A Lattice Approach to Pricing of Multivariate Contingent Claims

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```					A Lattice Approach to Pricing of Multivariate Contingent Claims with Regime Switching
M.I.M. Wahab and Chi-Guhn Lee1

Department of Mechanical and Industrial Engineering University of Toronto, Toronto, Ontario, M5S 3G8, Canada July 14, 2005

1

Corresponding author can be reached at cglee@mie.utoronto.ca

Abstract

Since the binomial lattice was introduced by Cox, Ross, and Rubinstein to value options on a single state variable with a single regime, extensions have been made in two directions: lattices for multiple state variables with a single regime and lattices for a single state variable with multiple regimes. In this paper we further generalize these extensions by developing a framework to construct a lattice for multiple correlated state variables with regime switching. The resulting lattice can be used to price contingent claims whose payoﬀ is a function of multiple underlying state variables with regime switching, provided these variables have a multivariate lognormal distribution with diﬀerent means, variances and correlations in diﬀerent regimes. We apply the proposed lattice approach to the valuation of options with multiple underlying assets with regime switching and compare the results with those of Monte Carlo simulation to demonstrate the accuracy of the proposed approach. Keywords: Option valuation, Contingent claims, Lattice, Regime switching, Multivariate

1

Introduction

Corporate ﬁnance as well as various investment practices has increasingly required tools for valuation of contingent claims whose values depend on multiple underlying sources. Sophisticated instruments are available around world to investors, as a number of examples listed in Stulz [18]. While the valuation of these securities is critical for corporate ﬁnance, non-trivial extensions of the famous Black-Scholes model [1] are often required. Closed-form solutions are rare as they cannot incorporate advanced features such as the early exercise of American options, path dependent options, and multiple asset options. In such cases the valuation relies on numerical methods. Lattice methods, along with Monte Carlo simulation and a few others, are one of the widely employed numerical approaches and use discrete-time and discrete-state approximation to compute derivative prices. Lattice approaches were ﬁrst proposed in Cox et al. [8] and have been the method of choice for models and securities that are more complicated for their simplicity. The binomial lattice introduced by Cox et al. [8] assumes that there exists a single underlying state variable that does not undergo any fundamental changes over time. To overcome limitations from these assumptions extensions have been made in two major directions: lattices for multiple assets and lattices for an asset with regime switching. Examples of the ﬁrst type of extension include Boyle [6], Boyle et al. [7], Kamrad and Ritchken [14] and Ekvall [9]. Though these extensions widen the range of applications for the binomial lattice, the accuracy of the model comes into question whenever the underlying variables undergo fundamental changes over the life of the contingent claims. Models developed on assumed ignorance of such dynamics are said to have a single-regime, whereas regime-switching models allow for changes in the means, volatilities and correlations of underlying state variables. The Black-Scholes option pricing model [1] is one of the most well known single-regime models, as is the binomial lattice by Cox et al. [8]. Regime-switching models have emerged as an alternative approach and allow for changes in 1

the means, volatilities and correlations of the variables over time. Possible causes of the dynamic behavior of the economic environment are discussed in [10] and [11]. Regime-switching models have been successfully used to capture the time series behavior of ﬁnancial variables such as short-term interest rates and exchange rates [2]. Bollen et al. [3] examine the ability of regime-switching models to capture the dynamics of foreign exchange rates and ﬁnd that the regime-switching model with independent shifts in mean and variance shows a closer ﬁt and produces more accurate variance forecasts than GARCH models. Gray [10] develops a generalized regime-switching (GRS) model for short-term interest rates and shows that, when state-dependent transition probabilities govern the switching between regimes a regime-switching model provides better volatility forecasts than either a constant-variance or a single-regime GARCH model. Hamilton [11] maximizes a loglikelihood function based on the probability of switching regimes to estimate parameters with constant moments in each regime. As brieﬂy reviewed, the existing literature addresses two directions of generalization of the standard binomial lattice: multiple underlying state variables and regime switching. However, to the best of our knowledge, no eﬀort has been made to combine these two directions of extension. Therefore, the aim of this paper is to ﬁll this gap by proposing a framework to construct a lattice for the valuation of contingent claims whose values depend on multiple underlying state variables with regime switching. The resulting lattice oﬀers ways to address a wider spectrum of applications. Especially, non-ﬁnancial applications such as operational ﬂexibility [4, 19], R&D decision [17], capacity expansion, and many others can now be analyzed since they often involve multiple pertinent stochastic processes and their decision horizons are typically too long to assume stationarity of the underlying processes. The paper is organized as follows. Section 2 introduces how to construct a lattice for multiple variables with regime switching. Section 3 presents how the lattice can be used for the valuation of

