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

Authors: Youngchul Han; Geonwoo Kim
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## 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.


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## Body

## 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 Section2, 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 ofk-state of the economy with regime switching, we first define that X(t) is a continuous-time Markov chain with finite k-state space χ≔(x1,x2,…,xk), which represents general market trends and economic conditions. And we assume that X(t) is observable. Then, under the risk neutral measure, the dynamics are the underlying asset with regime switching are (1)dSt=rXtStdt+σXtStdWt,where W(t) is a standard Brownian motion. For observed state k at time t, the interest rate r(xk)=rk and the volatility σ(xk)=σk are constants. In addition, we assume that A(t)=[aij(t)]i,j=1,…,k is the generator matrix of X(t) to be state dependent. Then functions of elements are continuous and bounded and are constants satisfying aij(t)≥0 for i≠j and ∑jaij(t)=0 for each i=1,…,k.We now introduce the tree method of Yuen and Yang for valuing options with regime switching. Letπu,πm,πd be the risk neutral probabilities of the underlying asset price up, middle, and down, respectively. We put N+1 uniform time grids between 0 and maturity T with time mesh Δt=T/N. Then we have(2)πueλσΔt+πm+πde-λσΔt=erΔt,πu+πdλ2σ2Δt=σ2Δt,where r is the risk free interest rate and λ should be greater than 1, so that the risk neutral probability measure exists.We choosee±σΔt as the size of the up and down move, where σ>maxi∈{1,…,k}σi. For each X(t)=xi,(i=1,…,k), let πui,πmi,πdi 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: (3)πuieσΔt+πmi+πdie-σΔt=eriΔt,πui+πdiσ2Δt=σi2Δt.If λi is defined as σ/σi for each i, we have λi>1 and πui,πmi,πdi can be expressed in terms of λi as (4)πmi=1-σi2σ2=1-1λi2,πui=eriΔt-e-σΔt-1-1/λi21-e-σΔteσΔt-e-σΔt,πdi=-eriΔt+eσΔt-1-1/λi2eσΔt-1eσΔt-e-σΔt.Here, by the suggestion of Yuen and Yang [16], we choose σ as (5)σ=maxi∈1,…,k⁡σi+1.5-1σ¯,where σ¯ is the mean of σi.LetVt,ni be the option value with strike K at node n and time step t when the state at time step t is i. Then the option values at each node can be calculated by backward induction algorithm with the terminal condition VN,ni=(SN,n-K)+ (call option) or VN,ni=(K-SN,n)+ (put option) for all i, where underlying stock price SN,n is given by SN,n=S0e(n-1-N)σΔt. 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: (6)Vt,ni=e-riΔt∑j=1kpijπuiVt+1,n+2j+πmiVt+1,n+1j+πdiVt+1,nj,where pij is given by (7)eAΔt=I+∑l=1∞ΔtlAll!=p11⋯p1k⋮⋱⋮pk1⋯pkk,with the identity matrix I and the generator matrix A.


## 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 Section3.1. 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.LetY(t)=ln⁡(S(t)). Then, when the state at time 0 is i, the option value can be computed by (8)Vi≔V0iy=e-riTEΛYT∣Y0=y,where Λ(·) is a payoff function at maturity T.We denote that the value of underlying asset isYN,j=Y(0)+hj for j=0,…,2N+1, where h=σΔt, and divide the interval [Y(0)-(1/2)h,Y(0)+2N+(3/2)h] at the maturity into 2N+1 nonoverlapping intervals of the uniform length h. Then the average option prices on each interval can be calculated by (9)V¯N,j=1h∫YN,j-0.5YN,j+0.5Λsds,j=0,…,2N+1,where YN,j=Y(0)+hj at the maturity.We consider the average option prices at timet with regime i. Then the average option prices satisfy the following relation: (10)V¯t,ji=e-riΔt∑m=1kpimπuiV¯t+1,j+2m+πmiV¯t+1,j+1m+πdiV¯t+1,jm,for j=0,…,2n+1.From relation (10), we can obtain the option price V¯0,0i at time t=0 with regime i. We further use the following scheme to reduce the approximation error of the option price (for more details, see the Appendix)(11)Vi≔V0,0i=-124V¯0,-1i+1312V¯0,0i-124V¯0,1i.Then V0,0i finally becomes the standard option price with regime i by the LARS method.For the American option, which allows early exercise of the option before the maturity, we find the optimal boundaryS∗ at each regime i by (12)max⁡Vtiy,Λty=Vtiy,ifS∗≤y,Λty,ifS∗≥y,where Vti(y) is the option price with regime i from the backward process of the lattice at time t and Λt(y) is the exercise price at time t. Then we have (13)∫Yn,j-h/2Yn,j+h/2max⁡Vtis,Λtsds=max⁡∫Yn,j-h/2Yn,j+h/2Vtisds,∫Yn,j-h/2Yn,j+h/2Λtsds,ifS∗≤Yn,j-h2orS∗≥Yn,j+h2,∫Yn,j-h/2S∗Λtsds+∫S∗Yn,j+h/2Vtisds,ifYn,j-h2<S∗<Yn,j+h2.IfS∗ is in the interval [Yn,j-2/h,Yn,j+2/h], the error between max⁡{∫Yn,j-h/2Yn,j+h/2‍Vti(s)ds,∫Yn,j-h/2Yn,j+h/2‍Λt(s)ds} and ∫Yn,j-h/2S∗‍Λt(s)ds+∫S∗Yn,j+h/2‍Vti(s)ds gets smaller when h goes to 0. Let V¯n,j=(1/h)∫Yn,j-h/2Yn,j+h/2‍Λt(s)ds. Then we can update the option price at node n with regime i by max⁡{V¯n,ji,Λ¯n,j}, and the American option price with regime i 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 priceVB1i of the barrier option with regime i by LARS method on the assumption that the inner barrier B1 is the true barrier B.(2)
Compute the priceVB2i of the barrier option with regime i by LARS method on the assumption that the outer barrier B2 is the true barrier B.(3)
Compute the price of the barrier option with regimei as (14)Vi=VB1i+B-B1VB2i-VB1iB2-B1.


