An Improved Binomial Lattice Method for Multi-Dimensional Option Problems

Transcription

1 An Improved Binomial Lattice Method for Multi-Dimensional Option Problems Working paper - University of Cyprus current version January, 2004 (first draft January, 2001) Andrea Gamba Department of Economics University of Verona (Italy) Lenos Trigeorgis Department of Public and Business Administration - University of Cyprus (Cyprus) Corresponding author: Andrea Gamba Department of Economics University of Verona Via Giardino Giusti, Verona (Italy) tel fax The authors are grateful to Matteo Tesser for computation assistance. We wish to thanks the participants of the 5th Annual International Conference on Real Options - UCLA, and of seminars at University of Cyprus and University of Pisa for their valuable comments. We are responsible for all remaining errors. 1

2 An Improved Binomial Lattice Method for Multi-Dimensional Option Problems Abstract We propose an improved binomial lattice approach for valuing complex option problems whose payoff depends on multiple state variables following correlated geometric Brownian processes. The proposed approach relies on two main ideas: a specific change of time scale as in the log-trasformed binomial approach by Trigeorgis (1991), and a change of basis of the asset span, to trasform them into uncorrelated processes. These features improve the efficiency of the multi-dimensional binomial algorithm. Moreover, the proposed algorithm monotonically converges to the true result and is most suitable to employ Richardson extrapolation for improved accuracy. We present various applications to financial and to real option pricing problems and show the flexibility and improved efficency over existing binomial approaches. Keywords: Option pricing, binomial lattice, multi-dimensional diffusion. JEL classification: G13 2

3 Introduction Complex options or contingent claims dependent on multiple state variables are common in financial economics, both in financial as well as in real investment valuation problems. 1 Closed-form solutions for such problems are available only in a few special cases, so numerical methods must be generally employed to price such multi-dimensional option problems. A number of approaches have been proposed to numerically tackle option valuation problems. These can be divided into three main categories: numerical solutions to partial differential equations (pde) such as finite difference methods (first introduced in finance by Brennan and Schwartz (1977)) and finite elements; Monte Carlo simulation methods (first introduced by Boyle (1977)); and lattice methods, first proposed by Cox et al. (1979) (CRR in what follows). Among these, lattice methods are generally considered to be simpler, more flexible and, if dimensionality is not too large, more efficient than other methods. In this paper we propose a binomial lattice extension to evaluating contingent claims whose payoff depends on multiple state variables that follow joint (correlated) geometric Brownian processes. A number of variations of the CRR binomial lattice method have been proposed to approximate the price of options on asset values following a geometric Brownian motion. Rendleman and Bartter (1979), Hull and White (1988, footnote 4) and Jarrow and Rudd (1983), for example, propose different versions of the binomial lattice approach with different choices of parameters for the up (and down) multiplicative steps and the (risk-neutral) probabilities. Boyle (1988) and Kamrad and Ritchken (1991) propose trinomial lattice algorithms whose accuracy depends on the choice of a stretch parameter that must be chosen up front. Improved accuracy can be achieved at the cost of increasing computational effort, so the efficiency needs to 3

4 be assessed. 2 Boyle (1988) and Kamrad and Ritchken (1991) also provide extensions of their trinomial lattice approach for problems with several underlying assets. Boyle et al. (1989) (BEG, from now on) also provide a straightforward extension of the CRR approach to several underlying assets. However, as they acknowledge, their scheme inherits the CRR algorithm weakness that can result in negative probabilities (and variances) if the size of the time step is large, or if the number of steps is small enough. Unfortunately, if the number of underlying assets is large, one can hardly afford a reasonably large number of steps because of the exponential complexity of the lattice approach with respect to the number of state variables. So it can happen that for certain values of the parameters, the probabilities of the up and down jumps in BEG s scheme (which is based on CRR s original choice of probability) can turn negative, giving inaccurate estimates of the option value. Ekvall (1996), extending the scheme by Rendleman and Bartter (1979), also proposes a modification of the BEG lattice model (called NEK) to overcome the flaw of possibly negative probabilities. Other lattice approaches have been proposed to cope with different stochastic processes (e.g. Nelson and Ramaswamy (1990)) or with time-varying variance-covariance structures (see Ho et al. (1995)) for the underlying asset values. 3 Trigeorgis (1991) proposed a log-transformed version of the binomial lattice approach for contingent claims dependent on a single state variable which claims several improved qualities including avoidance of negative probabilities. It also proves to be more efficient than other lattice schemes, while keeping the appealing simplicity of the CRR approach. We herein present a binomial lattice approach for valuing contingent claims dependent on multi-dimensional correlated geometric Brownian pro- 4

5 cesses. The approach relies on two main ideas: a transformation of the value dynamics, as in the log-transformed approach of Trigeorgis (1991); and a change of basis of the asset span to numerically approximate an uncorrelated dynamic for asset values. These transformations result is computational improvements over existing binomial lattice approaches (BEG and NEK). The proposed method is consistent (i.e., the means and the variancecovariance matrix of the approximating stochastic process are the same as the means and the variance-covariance matrix of the diffusion process for any time step), stable (i.e., the approximating errors are not amplified), and efficient (i.e., the computational cost for accuracy of a given approximation is lower than in other methods). Interestingly, the relative accuracy and efficiency benefits are grater the larger is the problem dimensionality. The paper is organized as follows. Section 1 describes the improved binomial lattice approach for approximating multi-dimensional geometric Brownian processes. Section 2 discusses the implementation aspects of our model. In Section 3 we illustrate several applications to option pricing problems with up to five stochastic assets. Conclusions are offered in Section 4. 1 The multi-dimensional lattice model This section describes an extension of the log-transformed binomial lattice approach by Trigeorgis (1991) to multiple correlated underlying assets. The basic idea is to approximate a multi-dimensional geometric Brownian motion with a binomial lattice by choosing the size and the probabilities of the jumps so that the characteristic function of the discrete distribution converges to the characteristic function of the continuous distribution. 4 As a first step, we employ the multi-dimensional extension scheme proposed in Boyle et al. (1989) in extending the CRR approach to multiple assets. As we will see, 5

