARS method are the same as the European call option. It is consistent with the understanding that the American call option is always not optimal to be exercised before expiration if there is no dividend being distributed. The prices of the American put option in the table are larger than those of the European put option, meaning that early exercise of the option is sometimes preferred and there may be some situations when we have to exercise the American put option before expiration. The convergence pattern of the American put option is similar to the European one. In the method of Y&Y, the convergence pattern of the American put option with regime 2 is unstable even if it is fast, whereas the LARS method gives stable convergence patterns.

Table 3: 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 4–7). Although the volatility of regime 2 is greater and has a higher chance to achieve a higher option value at expiration, the high volatility also increases the chance of hitting the barriers and thus eliminates its advantage. The convergence pattern of the down-and-out barrier option is very complicated. It might be the effect of quadratic approximation errors in pricing barrier options. It is difficult to get any conclusions from the numerical results. However, we can see that the prices of out-type options are oscillating and the differences still have a decreasing trend in absolute value apart from converging uniformly in one direction. For the up-and-out barrier option, the prices of the option are oscillating and do not converge uniformly (Tables 8–11). Therefore, it is hard to conclude the efficiency of the LARS method for calculating up-and-out barrier option.

Table 4: 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

Table 5: 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

Table 6: 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

Table 7: 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

Table 8: 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

Table 9: 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

Table 10: 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

Table 11: 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 12–15 show the price of the double barrier option with different numbers of time steps. The lower barrier is set as 70 and the upper barrier is set as 150. The prices of the double barrier option decrease progressively and oscillate. In double barrier option, the convergence patterns between LARS and LARSI methods are similar but there is small difference. In European double barrier option, the prices obtained by LARSI method also demonstrate better stability compared with LARS method. On the other hand, in American double barrier option, LARS method gives a more stable performance than LARSI method. Therefore, we can use both LARS and LARSI methods to price double barrier options efficiently.

Table 12: 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

Table 13: 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

Table 14: 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

Table 15: 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 16 summarizes the value of the European double barrier options with different barrier levels (lower barriers of 90, 80, 70, 60, and 50 and upper barriers of 110, 120, 130, 140, and 150) using 1000 time steps. When the difference between the upper barrier and lower barrier is smaller, the prices of the options are lower as there is a bigger chance of crossing the barrier and becoming out of value. The effect of barriers is more significant for regime 2 because the underlying asset with higher volatility in regime 2 have a greater chance of reaching the barrier level. When the difference between the barriers increases, its effect on the barrier options is reduced. Also, the options with a larger volatility in regime 2 have a higher price than the same options in regime 1. Their prices are lower than those of the European call option, which has prices of 12.7587 and 15.7677 with respect to the two regimes, respectively. And the prices of LARSI method in the two regimes are closer to LARS method.

Table 16: 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

Regime 1

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

Regime 2

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 17). The transition probabilities of regime 1 with 20 time steps in the three branches are 0.0224, 0.9689, and 0.0086, respectively. Note that most of the probabilities are distributed on the middle branch. On the other hand, the transition probabilities of regime 2 with 20 time steps in the three branches are 0.3754, 0.2235, and 0.4010, respectively. The value of the European option is positively related to the volatility and so the price in regime 1 decreases, while the price in regime 2 increases, when we compare the results with the previous numerical experiments. The pricing error in regime 1 is larger when we compare it with the numerical results in the previous example, since large is used in this lattice.

Table 17: 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 1–6 show the comparison of the convergence behaviors of various trinomial tree methods in pricing of options with barrier. In this experiment, “Basic” denotes the method of Y&Y, “LA” is the local average trinomial tree method, “LA interpolation” is the combination of local average trinomial tree method and interpolation method, “LA-RS” means the LARS method, and “LA-RS interpolation” means the LARSI method. LARSI method gives much closer values at smaller number of steps than other models in pricing various barrier options. The results show that the prices of options with barrier by LARSI method converge smoothly always faster than those of the other trinomial tree methods. As a result, LARSI method is more efficient than other models in pricing barrier options. The main reason is that LARSI method can fit derivatives’ specifications to suppress the oscillation problem.

(a) European down-and-out barrier call option, regime 1

(b) European down-and-out barrier call option, regime 2

Figure 1: Convergence of various models for calculation of European down-and-out barrier call option.

(a) European down-and-out barrier put option, regime 1

(b) European down-and-out barrier put option, regime 2

Figure 2: Convergence of various models for calculation of European down-and-out barrier put option.

(a) European up-and-out barrier call option, regime 1

(b) European up-and-out barrier call option, regime 2

Figure 3: Convergence of various models for calculation of European up-and-out barrier call option.

(a) European up-and-out barrier put option, regime 1

(b) European up-and-out barrier put option, regime 2

Figure 4: Convergence of various models for calculation of European up-and-out barrier put option.

(a) European double barrier call option, regime 1

(b) European double barrier call option, regime 2

Figure 5: Convergence of various models for calculation of European double barrier call option.

(a) European double barrier put option, regime 1

(b) European double barrier put option, regime 2

Figure 6: Convergence of various models for calculation of European double barrier put option.

5. Concluding Remark

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.

Appendix

Approximation for Exact Pricing Formula of the Options with Regime Switching

We assume that is a function continuous and times differentiable in an interval and for some and .

Let us denote for and consider the following equation:

where , and are constants. Then, from the Taylor series, (A.1) can be represented as

In order to obtain fourth-order accuracy, the following equations must be satisfied:

If we take , , and , we have the following approximation equation of :

For given regime , let us define the function

We then have

From (A.4) and , we finally obtain the following accurate approximation:

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research was supported by the National Research Foundation of Korea grant funded by the Korea government (MSIP) (NRF-2015R1C1A1A02037533).

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