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options and Section 4 compares the option value computed using the lattice with that from Monte Carlo simulation to demonstrate the accuracy of the option values from the proposed approach. Finally, Section 5 concludes the paper.

2

Multi-variate Regime-switching Models

In this section we describe a 5-step framework to construct a lattice. For ease of exposition, we assume that there are two underlying state variables (S1 and S2 ) each of which has two regimes throughout the paper unless otherwise stated. However, the approach proposed in the paper is valid in the general case of n variables and mi regimes, where i ∈ {1, 2, . . . , n}. The ﬁst step (Section 2.1) identiﬁes all combinations of regimes (called combined regimes) and new variables. The second step (Section 2.2) transforms a pair of variables in each combined regime into uncorrelated processes. The third step (Section 2.3) is to build a single lattice for each uncorrelated variable with regime switching and the fourth step (Section 2.3) is to combine these lattices into a higher dimensional lattice. Lastly, as we perform the necessary computations in the lattice, the value of the original correlated variables is recovered (Section 2.4).

2.1

Combined Regimes

The continuously compounded return η of a variable in a regime can be described by probability distributions for the known mean µ and variance σ 2 as follows. η ∼ N (µ, σ 2 ) Notice that state variables are governed by multivariate normal distributions with diﬀerent means, variances, and correlations in diﬀerent regimes. In the case of two state variables, as each of the two processes switches between two regimes over time, they will collectively experience four combinations of regimes: (1H, 2H), (1H, 2L), (1L, 2H), and (1L, 2L), where 1H (2H) and 1L (2L) 3

represent the high and low volatility regimes of the ﬁrst (second) variable, respectively. These combinations of regimes will be called “combined regimes” to distinguish them from the original regimes for individual variables. In the general case of n variables, the number of combined regimes will be N =
n i=1 mi ,

where mi is the number of regimes for variable i, and the set of these regimes can

represented in the vector notation as {(R1 , R2 , . . . , Rn ) | Ri is one of the regimes of variable i, i = 1, 2, . . . , n}. For convenience the 4 combined regimes are indexed from 1 to 4; regime 1 for (1H, 2H), 2 for (1L, 2L), 3 for (1H, 2L), and 4 for (1L, 2H), respectively. For example, in regime 3, variable 1 is in the high volatility regime and variable 2 is in the low volatility regime. Let pi be the probability t that the combined regime i will govern the processes at time t for i ∈ {1, 2, 3, 4}. Further, let Φt−1 be the information set used to determined pi (i = 1, 2, 3, 4). Thus, at any time t, the conditional t probability that each combined regime will govern the next observation is expressed as: pi = P r(λt = i|Φt−1 ), i = 1, 2, 3, t
3

p4 = 1 − t
i=1

P r(λt = i|Φt−1 ),

where λt is the latent indicator variable that denotes the combined regimes. The transition probability, or regime persistence, is the probability of staying in the same combined regime from one observation to the next. For two underlying variables where each variable has two regimes, assuming the Markovian transition, the sixteen transition probabilities are deﬁned by γαβ = P r(λt = α|λt−1 = β) for α, β = 1, 2, 3, 4. Hamiltion [11] uses constant transition probabilities in his regime-switching model, while Gray [10] uses time varying transition probabilities. For ease of exposition, we use constant transition probabilities in presenting the proposed lattice framework, but it is straightforward to extend the framework to the case of time varying transition probabilities. This is because the lattice approach 4

to regime-switching models has the ﬂexibility to allow parameters to take on diﬀerent values in diﬀerent regimes.