6 this extension suffers with some of the same drawbacks as in CRR (and BEG) because of the presence of correlation. If the multiple underlying assets were uncorrelated, this straightforward multi-dimensional extension of the log-transformed approximation would have the same appealing features as the one-dimensional case. But the presence of correlation necessitates another transformation. By changing the coordinate system of the asset span to approximate a vector of uncorrelated diffusion processes, we can then evaluate an option written on multiple assets while preserving the positive features of the log-transformed approach. Consider N correlated stochastic assets whose value dynamics, denoted 5 X = (X 1,..., X N ), follow N-dimensional geometric Brownian motions, under the equivalent martingale measure (EMM): 6 dx i X i = α i dt + σ i dz i X i (0) = x i i = 1, 2,..., N (1) where α i is the risk-adjusted drift of the i-th asset value, 7 and dz i are the increments of correlated Gauss-Wiener processes, such that E[dZ i dz j ] = ρ ij dt, i j, where ρ ij denotes the correlation between asset i and asset j. Consider a derivative security with maturity T and value F whose payoff depends on the above underlying asset values. Our goal is to compute the option value, F. Following the standard risk-neutral valuation argument of Black and Scholes (1973) extended to a multi-factor economy by Cox et al. (1985), the option value F in the multi-dimensional case must satisfy the pde: 1 2 N i=1 j=1 N 2 F ρ i,j σ i σ j X i X j + x i x j N i=1 α i X i F x i + F t rf = 0 with appropriate boundary and terminal conditions related to the payoff 6

7 function. Because in general an analytic solution to this multi-dimensional problem does not exist, 8 one must resort to a numerical solution by approximating the continuous-time dynamics in (1). We use a log-transformed binomial lattice. Taking the logarithm of the asset values, 9 Y i = log X i, the dynamics of Y = (Y 1,..., Y N ) is (by Itô s Lemma) dy i = ( α i 1 ) 2 σ2 i dt + σ i dz i, i = 1, 2,..., N. (2) Given the option maturity T, the time interval [0, T ] is divided into n subintervals of increments t = T/n. At dates {0, t, 2 t,..., T }, the discrete-time process approximating the continuous-time process {Y } is {Ŷ } = {(Ŷ1,..., ŶN)}. For a given n, we denote by {Ŷ n } the discretetime approximating process with n time steps. One needs to specify the parameters of the approximating process {Ŷ n } so that, as n gets large, {Ŷ n } converges (in distribution) to {Y }. In what follows, we specify the parameters of the approximating process for a given n (so that the dependence on n will be omitted to simplify notation). For the sake of simplicity, we first illustrate the process for the cases N = 1 and N = 2. For the one-dimensional case, N = 1, our model coincides with the log-transformed approximation proposed by Trigeorgis (1991). The approximation of the process in (1) for N = 1 reduces to Ŷ (t) = Ŷ (t 1) + hu(t) for t = 1,..., n, where {U(t)} is a set of independent and identically distributed (i.i.d.) binomial random variables with probability of an upward jump p. The pa- 7

8 rameters for the jumps sizes and probabilities are: µ = m σ 2 = α σ 2 1 k = σ t 2 h = k 2 + (k 2 µ) 2 p = 1 ( ) 1 + k2 µ, (3) 2 h where k is the time step size (in units of standard deviation), h is the state step (jump), and p is the probability of the up jump. Hence, the continuous-time process {X} is approximated by the discretetime process X(t) = X(t 1)e hu(t) for t = 1,..., n,, with X(0) = x, where X(t) = ey b (t). An appealing feature of this method, in contrast to CRR, is that by design 0 p 1 for any number of time steps, because h k 2 µ, with no need for external constraints on the parameters. 10 Then, the logtransformed approach allows for unconditional stability. Note that the time step is k (instead of t), as time is measured in units of variance. In what follows, we aim at extending the appealing features of this scheme to a multi-dimensional setting. 11 When N = 2, the approximating discrete process is Ŷ i (t) = Ŷi(t 1) + h i U i (t), t = 1,..., n, and i = 1, 2 where (U 1 (t), U 2 (t)) is set of a bi-variate i.i.d. binomial random variables: (1, 1) with probability p 1 (1, 1) with probability p 2 (U 1, U 2 ) = ( 1, 1) with probability p 3 (4) ( 1, 1) with probability p 4 8

9 where 4 i=1 p i = 1. In the spirit of one-dimensional Equation (3), the parameters are (i = 1, 2): µ i = α i σ 2 i 1 2, k i = σ i t, and hi = k 2 i + (k2 i µ i) 2. (5) Moreover, let R ij = k ik j h i h j, and M i = k2 i µ i h i. The state probabilities are: p 1 = 1 4 (1 + (Rρ + M 1M 2 ) + M 1 + M 2 ), p 2 = 1 4 (1 (Rρ + M 1M 2 ) + M 1 M 2 ), p 3 = 1 4 (1 (Rρ + M 1M 2 ) M 1 + M 2 ), (6) p 4 = 1 4 (1 + (Rρ + M 1M 2 ) M 1 M 2 ), where ρ = ρ 12 and R = R 12. These parameters are such that the first two moments of the distribution of the increment of the discrete-time process match the first two moments of the distribution of the increment of the continuous-time process, for any time step: 12 E[ Ŷi] = k 2 i µ i = ( α i 1 ) 2 σ i t = E[ Y ], i = 1, 2 (7) Var[ Ŷi] = k 2 i = σ 2 i t = Var[ Y ], i = 1, 2 (8) Cov[ Ŷ1, Ŷ2] = ρ 12 k 1 k 2 = ρ 12 σ 1 σ 2 t = Cov[ Y 1, Y 2 ]. (9) Hence, the approximation of the bi-variate geometric Brownian motion {X} is given by the process {( X 1, X 2 )} such that X i (t) = X i (t 1)e h iu i, t = 1,..., n, i = 1, 2, with U i as defined in (4). 9

