Pricing of point-to-point bandwidth contracts

Transcription

1 Math Meth Oer Res (2005) 61: DOI /s Pricing of oint-to-oint bandwidth contracts Jussi Keo Deartment of Industrial and Oerations Engineering, University of Michigan, 1205 Beal Avenue, Ann Arbor, Michigan , USA ( Manuscrit received: December 2003/Final version received: June 2004 Abstract. In this aer we consider the ricing of oint-to-oint bandwidth leasing contracts and otions. The underlying asset of these contracts is a oint-to-oint telecommunications connection. Due to the network structure the network caacity rices deend nonlinearly on each other. A leasing contract on a oint-to-oint connection can be seen as an otion because the seller of the connection selects the cheaest ath between the oints. Therefore, a bandwidth otion is a comound otion. Key words: Bandwidth, Network, Leasing contract, Otion ricing, Real otion 1. Introduction Term bandwidth corresonds to the amount of data transferred on a given transmission ath within a secified block of time. Thus, bandwidth is a synonym for telecommunications caacity, and it is measured in units of bits er second (bs). For examle, downloading a icture in one second requires more bandwidth than downloading a text age in one second. Large sound files, comuter rograms, and videos require even more bandwidth for accetable system erformance. Managed bandwidth services are connections between two or multile oints on a certain caacity. The bandwidth seller and buyer define the start The author is grateful to conference articiants at the INFORMS International 2001 Conference for helful comments. The author also thanks Petri Aukia, Mika Karjalainen, Kai Arte, Erkka Na säkkälä, Janne Lassila, Jeffrey K. MacKie-Mason, Michael Wellman, Leonard Kofman, Bozenna Pasik-Duncan, Giorgos Cheliotis, and Chris Kenyon for useful discussions. All remaining errors are mine. This aer was reviously circulated as Pricing of bandwidth derivatives under network arbitrage condition.

2 192 J. Keo date and end date of the service contract as well as the quality of service. Quality can be measured by ermissible errored seconds, severely errored seconds, and unavailable seconds over time [see Mayfield (2000)]. At the start date the buyer buys the connection from the seller at the rice secified in the service contract, and after the end date the owner can sell the connection to the next customer. Thus, the bandwidth service contracts are actually forwards [for forward contracts see e.g. Hull (1997)]. Caacity level, connection s start and end oints, and the quality of service secify the underlying asset of the forward. The start and end dates define the maturity and duration of the forward. It is estimated that about $25 billion worth of telecommunications caacity is bought and sold worldwide annually [see Ryan (2000)]. According to Mayfield (2000) there are about 7500 market articiants in US alone and roughly double the number of that internationally. Fletcher and DiClemente (2001) have groued the bandwidth buyers into five categories: 1) large existing carriers, 2) emerging telcos, 3) service roviders, e.g. ISPs, 4) dotcoms, which oerate through the Internet, and 5) enterrise customers, which need huge amounts of bandwidth in their everyday oerations. In the bandwidth market there are also caacity roviders of several magnitudes. The ricing of bandwidth contingent claims is similar to the ricing of corresonding electricity instruments in the sense that both these commodities are held for consumtion and they cannot be stored [see e.g. Kenyon and Cheliotis (2001) for a discussion about the roerties of bandwidth]. Therefore, the bandwidth contracts cannot be hedged by using the underlying asset like usual financial derivative instruments and arbitrage argument is ossible only between different bandwidth contracts. However, in the resent aer we do not use this kind of hedging argument because the bandwidth markets are currently illiquid. In the bandwidth markets there is a new arbitrage condition due to the structure of the network and to the market s otimal oint-to-oint routing selection. The buyer of a bandwidth is buying the oint-to-oint connection indeendent of the routing. Thus, the seller can rovide an alternative routing in case the direct routing between the oints costs more at the delivery time. Therefore, the risk of the seller is in a way bounded above and the seller s otimal routing selection leads to the network arbitrage condition, also called geograhical arbitrage condition [see, e.g., Chiu and Crametz (1999, 2000) and Uton (2002)]. This condition imlies that at each time a oint-to-oint caacity market rice has to be equal to the minimum caacity rice over all ossible routings between these two oints. That is, the bandwidth market rices are obtained by using the cheaest aths between the start and end oints. This new arbitrage condition has to be considered in the ricing of bandwidth contracts and, therefore, the ricing is artly different from the ricing of corresonding electricity instruments. The urose of this aer is to calculate oint-to-oint forward (leasing contracts) and otion rices under the network arbitrage condition. Note that in the bandwidth markets there are also other secial characteristics [see e.g. Kenyon and Cheliotis (2001), Crametz (1999, 2000), and Uton (2002)] but here we focus on the network arbitrage condition. We firstly model the network rices without the network arbitrage condition, i.e., the initial situation is close to the electricity market. Therefore, we assume similar stochastic rocesses for the fixed routing rices that are used in several electricity aers [see e.g. Deng, Johnson, and Sogomonian (2001), Keo and Ra sa nen (1999,