2.2

Process Transformation

In the regime switching model, a state variable in diﬀerent combined regimes has not only a diﬀerent mean and volatility but also diﬀerent correlations with the other variables. With two state variables, let ρ1 , ρ2 , ρ3 , and ρ4 be the correlations between the two variables in combined regimes 1, 2, 3, and 4, respectively. The traditional approaches to constructing a lattice for a single-variate process with a single regime, a single-variate process with regime switching, and for a multi-variate with a single regime are matching moments [6] and matching characteristic functions [7]. For a multivariate process with regime switching, however, these approaches fail to work due to the excessive number of moments or terms in characteristic functions. As the correlation factors approach 1 or −1, we experience negative branching probabilities, thereby nullifying the lattice, when using the traditional approaches. Therefore, we adopt the linear transformation proposed in Ekvall [9]. That is, we ﬁrst un-correlate the two processes in each combined regime so that the number of moments to match can be reduced. We then build a lattice using the traditional methods (we use moment matching in this paper but it is also trivial to use characteristic function matching). The lattice approach is based on risk-neutral pricing, and the risk neutral process is obtained by replacing the drift in the geometric Brownian process with the risk free rate (the original drift rates may remain in the case of non-ﬁnancial applications). Therefore, the risk neutral processes for a combined regime that has n correlated processes are given as: dSi = rSi dt + σi Si dzi , i = 1, 2, 3, . . . , n, (1)

where r is the risk free rate, and σi is the instantaneous volatility. We denote ρij as the correlation between Wiener processes dzi and dzj . First we transform these processes into Ito’s processes using 5

Ito’s Lemma, and then we transform the Ito’s processes into uncorrelated processes using Cholesky factorization. Let yi = lnSi , The Ito’s process is then dyi = r−
2 σi 2

i = 1, 2, 3, . . . , n.

dt + σi dzi , i = 1, 2, 3, . . . , n.

These n Ito’s processes can be written in vector from as follows: dy ∼ Nn r− σ2 dt, Ωdt , 2 (2)
σ2 2 )

where Nn (·, ·) denotes the n-variate normal distribution, dy = [dy1 , dy2 , dy3 , ..., dyn ]T , (r − [(r −
2 σ1

=

2

), (r −

2 σ2

2

), (r −

2 σ3

2

), ...., (r −

2 σn

2

)]T , T denotes the transpose of a vector or a matrix, and the

variance-covariance matrix is given by:



 2  σ1 σ2 ρ12 σ2 σ2 σ3 ρ23    . . . Ω=  . . .    . . .  σ1 σn ρ1n σ2 σn ρ2n .

2 σ1

σ1 σ2 ρ12 σ1 σ3 ρ13 ... σ1 σn ρ1n



 ... σ2 σn ρ2n     ... .   ... .    ... .  2 ... σn

Here we recall the distinct asset assumption made by Merton [16], which says that “distinct means that none of the assets’s return can be written as instantaneous linear combination of other assets.” This assumption implies that the variance-covariance matrix is symmetric and positive deﬁnite matrix. Therefore, Ω can be written as Ω = LLT (3)

Now we multiply expression (2) by L−1 and substitute equation (3) into (2). The processes can

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now be written as L−1 dy ∼Nn L−1 r − σ2 dt, L−1 (LLT )(L−1 )T dt , 2 σ2 dt, Idt , 2