10 Although the above approximation is an improvement over BEG s extension, which it follows, it is still possible for some values of the parameters to have negative probabilities in some states due to the presence of correlation (ρ) (see Figure 1 for a numerical example). It is obvious from (6) that correlations play an important role in the ability to have positive probabilities in all states. In the remainder of this section, we exploit this realization to develop an alternative, improved log-transformed binomial lattice approach for the multi-dimensional case. The basic idea is to transform the basis of the asset span to approximate uncorrelated asset dynamics. We accomplish this by changing the basis of R N, the asset span generated by the N-dimensional diffusion of asset returns Y = (Y 1,..., Y N ), so that the value of the option is dependent on an N-dimensional diffusion y = (y 1,..., y N ) obtained by a change of basis such that its components y i are uncorrelated. Of course, if we change the basis of the market space, we also have to change the payoff function accordingly. Denoting by Π(Y ) the original payoff of the option, and by W the matrix representing the change of basis, the expression of the adjusted option payoff with respect to the new basis is Π(y) = Π(W y). The dynamics of the returns y can then be approximated by a suitable multidimensional binomial lattice. This method maintains the stability feature of the one-dimensional approach of Trigeorgis (1991) for a wider choice of parameters than the log-transformed extension in (5) and (6). The economic rationale for the change of basis is the following. We want to price an option with payoff Π(Y ), where Y are the returns of N assets traded in the market, in a risk-neutral setting. If the financial markets are complete 13, we can generate N portfolios with the original assets: we denote w i = (w i1,..., w in ) the i-th portfolio, i = 1,..., N, where w ij 10

11 is the position in the j-th asset in portfolio i. These portfolios can be thought of as new synthetic assets spanning the (same) market space. Any contingent claim which is spanned by the original assets is spanned also by these synthetic assets. The N portfolios thus generated are selected so as to have uncorrelated returns. The option to be priced, dependent on the returns of these synthetic assets, is denoted by Π(y). Because the risk structure of the market is unchanged, 14 we can price the option using riskneutral valuation with respect to the original EMM by a simple change of basis. Consider asset return dynamics as in Equation (2), or dy = adt + σdz in a vector notation, where a = (a 1,..., a N ), with a i = α i σi 2/2, dz = (dz 1,..., dz N ), 15 1 ρ 12 ρ 1N ρ 12 1 ρ 2N Σ = ρ 1N ρ 2N 1 σ σ 2 0 and σ = σ N Suppose dy has a time-independent covariance matrix. The covariance matrix of dy is dy dy = σdzdz σ = σσσ dt = Ωdt. By definition, Ω is a symmetric positive definite matrix. Hence, it can be factorized using an N N matrix W such that W W = I N, with I N being the N-dimensional identity matrix, so that W ΩW = Λ, where Λ is the diagonal N-dimensional matrix (λ i ) with λ i > We denote by y = W Y the returns of the synthetic portfolios obtained by linear combinations of the original assets spanning the financial markets. The diffusion process of y is dy = Adt+BdZ, where A = W a and B = W σ. The covariance matrix of dy is dydy = Λdt, i.e., the components of y = (y 1,..., y N ) are uncorrelated: dy i dy j = 0 11

12 whenever i j and (dy i ) 2 = λ i dt. Let Π(X(t)) = Π(X 1 (t),..., X N (t)) be the payoff of the option. According to the change of variable Y = log X, the payoff is Π(X 1 (0)e Y 1(t),..., X N (0)e Y N (t) ). We can make the option dependent on y = W Y by changing the payoff function as follows: ( ) Π(y(t)) = Π X 1 (0)e (W y(t)) 1,..., X N (0)e (W y(t)) N where (W y(t)) i is the i-th component of Y (t) = W y(t). The risk-neutral expected value of the option payoff Π, denoted F, is equal to the risk-neutral expected value of Π. Hence, F (y(t)) = e r(t t) E y [ Π(y(T )) ] = e r(t t) E Y [Π(Y (T ))] = F (Y (t)), (10) where E y [ ] denotes the risk-neutral expectation with respect to ν y, the EMM of the process {y}, and E Y [ ] is the expectation w.r.t. ν Y, the EMM of the process {Y }. 17 The above is also true for American-type options (see the numerical results in Section 3). 18 The economic intuition behind the valuation equivalence underlying Equation (10) is that a change of basis does not change the risk structure of the market. The geometric intuition is the following: because the covariance matrix Ω is time-independent, the measure ν Y is invariant under a change of basis. Given this, we can numerically evaluate F (y) by approximating ν y with a discrete (binomial) distribution. First, we illustrate the resulting improved binomial scheme for the two- 12