3 Pricing of oint-to-oint bandwidth contracts ), Manoliu and Tomaidis (2000), and Oren (2001)]. Because the underlying bandwidth is non-storable, the forward rices are functions of the bandwidth market rice of risk. Thus, unlike in a storable commodity derivative market the risk attitude of bandwidth market s articiants affects the forward rices. We assume that all the bandwidth instrument rices are given by the same general ricing function. This imlies that the bandwidth instruments are in equilibrium with each other and they are riced with resect to the same market rice of risk, i.e., with resect to the same utility function. The risk ricing is imortant in bandwidth markets because the bandwidth rices are highly volatile [see e.g. Mayfield (2000)], according to Cass (2000) 20 40%. In the ricing function we model the network arbitrage condition as a feature of the forward contracts. That is, the bandwidth forward rices are solved by using the rocesses of fixed (straight) routing ointto-oint rices and the otion nature of the cheaest ath selection. Fixed routing bandwidth rices are used in the ricing models because their rocess arameters can be assumed to be constant and, therefore, they act as the risk factors of the bandwidth market. Further, this way we can better understand the effect of the network arbitrage condition on the bandwidth market rices. Arte and Keo (2003) illustrate arameter estimation for the resent aer s forward model and show how the framework is used in ractice. Due to the cheaest ath selection a bandwidth forward rice is equal to the forward rice with fixed routing minus the otion to change the oint-to-oint routing. This routing otion is a kind of exchange otion [for the ricing of exchange otions see e.g. Margrabe (1978)] and by using the ricing models of exchange otions we can derive analytical ricing formulas for bandwidth forwards. Therefore, we can also derive analytical formulas for the nonlinear deendencies between the market rices. According to the model, the network arbitrage condition increases ositive correlations between the caacity market rices, and the robability distributions of the market rices have short uer tails. This is natural since network routing smoothens the ointto-oint demands over the whole network and therefore rices move together [for routing see, e.g., Gune and Keo (2002)]. In addition to the cheaest ath selection, some bandwidth service contracts include also other otion tye characteristics for the arties of the contracts. For instance, the seller can have a right to disconnect the service for a redefined enalty ayment. These rights can be modeled as bandwidth otions and, therefore, the understanding of otion ricing is imortant in the bandwidth markets even though there do not yet exist traded otion contracts. Further, this hels the alication of real otion theory in telecommunications markets [see e.g. Alleman and Noam (1999)]. Bandwidth otions can be modeled as otions on bandwidth forwards, because at maturity the forward rice equals the underlying connection rice. This gives that the bandwidth otions are a kind of comound otions [for comound otions see Geske (1979) and Rubinstein (1992)] since the underlying bandwidth forward contracts can be seen as exchange otions. The otion rices are described in terms of bivariate and trivariate normal distributions because we have to integrate the forward rice function, which now includes cumulative normal distributions due to the routing otion, over the underlying oint-to-oint rice distribution and also over the alternative routing s rice distribution. This ends u with the ricing equation that includes the trivariate and bivariate robability distributions. Because the robability distribution of the underlying forward rice has a short