= Nn L−1 r −

where I is the (n × n) identity matrix. Substituting dΨ = L−1 dy, the expected value and variance of dΨ can be expressed as: dΨ ∼ Nn L−1 r − σ2 dt, Idt . 2 (4)

This gives n uncorrelated processes in a given combined regime. Once the same transformation is applied to all N (=
n i=1 mi )

combined regimes, one can consider the resulting processes as n

uncorrelated processes, where each process has N regimes, and where each process simultaneously switches between the N regimes. More speciﬁcally, after the process transformation we arbitrarily choose an uncorrelated process from each combined regime and consider it to be a regime for a new uncorrelated process. Since there are N combined regimes, the new uncorrelated process will have N regimes. We repeat the above step to deﬁne n new processes with N regimes for each. For example, suppose that we are given two correlated processes with two regimes for each. Using the transformation explained in this subsection we obtain two uncorrelated processes in each of the four combined regimes. We then arbitrarily choose one of the two processes in each combined regime and consider the collection of 4 uncorrelated processes to be a single new uncorrelated process with 4 regimes. Similarly, we can deﬁne another new uncorrelated process with four regimes. Notice that the n new uncorrelated processes switch regimes simultaneously unlike the original correlated processes.

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2.3

Lattice Construction

Bollen [2] proposes a pentanomial lattice for a single variable with two regimes, in which each regime is represented by a trinomial lattice with the middle branch being shared by the two regimes. Both pentanomial lattices are joined at each node because regime-switching may occur at any time. Corresponding regime switching probabilities have to be used as computation is performed in the lattice. The step sizes of the two lattices are adjusted to a 1:2 ratio so that the descendant nodes can be merged later in the lattice. Here, we generalize Bollen’s lattice approach [2] to tackle a single variable with N (=
n i=1 mi )

regimes; we use a trinomial lattice for each regime with the middle branch shared by all the regimes. In an eﬀort to minimize the size of the lattice, we adjust the step size to merge nodes that are to be extended later in the lattice. That means, if there are N regimes, the step size of N − 1 regimes must be adjusted so that all the nodes generated for N regimes are evenly spaced (see Figure 1). In order to determine the N − 1 regimes of which step sizes are to be adjusted, the step size of all N regimes are independently calculated as follows:
2 σk h + µ2 h2 , k

k = 1, 2, . . . , N,

where µk and σk are the mean and volatility in regime k, and h is the time interval between two layers of the lattice. Let φ1 , φ2 ,. . .,φN be the step size of regimes 1, 2, . . . , N , respectively (i.e., φk =
2 σk h + µ2 h2 for k = 1, 2, . . . , N ). After re-indexing the regimes so that φ1 < φ2 < φ3 , .., < k φl l

φN , we set φ = max(φ1 , φ2 , φ3 , . . . , φN ). Let φ = 2 3 N be as follows: φk =      φl ,   φ  l k ,  l

(i.e., φl ≥ l

φk k , ∀k),

then the step size φk should

if k = l, (5) if k = l.

The conditional probabilities for all the branches emanating from a node can be computed by match-

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Figure 1: Multinomial Lattice-Single Step

ing the ﬁrst and the second moments in the lattice and in the uncorrelated continuous processes. The conditional branch probabilities of the trinomial lattice for regime k = 1, 2, . . . , N are given by πφk ,u = πφk ,d = φ2 1 µk h k + φ , φl 2 2 (k ) k ll l φ2 1 µk h k − φ , φl 2 2 (k ) k ll l (6) (7) (8)

πφk ,m = 1 − πφk ,u − πφk ,d ,

where πφk ,u , πφk ,m , and πφk ,d are the conditional probabilities for the upward, the middle, and the downward branch of the trinomial lattice, respectively. πφk ,u is the probability that the value of the underlying variable increases by k φl , πφk ,m is the probability that the value does not change, l and πφk ,d is the probability that the value decreases by k φl . Notice that when k = l the probability l

