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Leasing Contract document sample
Estimation of Bandwidth Leasing Contract Prices Kai Arte EigenValue Ltd., Salomonkatu 17 A 11, FIN-00100 Helsinki, Finland; E-mail: kai.arte@eigenvalue.com Jussi Keppo Department of Industrial and Operations Engineering, University of Michigan, 1205 Beal Avenue, Ann Arbor, Michigan 48109-2117, USA; E-mail: keppo@umich.edu Abstract In this paper we present a parameter estimation method for a bandwidth leasing contract model. Leasing contracts are modeled by using underlying fixed routing capacity prices and network’s routing options. Therefore, the contract prices are non- linear functions of the fixed routing prices and the fixed routing volatilities affect the leasing contract prices. The estimation of the model parameters is based on the existing price data and we compute the implied parameters of the model. According to the presented estimation method we can calculate the exact price curve of a connection if we know the curves of the surrounding connections, because those price curves contain sufficient information on the estimated curve. This is important in practice, because telecommunication companies’ price data is usually scarce and, therefore, direct modeling of the price curves is difficult. Key words: Bandwidth, forward contract, estimation, network arbitrage. 1. Introduction Bandwidth as a commodity is unique. It has several properties familiar with electricity, the most important common property being non-storability [see Kenyon and Cheliotis (2001) for a discussion about the properties of bandwidth]. Further, the network structure and telecommunications companies’ optimal point-to-point routing selection introduce a new arbitrage condition called geographical or network arbitrage condition [see e.g. Chiu and Crametz (1999, 2000) and Reiman and Sweldens (2001)]. The network arbitrage condition states that the optimal price of any point-to-point capacity must be the minimum price over all possible routings. According to Keppo (2001) the network arbitrage condition can be considered as an option for the seller. The basic 1 contracts in the bandwidth markets are leasing contracts that are actually forwards [see e.g. Hull (2002) for the definition of forward contract] and they specify the underlying point-to-point connection and the price that is used in the future. The seller of a bandwidth leasing contract on a single connection is maximizing its revenues and, therefore, provides the cheapest routing, which might not be the direct one. Because telecommunications companies can seamlessly change the traffic routings of their networks, the methods of this paper can be used by utilizing their own price data. The bandwidth pricing model of Keppo (2001) considers a three-node network and derives analytical formulas for bandwidth leasing contracts. The leasing contract prices are expressed in terms of underlying fixed routing bandwidth prices that can be understood as the risk factors of the network. A simple extension to larger networks is considered in this paper. Similar risk factor based pricing and hedging have been done, e.g., in electricity and weather derivative markets, where the underlying asset is not a traded financial instrument [see e.g. Davis (2001) and Manoliu and Tompaidis (2000)]. The network arbitrage condition implies that network price curves (as a function of maturity) and processes are not simple. The leasing contract prices are nonlinear instruments of the underlying fixed routing prices and, therefore, their process parameters are not constant. The parameter estimation of the leasing contract processes is difficult – the problem is similar to the modeling of a regular stock option’s price process from the option’s historical data. This estimation of the stock option’s process is complicated, because the option’s expected drift and volatility depends nonlinearly on the maturity and on the difference between the stock and strike prices. Similarly, with a bandwidth leasing contract the drift and volatility depends nonlinearly on the maturity and the difference between the underlying fixed routing prices. These problems in the bandwidth price modeling give us the reason to identify the processes of the fixed routing prices that are independent of routing options and, therefore, they can be assumed to have constant process parameters. In this paper we present a parameter estimation scheme for Keppo’s (2001) bandwidth pricing model. We estimate the model parameters of all the connections at once by using the data from all underlying connections. This method is similar to fitting a parameterized yield curve, e.g., Nelson-Siegel model (1987) to market data. Our model is important in practice because telecommunications companies’ price data is usually 2 scarce and, therefore, direct modelling of price curves is difficult. We illustrate this estimation method by using a three-node network and real market quotes. The remainder of this paper is organized as follows. Section 2 of this paper summarizes the framework of Keppo (2001). Section 3 illustrates the estimation method with a numerical example. Section 4 outlines the expansion to general network structures and Section 5 concludes. 