13 dimensional case. With N = 2 and ρ = ρ 12, we obtain Λ = (λ i ), a twodimensional diagonal matrix, where λ 1,2 = 1 2 ( ) σ1 2 + σ2 2 σ (1 2ρ2 )σ1 2σ2 2 + σ4 2, and where W = ( λ1 σ 1 σ 2 σ 2 σ 1 ) / (ρc 1 ) ( ) λ2 σ 1 σ 2 σ 2 σ 1 / (ρc 2 ) 1/c 1 1/c 2 ( λi σ 2 2 c i = 1 + 2) ρ 2 σ1 2. σ2 2 The processes of the returns for the synthetic portfolios are dy i = A i dt + B i1 dz 1 + B i2 dz 2 i = 1, 2 where B = (B ij ) = W σ and A = W a. We approximate {y} with a discrete process: given the time interval [0, T ] and the number of steps n, the discrete approximating process is ŷ = (ŷ 1, ŷ 2 ) with dynamics ŷ i (t) = ŷ i (t 1) + l i U i (t) i = 1, 2 (11) t = 1,..., n where (U 1, U 2 ) is as defined in (4). To approximate the characteristic function of the continuous-time process, with a discrete-time process, we select the parameters (i = 1, 2) κ i = A i t, l i = λ i t + κ 2 i, L i = κ i /l i (12) 13

14 and probabilities p(s) = 1 4 (1 + δ 12(s)L 1 L 2 + δ 1 (s)l 1 + δ 2 (s)l 2 ) s = 1,..., 4, (13) where 1 if asset value i jumps up δ i (s) = 1 if asset value i jumps down (14) for i = 1, 2, and δ ij (s) = δ i (s)δ j (s), i, j = 1, 2, i j. With the above choices, the characteristic function of the discrete distribution matches the characteristic function of the continuous distribution for any time steps (while transformed asset returns are uncorrelated): E [ ŷ i ] = κ i = A i t i = 1, 2 Var [ ŷ i ] = l 2 i κ 2 i = λ i t i = 1, 2 (15) Cov [ ŷ 1, ŷ 2 ] = 0. With ρ ij = 0 for all i j, this model collapses to the one described in (5) and (6). In this case, the matrix W reduces to the identity matrix, I N. To illustrate the above, consider the following numerical example. Example 1. Let N = 2 and the parameters of the process in (1) be α 1 = 0.05, α 2 = 0.08, σ 1 = 0.02, σ 2 = 0.3, ρ = 0.9, and t = 0.1. Given these parameters, the covariance matrix is Ω =

15 The eigenvalues of Ω are λ 1 = and λ 2 = and W = With this change of basis, the parameters of the process in (11) are A = , B =, l 1 = and l 2 = Since L 1 = and L 2 = , the resulting probabilities from (13) are: p 1 = , p 2 = , p 3 = , p 4 = The above analysis can be generalized to the N-dimensional case, as follows. Concerning the matrices Λ and W, instead of having exact expressions as in the two-dimensional case, they will be computed numerically (with fair accuracy). The discrete approximating process, ŷ = (ŷ 1,..., ŷ N ) with dynamics ŷ i (t) = ŷ i (t 1) + l i U i (t) i = 1,..., N (11 ) would have the same parameters as in (12), for i = 1,..., N, and probabilities: p(s) = S N δ ij (s)l i L j + δ i (s)l i (16) 1 i<j N i=1 with s = 1,..., S, where S = 2 N is the number of states at the end of 15

16 each time step and δ i (s) δ ij (s) are defined in (14). By simple substitution of parameters, it can be confirmed that the first two moments of the discrete approximating distribution are equal to the first two moments of the continuous distribution, for any time step: 19 E [ ŷ i ] = κ i = A i t = E [ y i ] Var [ ŷ i ] = l 2 i κ 2 i = λ i t = Var [ y i ] i = 1,..., N i = 1,..., N Cov [ ŷ i, ŷ j ] = 0 = Cov [ y i, y j ] i j. 2 Implementation procedure This section discusses implementation of the numerical procedure to value an option according to the improved multi-dimensional binomial lattice approximation described in Section First, for a given option maturity T and number of time steps n, a multi-dimensional binomial tree of the evolution of future asset values is generated according to the parameters in (12) until the time horizon (T ) is reached. The number of operations and the amount of computing time needed is essentially the same as in other binomial lattice schemes (BEG and NEK). The routine for diagonalizing the covariance matrix is called for only once, and so it does not affect the complexity of the algorithm. Next, if the option is European, we need only evaluate the option payoff on the final nodes of the tree (at maturity T ), average (using the riskneutral probabilities determined from (16)) and discount (at the riskless rate) to obtain the current (t = 0) value of the option. If the option is American, we take into account the early exercise feature in a backward dynamic programming fashion by comparing, at each node of the lattice, at 16

17 the indicated times, the current payoff from early exercise with the option continuation value obtained by applying the risk-neutral valuation as above. Discrete dividend-like payments can be accommodated in a standard way (e.g. Hull and White (1988), and Trigeorgis (1991)). In our model, discrete non-proportional dividends paid by the underlying assets at known dates affect the valuation in a similar way as in the Trigeorgis (1991) and BEG algorithms. The presence of discrete dividends for the i-th asset can be accounted for by a shift of the nodes along the i-th dimension at the ex-dividend date. The displacement is different for each node because the asset dynamics is exponential, whereas the dividend is additive. Hence, the tree may not be recombining at the ex-dividend date. 21 Given the dividend vector D = (D 1,..., D N ), paid at some known date τ (the exdividend date), t τ < t + t, and the value of the option at t from (10), F (t, y(t)) = F ( t, X 1 (0)e (W y(t)) 1,..., X N (0)e (W y(t)) N ), the value of the option at τ is F ( ) τ, X 1 (0)e (W y(τ)) 1,..., X N (0)e (W y(τ)) N = F (τ, X 1 (0)e (W y(τ )) 1 D 1,..., X N (0)e (W y(τ )) N D N ) + where τ is a time just before the ex-dividend date. The main difference with respect to the BEG algorithm is that the displacement is given according to a different (rotated) coordinate set of axes. For an American option, the efficiency is improved (i.e., the computational effort is reduced for a given accuracy level) if instead of generating the whole lattice evolution from the outset, at each time step t, only asset values for t and t + 1 are generated when applying the risk-neutral dynamic programming valuation. 22 N i=1 D i 17