9

of middle branch is zero, which means that one of the N regimes has zero probability associated with its middle branch. A layer of the multinomial lattice for a single underlying variable with N regimes is shown in Figure 1 when the jump sizes of the regimes satisfy the conditions: (φ1 <
φ3 3 φ2 2

<

< ... <

φN −1 N −1

<

φN N ). n i=1 mi )

So far we have constructed a lattice for a single variable with N (=

regimes and now we

have to construct a lattice for n uncorrelated variables, each with N regimes, that are simultaneously switching between the N regimes. We simply combine n multinomial lattices, each of which has been constructed for an uncorrelated variable, to form an n + 1 dimensional lattice. The correlation between any pair of variables in the resulting lattice will be zero due to the transformation performed on each variable as in Equation (4). In case of two processes with two regimes for each process, a nonanomial lattice is built for each variable, in which nine branches emanate from each node of the lattice. Assembling two nonanomial lattices leads to a higher dimensional lattice for the two uncorrelated processes as shown in Figure 2. Notice that while combining two nonanomial lattices may result in 81 branches (9 × 9) emanating from a node, the combined lattice shown in Figure 2 has only 33 branches (only 29 nodes are shown since 4 nodes are stacked on top of each other at the center). This is because the two uncorrelated processes should switch regimes simultaneously. Recall that when there are two variables and two regimes for each variable, we have four combined regimes: combined regime 1 (1H,2H), 2 (1L, 2L), 3 (1H, 2L), and 4 (1L, 2H). A pair of uncorrelated
A processes are obtained in each regime after the transformation given in Equation (2). Let ψi B and ψi denote the new processes in regime i for i = 1, 2, 3, 4. Then any arbitrary selection of a

single process from each regime deﬁnes four regimes of a new uncorrelated process. For example, a
A B B A collection of processes {ψ1 , ψ2 , ψ3 , ψ4 } will deﬁne a new process ψ 1 and the four distinct processes A B B A ψ1 , ψ2 , ψ3 , ψ4 represent a single process in four diﬀerent regimes. This will uniquely determine B A A B the second process ψ 2 that has a collection of four regimes {ψ1 , ψ2 , ψ3 , ψ4 }. Such selection is

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Figure 2: Child Nodes of a Parent Node

not unique but any arbitrary selection will do. One should keep it mind that at any time the two
A uncorrelated variables must switch regimes simultaneously. That is, as ψ 1 switches from regime ψ1 B B A to ψ3 , the second process ψ 2 must switch from regime ψ1 to ψ3 simultaneously. Therefore, there

are only 33 possible regime combinations for the new processes ψ 1 and ψ 2 . The spatial pattern of the 33 nodes is clearly shown in Figure 2, where we index the uncorrelated processes derived from (1H, 2H) as 1 and 2, from (1L, 2L) as 3 and 4, from (1H, 2L) as 5 and 6, and from (2L, 2H) as 7 and 8. When there are n variables with m regimes for each variable, the number of child nodes emanating from a single node is 3n (mn − 1) + 2n . The spatial pattern of child nodes shown in Figure 2 is under the following conditions: φ7 φ5 φ3 < < < φ1 4 3 2 and φ8 φ6 φ4 < < < φ2 . 4 3 2 (9)

The conditional branch probabilities in the ﬁnal lattice is computed as the product of the con11

ditional branch probabilities of the two corresponding branches from the two nonanomial lattices. Numbers in the parenthesis at each child node of the lattice in Figure 2 are conditional probabilities from the two nonanomial lattices, which can be calculated using Equations (6) through (8). The ﬁrst subscript of the probability is the step size of the branch, and the second subscript is the position of the branch in the nonanomial lattice. The conditional branch probability of the branch in the product lattice (or three dimensional lattice) is simply the product of these two probabilities in parenthesis. Step sizes in the vertical and horizontal directions in Figure 2 are determined by Equation (5). Given the conditions in (9), the vertical step size is one-fourth of φ1 , which is the largest step size in the ﬁrst nonanomial lattice; the horizontal step size is one-fourth of φ2 , which is the largest step size in the second nonanomial lattice.