2. Framework In Keppo (2001) pricing theory for bandwidth instruments under the network arbitrage condition is presented. The model is developed in a three-node network, where all nodes are interconnected. In this section we shortly present the model and discuss its implications. The key idea is to use fixed-routing point-to-point capacity prices as the underlying assets of the bandwidth instruments and then the network arbitrage condition is modeled as a characteristic of these instruments. That is, at the delivery date of a derivative instrument the underlying point-to-point connection’s market price is equal to the minimum price over all possible routings. We denote the fixed (or direct) routing prices by Si for all i ∈ {1,2, 3} and the market prices by Yi, respectively. The relationship between these prices is given by Yi (t ) = min [Si (t ), X i (t )] , (2.1) where Yi(t), Si(t), and X i (t ) = ∑ k ∈{1,2,3}−{i } Sk are the i’th connection’s spot, direct routing, and alternative routing prices at time t. For instance, if i = 1 then the first connection’s alternative routing price X1(t ) = S 2 (t ) + S 3 (t ) . Equation (2.1) implies that the spot prices satisfy the network arbitrage condition. Bandwidth leasing contract is a contract where the buyer and seller have agreed upon the terms of delivery and price. These terms include delivery period or duration, geographic location (source and destination points), capacity, and quality. However, even though the terms have been agreed upon, the seller still holds the option of choosing the routing between the source and the destination points as is indicated in (2.1). This optionality can be seen as an exchange option and it leads to the network arbitrage condition. 3 For simplicity, we assume that the leasing contract prices are equal to the expected spot prices, i.e., we assume that the expectation hypothesis holds. Thus, we have Yi (t,T ) = E Yi (T ) Ft for all i ∈ {1,2, 3}, t ∈ [0,T ] , (2.2) where Yi(t,T) denotes the i’th connection’s T-maturity price at time t and Yi(⋅,T) is adapted to the filtration {Ft }t ≥0 , Ft is the σ-algebra at time t (i.e. entire information set available at time t), and the expectation is taken with respect to the objective measure P because, for simplicity, we assume that it equals the pricing measure Q. Because the spot prices satisfy the network arbitrage condition, equation (2.2) implies that also the leasing contracts satisfy the condition. In order to calculate analytical leasing contract prices we assume that the S-processes of equation (2.1) are driven by Brownian motions but could equally well be expanded to include jumps1. In this paper the fixed routing price process is given as follows dSi (t,T ) = Si (t,T ) σidBi (t ) for all i ∈ {1,2, 3} (2.3) where Si (t,T ) = E Si (T ) Ft = Si (t )exp (αi (T − t )) , αi and σi are constant. Thus, dSi (t,T ) is the change of the expected fixed routing price. This implies that lognormal distribution and geometric Brownian motion are assumed. This guarantees non-negative prices and in Keppo (2001) more general processes are used. Note that since according to (2.1) and (2.2) all the S-processes affect the leasing contract prices, the network’s leasing contract prices would correlate with each other even though the S-processes were uncorrelated. Setting i = 1 and calculating the expectation in (2.2) by using (2.3) and (2.1) gives [see Keppo (2001) for details] Y1 (t,T ) = S1 (t,T ) Φ (−z 1 (t,T )) + X1 (t,T ) Φ (z 1 (t,T ) − σY1 T − t ) , (2.4) where Φ denotes the cumulative standard normal distribution, X1(t,T) = S2(t,T) + 2 2 S3(t,T), σY1 = σ1 + σX − 2ρS ,X σ1σX , σ1 is the volatility of S1, 2 2 2 2 σX = ω1,2σ2 + ω1,3σ3 + 2ω1,2 ω1,3ρ2,3σ2σ3 is the volatility of X1(t,T), ρS ,X = ω1,2 ρ1,2σ2 + ω1,3 ρ1,3σ3 is the correlation between S1 and X1, ρi, j is the correlation 1 σX between Si (t ) and S j (t ) , and z 1 (t,T ) = ln ( X11((tt,,T )) ) + ½σY1 (T − t ) (σY1 T − t ) . S 2 T 1 Then the derivative pricing formulas could be solved by using the Fourier transform method [see e.g. Duffie, Pan, and Singleton (1999)] and, therefore, they would differ from Keppo (2001). 