Discrete Dynamics in Nature and Society

Volume 2016, Article ID 2474305, 14 pages

http://dx.doi.org/10.1155/2016/2474305

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

^{1}Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea^{2}Department 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 havewhere 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]).