18 As suggested by Boyle et al. (1989), Richardson extrapolation (RE) can be a practical method to obtain accurate approximations of exact values while saving on computing time. In particular, we can use a four-point RE, fitting option values (as a function of T/n = t) with a cubic polynomial. RE provides accurate estimates of option values as long as the sequence of points is monotonic. In this respect, the NEK approach, although efficient, can provide flawed RE estimates because the sequence of evaluated points is oscillating and not monotonic (see Table 2). Our approach is also advantageous in this respect as seen in the next section. 3 Applications and comparative numerical results In this section we provide the results of three sets of numerical applications that illustrate the accuracy and efficiency of the proposed approach, compared to the BEG and NEK schemes. The first application is financial, concerning valuation of European optionn on three or four assets, using the numerical results of Boyle et al. (1989) and Ekvall (1996) as benchmarks. The second application is a real option valuation problem involving the best of two product standards described in Lint and Pennings (2001) and extended in Martzoukos and Trigeorgis (1999) to four sources of uncertainty. We provide numerical valuations and compare them with those presented in Martzoukos and Trigeorgis (1999), obtained with BEG s scheme. The third application deals with the American option on the maximum of five assets presented in Broadie and Glasserman (1997). The most accurate results presented by Broadie and Glasserman in this case are downward biased because they approximate an American option with a semi-american option with only four exercise dates. We therefore also benchmark our results against those obtained by Least-Squares Monte- 18

19 Carlo Simulation (LSM) proposed by Longstaff and Schwartz (2001) with more exercise dates. 3.1 European options three or four assets As a first example, we evaluate European options on the maximum, the minimum, and on the (arithmetic) average of three asset values. For simplicity, the assets do not pay any dividends. The parameters of the valuation problem are: initial values, X i (0) = 100; volatilities, σ i = 0.2, for i = 1, 2, 3; correlations, ρ ij = 0.5 i j, i, j = 1, 2, 3; risk-free rate r = 0.1; maturity, T = 1; exercise price, K = 100. Table 1 shows the estimates of option values given in Boyle et al. (1989) (improved by RE with n = 40, 80, 120, 160) and the relative absolute errors with respect to the accurate value. Note that the numerical option-price estimates obtained by the various binomial lattice approaches (BEG, NEK and LT 23 ) converge as the number of steps grows. In terms of accuracy, the proposed approach is generally more accurate than BEG s scheme. For the options on the max or min, LT has the same accuracy for as few as 20 time steps as BEG s approach for more steps. 24 Generally, the improved (uncorrelated) binomial model (LT2) has the same rate of convergence as Ekvall s NEK model. Moreover, as shown in Table 2, with a four-point Richardson extrapolation the NEK algorithm may provide severely flawed numerical estimates because it does not monotonically converge to the true value. This problematic behaviour is not observed in the improved binomial approach (LT2) due to a monotonic convergence. Table 3 replicates Ekvall (1996, Table 2) with European call and put options on the maximum or minimum of four assets. This case is interesting because analytic solutions are available (see Johnson (1987) and the exten- 19

20 sion provided in Ekvall (1996, Appendix B). The parameters of the problem are: initial value, X i (0) = 10 for i = 1,..., 4; risk-free rate r = 0.1; maturity, T = 1; exercise price, K = 10, with volatilities and correlations within a given range. Again, the LT (uncorrelated) algorithm is as accurate as the NEK approach. Figures 2A and 2B show the convergence rate of the various binomial lattice approaches for (European and American) options on the max of two assets, and European options on the maximum, or minimum of three assets. The convergence rate of LT2 is faster than the BEG and LT1 methods and is generally roughly the same as for NEK. The main difference between these two approaches is that the NEK approach presents an oscillating pattern, which is problematic when implementing Richardson Extrapolation, as discussed above. 3.2 Best of alternative product standards with four risky assets The second application is from real options and involves the choice of the best of two product standards under uncertainty. The basic problem is discussed in Lint and Pennings (2001) and has been extended by Martzoukos and Trigeorgis (1999) to more sources of uncertainty. As a byproduct, the example permits to see the influence of various parameter values on the accuracy of the proposed numerical method. Philips Electronics was considering the development of two product standards (digital vs analogic) for the VCR within a given time horizon. The standard that finally would prevail was uncertain. By investing in both technologies, the firm acquired an option on the best of the two alternatives (product standards). Each asset is the market value of the resulting cash 20

21 flows if that standard is implemented. The cost of introducing each standard at the final horizon (i.e., the exercise price of that option) is uncertain. The four assets (two product standards and their costs) are correlated. Frm the perspective of the date the decision was made, the problem is to evaluate an American option on the best of the two uncertain technologies with payoff max{v 1 C 1, V 2 C 2, 0}, where V i is the market value of i-th risky asset (i.e., the value of cash flows obtained by product standard i), and C i is the cost to introduce standard i, i = 1, 2. These variables are assumed to follow correlated geometric Brownian processes. The base-case parameters are: initial asset values, V i (0) = 100; initial cost values, C i (0) = 100; dividend yields, δ i = 0.1; volatilities, σ i = 0.2, for i = 1,..., 4; correlations, ρ ij = 0.5, i j, i, j = 1,..., 4; risk-free r = 0.07; maturity, T = 2. Besides the valuation results for the above base-case (see Table 4A), we present sensitivity results with respect to the impact of higher volatility, lower correlation, longer option maturity, different investment scale on the option value, and for different choices of parameters. The results again show that the LT (uncorrelated) cheme, for the same number of steps, is more accurate than BEG and comparable with NEK. 25 To compare numerical results with exact (analytic) solutions, we consider also the case of equal non-stochastic development costs for both technologies (C 1 = C = C 2 ). If dividend yields are also zero (δ V1 = 0 = δ V2 ), we obtain a European 26 option on the maximum of two risky assets, as in Stulz (1982); if dividend yields are not zero, an extension of Stulz formula for the European option on the maximum of two assets is given in Martzoukos and Trigeorgis (1999). 27 Again, the proposed model (LT2) and NEK are the most efficient in all cases. The accuracy of LT and NEK with 12 or 24 steps is comparable to the 21