4 Equation (2.4) gives the first leasing contract price in terms of the underlying S- processes. The other connections are calculated in the same way. The first part on the right hand side of equation (2.4) corresponds to the price under S1(T) ≤ X1(T). In this case at maturity Y1(T) = S1(T) and, according to the network arbitrage, the direct routing is used. The latter part on the right hand side of equation (2.4) corresponds to the price with S1(T) > X1(T). In this case the alternative routing is used at time T. In equation (2.4), Y1(t,T) is the price of T-maturity contract with instantaneous duration, because the underlying asset is the spot price. For extension to general duration see Keppo (2001). There are a few interesting implications to be pointed out. As can be seen from equation (2.4) the bandwidth leasing contract price is affected by the volatility of S and X. In this sense it does behave like an option on a stock. Furthermore, the possibility of alternative routing should truncate the upper tail of the price distribution. Equation (2.4) implies that the parameters of a leasing contract’s price process are not stable because the contract includes the routing option. That is, the price’s expected drift and volatility depend on the S-prices and on the maturity T-t, which change all the time. The underlying S-processes are independent of these routing options and, therefore, their process parameters can be assumed to be constant. This gives the fact that the estimation of the S-process parameters is easier than the estimation of the Y-process parameters. These S-process parameters can be calculated from the observed leasing contract prices because, according to equation (2.4), the price curve contains information on the expected values and volatilities of the underlying S-prices. Thus, this is analogous to computing the implied volatility from option prices. 3. Estimation We consider an example triangle network that consists of three interconnected nodes. The selected nodes are Chicago, Dallas, and New York. The capacity of the connections is DS-3 (44.736Mbps). The basis of the price curves is in actual sell offers and, therefore, they are indications for the market prices. The prices are monthly recurring payments (MRP) (Source: TFS Telecom, a division of TFS Energy L.L.C, September 4, 2001). The data points and the corresponding estimated price curves are drawn in Figure 1. For Y1 (Chicago – New York) the data points are represented by squares and the model 5 estimate by solid line, for Y2 (Chicago – Dallas) by rhombs and dashed line, and for Y3 (Dallas – New York) by triangles and broken line, respectively. The estimation method is illustrated in Appendix. We estimate all the price curves by using the data from Chicago – New York and Chicago – Dallas connections. Price ($) Price ($) 7000 SY, ,S2 1 1 Y2 7 000 S3 3 Y 6000 6 000 5000 5 000 4000 4 000 3000 3 000 2000 2 000 1000 1 000 0 0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 Time to maturity (years) Time to maturity (years) Figure 1. The data points and the model-estimated price curves. Y1 (Chicago – New York, squares and solid line), Y2 (Chicago – Dallas, rhombs and dashed line), and Y3 (Dallas – New York, triangles and broken line). From Figure 1 we see that the model curves capture the major features of the data and the network arbitrage condition holds for every instant. Our pricing model can be seen as a parameterization for network price curves ala Nelson-Siegel model (1987) in interest rate markets. In our model, the parameterization depends on the used S- processes and in this paper geometric Brownian motion processes were employed. As can be seen from Table 1 the model tries to fit the data with remarkable drifts and volatilities. The correlation matrix is non-singular and suggests that our model has identified three sources of uncertainty. Table 1. The parameter estimates of connections S1 (direct routing price for Chicago – New York), S2 (direct routing price for Dallas – New York), and S3 (direct routing price for Chicago - Dallas). S1(0) S2(0) S3(0) α1 α2 α3 σ1 σ2 σ3 1,2 1,3 2,3 0.866 0.000 0.228 2,963 9,167 7,233 3.465 4.444 1.789 6.185 0.6989 0.7047 7 1 6 6 Note that in the estimation we used price data from two first connections and we were able to solve the price curve of the third connection. Thus, our example implies that those two price curves contain enough information on the third price curve. Earlier this kind of estimation problem has been solved by using a cheapest path algorithm (i.e. a shortest path algorithm), which generally gives only the lower and upper boundaries for the third price curve. Thus, the shortest path algorithms are not able to identify the unique price curve. By using our network model, we can create a leasing contract on the third connection by using the contracts of the two first connections. Further, this leasing contract price is inside the lower and upper prices implied by the cheapest path algorithm. The model parameters can also be estimated using data from a single connection and equation (2.4). Effectively this means modeling the alternative routing with a single uncertainty and estimating the parameters of the direct and alternative routing processes (S- and X-processes, respectively) directly from the observations of that connection. Naturally, in this case the number of estimated parameters is smaller than in our above example. 