2.4

Inverse Transformation

In an eﬀort to keep the lattice as small as possible, the step size is set deliberately to allow nodes to merge as the lattice is expanded. As a result a node may represent a process in diﬀerent combined regimes and we need to know which regime a given node represents to compute values such as the expected option price. The standard rollback procedure, employed to compute the expectations conditional on the current regime, requires values for n processes for a given node. Although nodes store values for the n processes, these values are for the transformed processes given in Equation (4) instead of the original processes given in Equation (1). Therefore, an additional step is required to compute the value of the original processes from the values of the transformed processes. Suppose that we are given a vector Ψ (= (ψ1 , ψ2 , . . . , ψn )T ) for a given combined regime, that consists of values for n transformed processes. From the transformation given in Equation (4) we

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know y = LΨ. (10)

Consequently the value of the original correlated state variable in the combined regimes Si = eyi for i = 1, 2, . . . , n. Since the transformation given in (4) is based on the mean and the volatility (matrix L depends on the variance-covariance matrix Ω deﬁned for each combined regime), which are diﬀerent under diﬀerent combined regimes, the inverse transformation should use the appropriate mean and volatility corresponding to the combined regime for which the values are recovered. For example, suppose that we need the value of the original process S1 in the high volatility regime and that of the original process S2 in the low volatility regime, i.e., the two variables are in combined
A B regime 3, when a given node has two values ψ3 and ψ3 for two uncorrelated processes. The inverse A B transformation is simply L(ψ3 , ψ3 )T , where T stands for transpose of a vector or a matrix, L is

the lower triangular matrix obtained by the Cholesky factorization from the variance-covariance matrix Ω for the ﬁrst original process in the high volatility regime and the second original process in the low volatility regime. The last tricky point that needs to be emphasized is the initial regime of the state variables. To truly make the inverse transformation in Equation (10) the inverse of the transformation in Equation (4), the two matrices L and L−1 should be derived from a common variance-covariance matrix Ω, that is unique for each combined regime. For example, if we want to recover S1 in the
A B high volatility regime and S2 in the low volatility regime from ψ3 and ψ3 stored in a given node,

the transformation should use L−1 derived from Ω for S1 in the high volatility regime and S2 in the low volatility regime and the inverse transformation should use L. This implies that we must separately compute four lattices to represent each of the four possible initial regimes. Fortunately, this is not the case since our multinomial lattice was constructed such that processes in diﬀerent combined regimes (i.e., the four distinct lattices implied above), are simply some constant shift 13

from one another. Recall from Equation (5) that the step sizes of the lattice are set such that the nodes are equally spaced for all combined regimes and that the diﬀerence in value of an uncorrelated process (or transformed state variable) between two combined regimes is some constant. Figure 2 illustrates this where each step in the vertical direction is a constant Φ1 /4 and each step in the horizontal direction is a constant Φ2 /4. Therefore, once we know ψ1 and ψ2 , we can simply add a constant to, or subtract a constant from the values stored in a given node, for a given combined regime, to obtain the corresponding values in the other combined regimes.

3

Option Valuation

In this section we brieﬂy recap risk-neutral valuation and overview the standard procedure to value the option for a simple binomial lattice. Then, we explain the option valuation procedure for the (n + 1) dimensional lattice that we constructed in Section 2.3. Options are valued in the risk-neutral world, where the expected return is the risk free rate. This is implied by the Black-Scholes partial diﬀerential equation [1], which is independent of risk preferences. In the risk-neutral world, the drift of the underlying asset is replaced by the risk free rate minus any continuous dividend yield, and all future payoﬀs are discounted at the risk free rate. In regime-switching models, the volatility of the underlying variables changes over time and hence risk-neutral valuation fails. Bollen [2] indicates that there are two ways to deal with this problem: to assume that the additional regime risk is not priced, or to specify a risk adjusted rate in the model to allow risk-neutral discounting. To simplify the valuation he assumes that the regime risk is not priced, so that the risk-neutral valuation can be used. He also indicates that this assumption is equivalent to the assumption made by Hull and White [13] that the risk of stochastic volatility is not priced. We will also use Bollen’s assumption to value options with regime switching. To brief the standard option valuation procedure in a lattice, consider a binomial lattice that 14