## 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.LetY(t)=ln⁡(S(t)). Then, when the state at time 0 is i, the option value can be computed by (8)Vi≔V0iy=e-riTEΛYT∣Y0=y,where Λ(·) is a payoff function at maturity T.We denote that the value of underlying asset isYN,j=Y(0)+hj for j=0,…,2N+1, where h=σΔt, and divide the interval [Y(0)-(1/2)h,Y(0)+2N+(3/2)h] at the maturity into 2N+1 nonoverlapping intervals of the uniform length h. Then the average option prices on each interval can be calculated by (9)V¯N,j=1h∫YN,j-0.5YN,j+0.5Λsds,j=0,…,2N+1,where YN,j=Y(0)+hj at the maturity.We consider the average option prices at timet with regime i. Then the average option prices satisfy the following relation: (10)V¯t,ji=e-riΔt∑m=1kpimπuiV¯t+1,j+2m+πmiV¯t+1,j+1m+πdiV¯t+1,jm,for j=0,…,2n+1.From relation (10), we can obtain the option price V¯0,0i at time t=0 with regime i. We further use the following scheme to reduce the approximation error of the option price (for more details, see the Appendix)(11)Vi≔V0,0i=-124V¯0,-1i+1312V¯0,0i-124V¯0,1i.Then V0,0i finally becomes the standard option price with regime i by the LARS method.For the American option, which allows early exercise of the option before the maturity, we find the optimal boundaryS∗ at each regime i by (12)max⁡Vtiy,Λty=Vtiy,ifS∗≤y,Λty,ifS∗≥y,where Vti(y) is the option price with regime i from the backward process of the lattice at time t and Λt(y) is the exercise price at time t. Then we have (13)∫Yn,j-h/2Yn,j+h/2max⁡Vtis,Λtsds=max⁡∫Yn,j-h/2Yn,j+h/2Vtisds,∫Yn,j-h/2Yn,j+h/2Λtsds,ifS∗≤Yn,j-h2orS∗≥Yn,j+h2,∫Yn,j-h/2S∗Λtsds+∫S∗Yn,j+h/2Vtisds,ifYn,j-h2<S∗<Yn,j+h2.IfS∗ is in the interval [Yn,j-2/h,Yn,j+2/h], the error between max⁡{∫Yn,j-h/2Yn,j+h/2‍Vti(s)ds,∫Yn,j-h/2Yn,j+h/2‍Λt(s)ds} and ∫Yn,j-h/2S∗‍Λt(s)ds+∫S∗Yn,j+h/2‍Vti(s)ds gets smaller when h goes to 0. Let V¯n,j=(1/h)∫Yn,j-h/2Yn,j+h/2‍Λt(s)ds. Then we can update the option price at node n with regime i by max⁡{V¯n,ji,Λ¯n,j}, and the American option price with regime i 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 priceVB1i of the barrier option with regime i by LARS method on the assumption that the inner barrier B1 is the true barrier B.(2)
Compute the priceVB2i of the barrier option with regime i by LARS method on the assumption that the outer barrier B2 is the true barrier B.(3)
Compute the price of the barrier option with regimei as (14)Vi=VB1i+B-B1VB2i-VB1iB2-B1.