4. General Network Structure In this section we discuss an extension of the three connections’ estimation scheme to general network structures with a simple example. We consider the four-node network illustrated in Figure 2. We assume that there exist price data on five connections. Further, we assume that there do not yet exist contracts on the last connection, connection 6, and, hence, no price observations are available on that connection. S5 S4 S1 S6 S2 S3 Figure 2. The four-node network and the fixed routing network prices 7 We identify two triangles formed by connections {1, 2, and 3} and {1, 4, and 5}. We may estimate the parameters of the underlying fixed-routing price Si for all i ∈ {1,2,..., 5} . Thus, the network arbitrage condition holds in each of these sub networks. However, we may end up with more than one estimate for the process parameters of S1, which can be understood as a modeling error. That is, if the network arbitrage condition holds and if we have the correct stochastic processes for the fixed and alternative routing prices the modeling error should be zero. In practice, the existence of the modeling error is a common situation with a general network structure, and in this case we use a weighted average of the different parameter estimates to model the S1-process. Furthermore, we can produce an estimate for the fixed-routing price process S6, which is not readily observed. Using the estimation scheme for the triangles {2, 5, and 6} and {3, 4, and 6} with the price data of connections 2, 3, 4 and 5 we end up with two estimates for the S6’s process parameters. We select a consensus of these two estimates by averaging them. This effectively gives us an estimate of the price process of S6, i.e., the implied parameters (S 6 (0), α6 , σ6 , and ρi,6 , i ∈ {2, 3, 4, 5}) . In a similar manner, we can estimate the process parameters of the alternative routing of connection 6, X6. By using the estimates of S6 and X6 we can price any contract on this connection and this pricing considers the network effects. 5. Conclusions We derived a method for estimating bandwidth leasing contract prices. The model can be used to price bandwidth instruments on previously untraded connections if we have price data from two neighboring connections and if the connections form a three-node network. We illustrated this in our example, where the whole price curve of an untraded connection was estimated. Earlier this kind of estimation problem has been solved using a cheapest path algorithm, which generally gives only the lower and upper boundaries for an untraded price curve. The basic ideas of the estimation methods can be extended to consider general network structures. In this case the estimation simplifies to estimating the process parameters in several three-node sub networks, thus again utilizing the estimation method for three-node networks. 8 Appendix We denote the observed leasing contract prices with maturity T at time t by Fi(t,T). The corresponding model predicted price is denoted by Yi(t,T). Given the parameters that define the risk factor processes Si we can calculate the leasing contract prices Yi(t,T;θ) for all feasible T, where θ is the parameter vector that depends on the selected stochastic processes for the direct routing and the alternative routing. We choose the parameters so that the predicted contract prices fit the observed prices as well as possible. Thus, we formulate the objective function as follows J (θ ) = ∑ Yi ( t , T;θ ) − Fi ( t , T ) , 2 i where T = [T1 … TN], Fi(t,T) = [Fi(t,T1) … Fi(t,TN)], Yi(t,T;θ) = [Yi(t,T1;θ) … Yi(t,TN;θ)], θ ∈ Θ is the parameter vector for the S-processes and their correlations, and Θ is the space of feasible parameter values. Formally our minimization problem is minJ (θ ) subject to θ ∈ Θ . θ We use the second norm in the objective function, because the least squares estimation is widely studied and there exist several algorithms for solving such problems. The objective function is nonlinear and nonconvex with respect to the parameter vector . This can create numerical instability and solving difficulties. Theory and algorithms for nonlinear least squares estimation can be found, e.g., in Marquardt (1963). The global optimization of nonlinear least squares problems is studied, for instance, in Bliek et al. (2001). We use a geometric Brownian motion assumption for Si(·) and Xi(·) in the examples of this paper. Given the geometric Brownian motion assumption we need to estimate only three parameters for each process and their correlations. These parameters are the current states, the drift terms and the volatilities. Acknowledgements The authors are grateful to Janne Lassila, Erkka Näsäkkälä, Michael G. Sullivan, and Chris Kenyon for valuable comments on this paper. 9 References Bliek, C., P. Spellucci, L.N. Vicente, A. Neumaier, L. Granvilliers, E. Monfroy, F. Benhamou, E. Huens, P. Van Hentenryck, D. Sam-Haroud, and B. 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