22 one given by BEG with a much higher number of steps. The usual remarks apply as far as RE is concerned American options on five assets Broadie and Glasserman (1997) (BG, thereafter) provide estimates of an American option on the maximum of five assets. In the same article (Tables 5 and 6), they provide also confidence bounds for the option price using simulated trees with a small number of allowed early exercise dates (4 dates) and a large number (50) of branches. Although they present accurate estimates with two assets (based on Kamrad and Ritchken (1991)), in the five asset case there is no such benchmark and hence no assessment of the relative error is available. 29 A main limitation in BG s valuation is that it provides downward biased estimates of the American option, since they compute sub-optimal values with restricted early exercise dates. 30 To provide a more appropriate benchmark for assessing the accuracy of our improved binomial model, we first present numerical results for the two-dimensional case in Broadie and Glasserman (1997, Table 3) based on the Least-Squares Monte Carlo simulation 31 approach (LSM) proposed by Longstaff and Schwartz (2001). We compare our results to both the results provided by LSM and the confidence bounds given by Broadie and Glasserman to establish the accuracy of our method. We then present the five-dimensional case where there are no known accurate results, comparing our results to the estimates provided by LSM and NEK. Finally, we show that the proposed model provides accurate estimates for the American option price on five assets with a fairly small number of steps. The five-asset case parameters are: initial asset values, X i (0) = 100; dividend yields, δ i = 0.1; volatilities, σ i = 0.2, for i = 1,..., 5; correlations, ρ ij = 0.3, 22

23 i j, i, j = 1,..., 5; risk-free rate, r = 0.05; maturity, T = 1; exercise price, K = 100. The option payoff is max{x 1 K,..., X 5 K, 0}. In Table 5A, with two assets, the lattice algorithms give results within the BG confidence bounds. 32 LSM simulation also gives results within these bounds, though also downward biased. Generally, the option estimates provided by (uncorrelated) LT are close to the values given by BEG and NEK, though the LT algorithm converges faster (as already observed in Figure 2A). For the five-asset case (Table 5B), the results obtained by the three lattice algorithms are almost the same, though sometimes (e.g., S 0 = 130) are outside the BG confidence bounds. Figure 3 presents the convergence rates for the three binomial lattice methods. The values obtained by these lattice methods are always closer to the high estimator than to the low estimator in BG s bounds. LT provides option estimates very close to the most accurate values with very few steps. 33 As in previous cases, NEK exhibits oscillating patterns. The LSM simulation provides downward-biased estimates within BG s confidence bounds. In terms of computational cost, the LT estimates we provide require less computational time than the LSM. 34 Though BG confidence bounds can be obtained relatively fast, BG s algorithm must be applied with a higher number of exercise dates to obtain comparable results and make efficiency comparisons on equal footing. 4 Conclusions In this paper we propose an improved binomial lattice approach for valuing contingent claims dependent on multi-dimensional correlated geometric Brownian processes. We relied on two basic ideas: a change of time scale (measuring time in units of volatility), and a change of the basis of the asset span to approximate uncorrelated geometric Brownian motions. This 23

24 approach proves to be consistent, stable and efficient, and is suitable for applying Richardson Extrapolation. We provided an extensive set of numerical applications, from both financial as well as real options pricing problems, in order to compare our results to those of other binomial lattice approaches (BEG and NEK) and, where available, to analytic solutions. The applications covered European and American options on three, four, and on five stochastic assets. Because no lattice benchmark is available for options on five assets, we compare our results to Broadie and Glasserman (1997) simulated tree algorithm and to Longstaff and Schwartz (2001) Least-Squares Monte Carlo simulation. In all cases our approach proves to be at least as efficient as the best of the other methods (and in some cases it dominates the other approaches). Unlike NEK (and other approaches) our results converge monotonically. This appealing feature makes it fruitful to employ Richardson extrapolation to improve accuracy, avoiding unnecessary computing. 24

25 References Amin, K. and Khanna, A. (1994). Convergence of American option values from discrete- to continuous-time financial models. Mathematical Finance, 4: Barone-Adesi, G. and Whaley, R. (1987). Efficient analytic approximation of American option values. Journal of Finance, 42: Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81: Boyle, P. P. (1977). Options: a Monte Carlo approach. Journal of Financial Economics, 4: Boyle, P. P. (1988). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 23:1 12. Boyle, P. P., Evnine, J., and Gibbs, S. (1989). Numerical evaluation of multivariate contingent claims. Review of Financial Studies, 2: Brekke, K. A. and Schieldrop, B. (2000). Investment in flexible technologies under uncertainty. In Brennan, M. J. and Trigeorgis, L., editors, Project Flexibility, Agency, and Competition, pages 34 49, New York, NY. Oxford University Press. Brennan, M. and Schwartz, E. (1977). The valuation of American put options. Journal of Finance, 32: Broadie, M. and Detemple, J. (1996). American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies, 9:

26 Broadie, M. and Glasserman, P. (1997). Pricing American-style securities using simulation. Journal of Economic Dynamics and Control, 21: Chen, R., Chung, S., and Yang, T. T. (2002). Option pricing in a multiasset, complete market economy. Journal of Financial and Quantitative Analysis, 37(4): Constantinides, G. M. (1978). Market risk adjustment in project valuation. Journal of Finance, 33: Cortazar, G. and Schwartz, E. S. (1994). The valuation of commodity contingent claims. Journal of Derivatives, 1: Cortazar, G., Schwartz, E. S., and Salinas, M. (1998). Evaluating environmental investments: A real options approach. Management Science, 44: Cox, J., Ingersoll, J. E., and Ross, A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 53: Cox, J., Ross, A., and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7: Ekvall, N. (1996). A lattice approach for pricing of multivariate contingent claims. European Journal of Operational Research, 91: Geltner, D., Riddiough, T., and Stojanovic, S. (1995). Insight on the effect of land use choice: The perpetual option on the best of two underlying assets. Technical report, Massachussets Institute of Technology. Harrison, J. and Kreps, D. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20:

27 He, H. (1990). Convergence from discrete- to continuous-time contingent claims prices. Review of Financial Studies, 3(4): Ho, T. S., Stapleton, R. C., and Subrahmanyan, M. G. (1995). Multivariate binomial approximations for asset prices with nonstationary variance and covariance characteristics. Review of Financial Studies, 8: Hodder, J. E. and Triantis, A. J. (1990). Valuing flexibility as a complex option. Journal of Finance, 45: Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance, 42: Hull, J. and White, A. (1988). The use of the control variate technique in option pricing. Journal of Financial and Quantitative Analysis, 23: Jarrow, R. and Rudd, A. (1983). Option Pricing. Irwin, Homewood, IL. Johnson, H. (1987). Options on the maximum or the minimum of several assets. Journal of Financial and Quantitative Analysis, 22: Kamrad, B. and Ritchken, P. (1991). Multinomial approximating models for options with k state variables. Management Science, 37: Lint, O. and Pennings, E. (2001). An options approach to the new product development process: a case study at Philips Electronics. R & D Management, 31(2): Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-squares approach. Review of Financial Studies, 14:

28 Madam, D. B., Milne, F., and Shefrin, H. (1989). The multinomial option pricing model and its Brownian and Poisson limits. Review of Financial Studies, 2(2): Martzoukos, S. H. and Trigeorgis, L. (1999). General multi-stage capital investment problems with multiple uncertainties. Working paper, University of Cyprus. McDonald, R. and Siegel, D. (1984). Option pricing when the underlying asset earns a below-equilibrium rate of return: a note. Journal of Finance, 39: Nelson, D. B. and Ramaswamy, K. (1990). Simple binomial processes as diffusion approximations in financial models. Review of Financial Studies, 3: Rendleman, R. and Bartter, B. (1979). Two-states option pricing. Journal of Finance, 34: Schwartz, E. S. (1982). The pricing of commodity linked bonds. Journal of Finance, 37: Stulz, R. M. (1982). Options on the minimum or the maximum of two risky assets: Analysis and applications. Journal of Financial Economics, 10: Trigeorgis, L. (1991). A log-transformed binomial numerical analysis method for valuing complex multi-option investments. Journal of Financial and Quantitative Analysis, 26:

29 Notes 1 On the financial side, option pricing models for contracts on several underlying asset values have been presented by Stulz (1982), Johnson (1987), Boyle (1988), and Boyle et al. (1989) regarding options on the maximum or the minimum of several asset values and by Hull and White (1987), Schwartz (1982)) concerning problems with more than one state variable. Multi-factor real options have been studied by Hodder and Triantis (1990), Cortazar and Schwartz (1994), Geltner et al. (1995), Cortazar et al. (1998), Martzoukos and Trigeorgis (1999), Brekke and Schieldrop (2000), and others. 2 For a thorough comparison of lattice approaches in the one-dimensional case, see Broadie and Detemple (1996). They show that, for one-dimensional problems, trinomial methods are slightly more efficient than traditional binomial methods. 3 A related but different branch of research is the one dealing with the complete markets property of the CRR approach. Madam et al. (1989), He (1990) and Chen et al. (2002) extend this property to multi-dimensional option problems. The main contribution of these extensions is to provide economically satisfactory solutions to such option pricing problems. With respect to this branch of research, our contribution is focused on increasing the efficiency of the binomial lattice method. 4 As shown in Boyle et al. (1989) and in Ekvall (1996), this is equivalent to matching the vector of means and the variance-covariance matrix of the continuous distribution with the corresponding moments of the discrete distribution. This is the actual approximation criterion we employ in what 29

30 follows. 5 The symbol denotes matrix transposition. 6 For the sake of brevity, we just assume that the conditions ensuring the existence of the EMM hold. For a reference, see for instance Constantinides (1978), Harrison and Kreps (1979), Cox et al. (1985). 7 If the risk premium, RP, is known, α = g RP, where g is the actual growth rate. If the asset is a traded security (or commodity) with a proportional dividend (or, convenience) yield, δ, and r is the risk-free interest rate, α = r δ. If the asset earns a below-equilibrium rate of return, α = r δ, where δ is the rate of return shortfall (see McDonald and Siegel (1984)). 8 Exceptions are available when the contingent claim is of European type. In Section 3, we will present and use some of them as a benchmark to test the accuracy of the numerical solutions found using our lattice approach. 9 The idea of approximating the log- of the asset value, instead of the asset value itself when it follows a GBM was first introduced in Cox et al. (1979) and then extended to a more general setting by Nelson and Ramaswamy (1990). The main advantage is that doing so, the binomial tree is simple, i.e. recombining. Once the binomial lattice for the log-value is obtained, the value of the asset can be computed by inverting (i.e, by taking the exponential of) the logarithm. 10 As noted in Trigeorgis (1991), this is an improvement over the CRR approach in Cox et al. (1979). If the volatility or the number of steps used are small relative to the drift parameter the CRR approach may become unstable because the probability (and the variance) can turn negative. 30