## 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 priceS0 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 (15)-0.50.50.5-0.5,-111-1,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.
S 0 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 15.4569We 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 is4% and the volatility of stock is 0.25; in regime 2, the risk free interest rate is 6% 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 (16)-0.50.50.5-0.5.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.Table2 shows that the prices of the European call and the European put options converge rapidly. “N” 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.
N 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   10.2906The result of the American option is similar to that of the CRR model (Table3). 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.
N 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   10.8972For 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 (Tables4–7). 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.
N 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   9.6986Table 5
Pricing the European down-and-out barrier put option with the LARS-type method.
N 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   0.0400Table 6
Pricing the American down-and-out barrier call option with the LARS-type method.
N 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   9.6986Table 7
Pricing the American down-and-out barrier put option with the LARS-type method.
N LARS method LARSI method Regime 1 Regime 2 Regime 1 Regime 2 Price Difference Ratio Price Difference Ratio Price Difference Ratio Price Difference Ratio 2 0 6.3763 −1.3074 −1.0040 6.8947 −1.5517 −0.9916 0.8840 2.1826 0.6745 0.9394 2.2911 0.6740 40 5.0690 1.3126 0.1710 5.3429 1.5387 0.1778 3.0667 1.4722 0.5649 3.2305 1.5442 0.5954 80 6.3816 0.2245 −0.9941 6.8817 0.2736 −1.0116 4.5389 0.8316 0.5833 4.7747 0.9195 0.6112 160 6.6061 −0.2232 −1.0030 7.1553 −0.2768 −0.9936 5.3705 0.4851 0.6695 5.6942 0.5620 0.6775 320 6.3829 0.2239 0.9866 6.8785 0.2750 1.0062 5.8556 0.3247 0.6666 6.2561 0.3807 0.6768 640 6.6067 0.2209 −0.9993 7.1535 0.2767 −1.0017 6.1803 0.2165 0.6214 6.6369 0.2577 0.6431 1280 6.8276 −0.2207 −1.0003 7.4302 −0.2772 −0.9991 6.3968 0.1345 0.7080 6.8945 0.1657 0.7073 2560 6.6069 0.2208  7.1530 0.2769  6.5313 0.0952  7.0603 0.1172 5120 6.8277   7.4299   6.6265   7.1775Table 8
Pricing the European up-and-out barrier call option with the LARS-type method.
N 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   0.0291Table 9
Pricing the European up-and-out barrier put option with the LARS-type method.
N 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   5.5918Table 10
Pricing the American up-and-out barrier call option with the LARS-type method.
N 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   7.3064Table 11
Pricing the American up-and-out barrier put option with the LARS-type method.
N 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   6.0283For 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.Tables12–15 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.
N 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   4.2362Table 13
Pricing the European double barrier put option with the LARS-type method.
N LARS method LARSI method Regime 1 Regime 2 Regime 1 Regime 2 Price Difference Ratio Price Difference Ratio Price Difference Ratio Price Difference Ratio 2 0 12.6678 0.9570 −0.9571 9.2339 1.0463 −0.9747 11.8281 0.0825 0.5818 8.2848 0.1559 0.3274 40 13.6248 −0.9160 0.4357 10.2802 −1.0198 0.3518 11.9106 0.0480 0.6612 8.4407 0.0511 0.6639 80 12.7088 −0.3991 −1.0572 9.2604 −0.3587 −1.1282 11.9586 0.0317 0.3365 8.4918 0.0339 0.4967 160 12.3098 0.4219 −1.0389 8.9017 0.4047 −1.1260 11.9903 0.0107 0.6280 8.5257 0.0168 0.5402 320 12.7317 −0.4383 0.1409 9.3064 −0.4557 0.1572 12.0010 0.0067 0.5624 8.5425 0.0091 0.5195 640 12.2934 −0.0617 −1.0474 8.8507 −0.0716 −1.0811 12.0077 0.00368 0.5285 8.5516 0.0047 0.4499 1280 12.2316 0.0647 −0.9775 8.7791 0.0774 −0.9625 12.0115 0.0020 0.3737 8.5563 0.0021 0.3817 2560 12.2963 −0.0632  8.8565 −0.0745  12.0135 0.0007  8.5585 0.0008 5120 12.2331   8.7820   12.0142   8.5593Table 14
Pricing the American double barrier call option with the LARS-type method.
N 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   14.9199Table 15
Pricing the American double barrier put option with the LARS-type method.
N 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   23.1422We now consider a few more examples. First, we compare the prices between different barrier levels in double barrier option. Table16 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 7.1335 e - 04 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 1.2048 e - 04 0.0086 0.0322 0.0416 0.0430 4.9945 e - 05 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 4.4097Second, 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 (Table17). 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 σ is used in this lattice.Table 17
Pricing the European call option with the LARS method: great deviation in volatility.
N 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   19.9214Figures1–6 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.Figure 1
Convergence of various models for calculation of European down-and-out barrier call option.
(a)
European down-and-out barrier call option, regime 1 (b)
European down-and-out barrier call option, regime 2Figure 2
Convergence of various models for calculation of European down-and-out barrier put option.
(a)
European down-and-out barrier put option, regime 1 (b)
European down-and-out barrier put option, regime 2Figure 3
Convergence of various models for calculation of European up-and-out barrier call option.
(a)
European up-and-out barrier call option, regime 1 (b)
European up-and-out barrier call option, regime 2Figure 4
Convergence of various models for calculation of European up-and-out barrier put option.
(a)
European up-and-out barrier put option, regime 1 (b)
European up-and-out barrier put option, regime 2Figure 5
Convergence of various models for calculation of European double barrier call option.
(a)
European double barrier call option, regime 1 (b)
European double barrier call option, regime 2Figure 6
Convergence of various models for calculation of European double barrier put option.
(a)
European double barrier put option, regime 1 (b)
European double barrier put option, regime 2


## 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.



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*Source: 2474305-2016-10-03.xml*