represents a single variable with one regime. For an European option, the value at all the terminal nodes (at expiration of an option) are calculated as the maximum of zero and the proceeds of the option’s exercise. The option value in earlier periods can be computed as the maximum of zero and the expected discounted value of the option. In order to calculate the expectation at each node, branch probabilities are used. This procedure is continued from all the terminal nodes to the root node in a roll-back manner. For an American option, at each node, the proceeds of early exercise is compared with the discounted expected value of the option and whichever oﬀers the maximum is used for further computation. In the regime-switching model, unlike the binomial lattice, time-varying regime probabilities and the possibility of switching regimes at each node cause diﬃculties in computing expected values. This hurdle can be overcome by computing conditional option values at each node for each combined regime, where the conditioning information is the regime that governs the prior observation. These conditional option values are computed in a roll-back manner from the terminal nodes to the root. With N (=
n i=1 mi )

combined regimes for each state variable, we need to compute N conditional

option values at each node. As the computation rolls back as explained earlier, the conditional option values clearly depend on regime persistence. To illustrate the procedure, we introduce the expression to value an American call option using our proposed lattice. For an American option early exercise proceeds are compared with the expected discounted value. An American call option value at time t conditional on combined regime α is:    N   c(t, α) = max θt − K, e−rf h  γαβ E[c(t + 1, β)] , α = 1, 2, 3, . . . , N,  
β=1

(11)

where K is the exercise price, γαβ is the transition probability (regime persistence) from combined regime α to β, rf is the risk free interest rate, and θt is value of the underlying variables at time t. Notice that the call option at time t, which is conditional on regime α, is related to conditional option values at time t + 1. Since we have more than one underlying variable, θt is a predeﬁned function 15

of underlying variables, such as maximum, minimum, and average of the underlying variables, at time t. The expectation of the future conditional option values are computed based on the conditional branch probabilities of the appropriate set of branches. The regime persistence aﬀects the conditional option value by weighing the expectation of future conditional option values. The rollback computation eventually yields N conditional option values at the root node, each corresponding to one combined regime. If one knows the current regime, then the option value for the regime can be used. Otherwise, the probability distribution for the current regime may be available so that the expectation of the option value can be computed. In order to accurately approximate a continuous process by a discrete lattice, the time increment h of the lattice should be reduced. Unlike the standard lattice approach where the stochastic process can be scaled up or down, in regime switching models, the persistence of a regime (or regime switching probabilities) at successive observations has to be adjusted as the time between observations is scaled up or down. Obviously, as the time interval between two consecutive observations decreases, it is less likely to make a regime switch. There are two ways to adjust the transition probabilities. If the generator matrix of the Markov chain G, which describes the regime switching process, is known, then exp(Gh) = P (h), where P (h) is the transition probability matrix for the length of the time step h. In general, P (h) = I +
∞ hn Gn n=1 n!

and for a small length of the time

step h, P (h) = I + Gh, where I is the identity matrix. The other way is for when the new time increment h is a power of 2 of the original increment h, i.e., h = 2a · h, a ∈ {. . . , −2, −1, 1, 2, . . .}. For a > 1, we can simply compute the multi-step transition probabilities from the given one-step probabilities. For a < 0, we need to solve a quadratic system to cut the increment in half until the desired time increment is achieved. Note that in the regime-switching model, assuming the chain is irreducible, the quadratic system is guaranteed to have positive eigenvalues so that the square root always exists.