31 11 Trigeorgis (1991) presents an extensive numerical comparison with other methods for the American put option for a wide choice of parameter values. The log-transformed method is generally shown to be more accurate and more efficient than other methods, providing estimate errors within 1% of the true value with much less time steps. Following Barone-Adesi and Whaley (1987), he considers the value obtained by a finite difference approximation of the Black and Scholes pde for the American put as the benchmark. Trigeorgis (1991) shows that the log-transformed binomial lattice algorithm is as accurate with n = 50 time steps as the CRR algorithm with n = 500 steps. 12 The parameters in (5) and (6) are found as a solution of the system of equations (7), (8) and (9). 13 Financial markets are complete if there are as many non-perfectly correlated traded assets as there are sources of uncertainty. In complete markets any risk can be hedged with a suitable portfolio strategy, and hence a contingent claim can be valued by replication if there there cannot be lasting arbitrage opportunities. In complete markets there is only one EMM. The contingent claim can be valued using risk-neutral valuation by taking expectation of its terminal payoff with respect to the EMM. The valuation remains valid in the more general case of valuing a contingent claim which is redundant with respect to the asset span, according to the EMM. 14 The market spanned by the synthetic assets is the same as the one where the asset span is generated by the primitive securities. The only thing that changes is the representation of returns. 15 With the usual rules: dtdz i = 0, (dt) 2 = 0, dz i dz j = ρ ij dt. 31

32 16 Given the square matrix Ω, the columns of W are the eigenvectors and the diagonal of Λ gives the eigenvalues of Ω, that is ΩW = ΛW. The matrix W represents a change of basis in R N and, from an economic viewpoint, it is a set of N portfolios of the original assets having uncorrelated returns. 17 Equation (10) can be derived as follows F (Y ) = = e rt Π(Y T )ν Y (dy T ) = e rt Π(W y T )ν Y (W dy T ) e rt Π(yT )ν y (dy T ) = F (y) where ν y (dy) = ν Y (W dy T ) and integration is over the support of Y (t) and y T, respectively. 18 The argument is similar to the one used in Amin and Khanna (1994) for the BEG algorithm. 19 In the two-dimensional case, it can be shown that the probabilities of the improved binomial approximation are positive and less than one for any parameter values. To see this, observe that Equation (13) can be written also as p(s) = (1 + δ 1 (s)l 1 ) (1 + δ 2 (s)l 2 ) /4. Because L i < 1 for all i, from Equation (12), 0 < p(s) < 1 for all s = 1,..., 4. The fact that the probability of the improved binomial approach is always positive makes the method unconditionally stable. Numerical experiments confirm that the probabilities are positive (and the algorithm stable) for a wider choice of parameters than the BEG. 20 The implementation procedure is similar to the BEG algorithm (see Boyle et al. (1989)) and other proposed binomial lattice algorithms. 21 As suggested in Trigeorgis (1991), in order to make the tree recombining, 32

33 the value of the option at the cum-dividend nodes (i.e., just before the exdividend date) can be found by interpolating the value obtained at the ex-dividend nodes. 22 The model can be applied also if there is a trigger event for the contingent claim (like a barrier for barrier options) as long as the trigger can be written in terms of the state variables. Since the applied transformations are one-to-one, if the trigger is X = ( X 1,..., X N ), we simply check it in the transformed space by checking the trigger ȳ = W Ȳ, where Ȳi = log X i. 23 LT stands for the improved multi-dimensional log-transformed binomial lattice model (Section 1). For illustration purposes, we provide also the results obtained with the first model, based on Equations (5) and (6). We will denote by LT1 the results obtained with this first approximation, and by LT2 the results with the improved (uncorrelated) log-transformed extension from Equations (12) and (16). 24 The only exception is the case of the put option on the average. 25 The numerical results for variations on the base case are available from the authors. When correlations ρ ij = 0 for all i j, the results obtained with LT1 and LT2 are the same. 26 If both dividend yields are zero, the American and the European option results are the same. 27 The European option C max on the maximum of two risky assets with cost C, δ Vi = δ i 0, σ Vi = σ i, i = 1, 2 is: C max = C BS (V 1, C, δ 1, r, T )+C BS (V 2, C, δ 2, r, T ) C min (V 1, V 2, C, δ 1, δ 2, r, T ) (17) 33

34 where C BS is the value of the European call option according to Black and Scholes formula, and C min is the value of the European option on the minimum of two assets: C min = V 2 e δ2t ( N [d 1 + σ 2 T, log(v1 /V 2 ) + (δ 2 δ 1 σ 2 /2)T ) /(σ ] T ), ρ 1 + V 1 e δ1t ( N [d 2 + σ 1 T, log(v2 /V 1 ) + (δ 1 δ 2 σ 2 /2)T ) /(σ ] T ), ρ 2 Ce rt N [d 1, d 2, ρ 12 ] where N [z 1, z 2, ρ] is the bivariate cumulative Normal distribution integrated up to z 1 and z 2 for two variables with correlation ρ; ρ 1 = (ρ 12 σ 1 σ 2 ) /σ; ρ 2 = (ρ 12 σ 2 σ 1 ) /σ; σ = ( σ σ 2 2 2ρ 12 σ 1 σ 2 ) 1/2, d i = [ log(v i /C) + (r δ i σ 2 i /2)T ] /(σ i T ), i = 1, The same behaviour observed in Figures 2A and 2B is observed also in this case. Hence, the plots are not included but are available from the authors. 29 In Broadie and Glasserman (1997, page 1339) they note that relative errors are not reported because the true value is unknown. [... ] With k = 5 [i.e., with five assets] the computations are prohibitive for n as small as 50. And even if the computations could be done, the resulting value would not be very accurate. Later, in their Conclusions, they suggest using parallel machines or networks of workstations to evaluate truly American options. 30 In fact, their computed confidence range is actually lower than the true confidence range for an American option. This is apparent even (especially) 34