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4

Numerical Examples

In this section we value several rainbow European calls and American puts to demonstrate the accuracy of the option value computed using the proposed framework. There are two correlated state variables each of which has two regimes. Monte Carlo simulation is used, in parallel to the lattice-based computations, to provide benchmarks. The annual volatilities and the correlation factors between the two variables in four combined regimes are given in Table 1. The initial values Table 1: Volatilities and Correlation factors (2 variables and two regimes each)

Asset 1 High Vol. (=50%) Asset 2 High Vol. (=20%) Low Vol. (=10%) 0.2 -0.1 Low Vol. (=15%) -0.15 0.4

of the both state variables are \$100. The risk free interest rate is 5% annually and the initial regime probability of each regime is assumed to be 0.25 (i.e., all the regimes are equally likely). The weekly transition probabilities, or regime persistence, are given in  0.99927 0.00003 0.00070    0.00210 0.17179 0.01639    0.17414 0.04917 0.72260  0.00010 0.07197 0.00160 a matrix as follows:  0.00001   0.80972    0.05409   0.92632

We follow the procedure given in Hull (pp. 412-413) [12] and Boyle [5] to complete the Monte Carlo simulation runs for European calls. But for American puts, we use the least-squares approach as described in Longstaﬀ and Schwartz [15]. We also employ the method of antithetic variates [12], which is popular for its simplicity to speed up the convergence, in our simulation runs. The time increment of the lattice is one eighth of a week and all four combined regimes are equally likely at the beginning. A total of 40,000 simulation sample paths are used in each simulation run, and one-forth of the 40,000 simulations paths begin with each regime. Table 2 presents the European 17

call option values, using Monte Carlo simulation and the lattice approach proposed in this paper, when the option payoﬀ depends on the arithmetic average, maximum, and minimum of the two underlying state variables. The maturity of all the European call option is 12 weeks. Table 3 presents the values of the American put option with the maturity of 6 weeks. As evident in these tables, the option values from the proposed lattice approach are fairly close to those from Monte Carlo simulation. Table 2: European call option value on the average, max. and min. of underlying variables

Maturity in weeks

European call option values Underlying Variables Mean of Underlies Exercise price Lattice Simulation Error Lattice Max. of Underlies Simulation Error Lattice Min. of Underlies Simulation Error 95 7.494 7.449 0.045 13.394 13.444 0.050 3.424 3.347 0.077 100 4.210 4.194 0.016 9.175 9.231 0.056 1.451 1.419 0.032 105 2.181 2.165 0.016 5.910 5.968 0.058 0.536 0.505 0.031

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Table 3: American put option values on the average, max. and min. of underlying variables

Maturity in weeks

American put option values Underlying Variables Mean of Underlies Exercise price Lattice Simulation Error Lattice Max. of Underlies Simulation Error Lattice Min. of Underlies Simulation Error 98 1.516 1.515 0.001 0.540 0.503 0.037 4.102 4.083 0.019 100 2.368 2.337 0.031 1.066 0.999 0.067 5.423 5.415 0.008 102 3.553 3.461 0.092 2.166 2.086 0.080 7.030 6.986 0.044

6

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5

Conclusions

The lattice proposed in this paper can be seen as a direct generalization of both the lattice with regime switching and the lattice for multiple state variables. With this elevated level of generality, the proposed lattice can be used in a wider range of ﬁnancial and non-ﬁnancial applications. In particular, the valuation of non-ﬁnancial options such as manufacturing ﬂexibility often requires multiple state variables over a much longer maturity, during which period it would be impractical to assume a single distribution for all variables. Despite the ﬂexibility and the simplicity of the proposed framework, challenges remain for further investigation in the future. The most critical hurdle is that the approach is computationally demanding. Although every eﬀort is made to reduce the size of the lattice, it could become intractably large as the number of variables and the regimes per variable increase.

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