4 194 J. Keo uer tail, the bandwidth call otion is cheaer than the corresonding otion rice imlied by the Black-76 model [see Black (1976)] that is a usual commodity otion ricing formula. Bandwidth ricing is considered, e.g., in Kenyon and Cheliotis (2001) that considers a jum-diffusion model for bandwidth sot rices. Present aer uses only continuous uncertainties but can be extended, e.g., to the jum rocesses by using transformation analysis [for the transformation analysis see e.g. Duffie, Pan, and Singleton (2000)]. The ossible drawback of our continuous uncertainty assumtion is a reduced accuracy. However, the advantage is that we are able to obtain analytical ricing formulas that are easily imlemented to everyday industry ractice [see Arte and Keo (2003)]. These analytical results enable the ricing and analyzing of huge ortfolios within a short time eriod and this is imortant in ractice since telcos ortfolios may include thousands of different instruments. There are several other aers that have considered bandwidth ricing. Rasmusson (2002) resents a new design for bandwidth markets. Reiman and Sweldens (2001) analyze the calculation of network arbitrage free bandwidth forward rices. Uton (2002) studies methods for ricing and agent behavior in bandwidth markets by using financial models and stochastic control. Guta, Kalyanaraman, and Zhang (2003) calculate bandwidth sot rices by using a nonlinear ricing method. Courcoubetis, Kelly, Stamoulis, and Manolakis (1998) analyze bandwidth allocation and ricing model, where each user is assumed to select his/her willingness-to-ay so as to maximize his/her net benefit, i.e., the difference of the utility induced to the user by the quality of service (QoS) received minus his/her willingness-to-ay. In the resent aer we do not consider QoS and assume that the QoS is the same for all the oint-to-oint connections. Keon and Anandalingam (2003) consider ricing of multile services with guaranteed QoS levels and a single telecommunications network. Courcoubetis, Kelly, Siris, and Weber (2000) study simly usage-based charging schemes for bandwidth networks. Songhurst and Kelly (1997) consider the issues of network interaction that are inherent in aroriate usage-sensitive charging schemes. The stability and fairness of rate control algorithms for communication networks are studied in Kelly, Maulloo, and Tan (1998). Courcoubetis, Dimakis, and Reiman (2001) analyze ricing in a best effort network and they also consider otion ricing. Our otion ricing model can be seen as an extension to that since we add the network arbitrage condition to their framework. MacKie-Mason and Varian (1995a, 1995b) and Paschalidis and Tsitsiklis (2000) study congestion deendent ricing. Odlyzko (2001) discusses Internet ricing and the history of telecommunications. The financial otion ricing theory under continuous uncertainties is develoed, e.g., in Black and Scholes (1973), Merton (1973), and Black (1976), and the hedging of financial instruments is considered in many aers [see e.g. Bertsimas, Kogan, and Lo (2001), Duffie and Jackson (1990), Lioui and Poncet (1996), and Keo and Peura (1999)]. General financial contingent claims ricing theory is derived in Harrison and Kres (1979), Harrison and Pliska (1981), Kres (1981), and Cox and Huang (1986). Utility function based financial asset ricing is considered, e.g., in Davis (1998, 2001) and Cochrane (2002). Our ricing function for bandwidth contracts can be viewed as an examle of this kind of utility aroach.

5 Pricing of oint-to-oint bandwidth contracts 195 To sum u, the main objectives of our model are the following: Analytical ricing models for forwards and otions under the network arbitrage condition. Nonlinear deendencies of bandwidth rices due to the network structure. The rest of the aer is divided as follows: Section 2 introduces the bandwidth rice models used in the aer. The stochastic rocesses for the caacity rices are defined and the rocesses are then alied to the forward ricing roblem in Section 3. Section 4 uses the forward rices in the ricing of bandwidth otions. Section 5 illustrates the derived models with numerical examles and finally Section 6 concludes. 2. Model For simlicity, we consider a network of three oint-to-oint caacity rices. All the oint-to-oint connections have the same caacity level and quality. For examle, the oints are New York, Los Angeles, and Atlanta, and the caacity is OC-3 ( Mbs). The extension to more general networks is discussed in Arte and Keo (2003). Figure 1 illustrates the network structure. The S-rices of Figure 1 are the bandwidth rices by using the caacity between the oints and without the network structure, i.e., the routing is fixed. For instance, S 1 is the bandwidth rice between the u and left oints by using the direct routing between these two oints. Price S 1 does not necessary equal the market rice between the u and left oints because if we have S 2 + S 3 S 1 then the market uses the longer routing and, therefore, the market rice equals S 2 + S 3. This cheaest ath selection is called network arbitrage condition [see e.g. Uton (2002)], i.e., it can be seen as the otimal behavior of telecommunications comanies. Let us denote the market sot rices by Y 1, Y 2, and Y 3. Then we have Y 1 ðtþ ¼min½S 1 ðtþ; S 2 ðtþþs 3 ðtþš Y 2 ðtþ ¼min½S 2 ðtþ; S 1 ðtþþs 3 ðtþš for all 2½0; sš: ð2:1þ Y 3 ðtþ ¼min½S 3 ðtþ; S 1 ðtþþs 2 ðtþš Fig. 1. Network rices

6 196 J. Keo According to (2.1) sot rices are free of network arbitrage (Y 2 +Y 3 Y 1, Y 1 +Y 3 Y 2, Y 1 +Y 2 Y 3 ). Currently, in the bandwidth commodity market there does not exist ure sot rices and all the instruments in the market are derivative instruments. Therefore, in ractice the sot rice refers to the forward rice (leasing contract rice) with shortest maturity and duration. At resent, this means one-month forward rice that starts after two months. From now on, we assume that the sot rices can be aroximated with the forwards of shortest maturity and duration. From (2.1) we see that even though S 1, S 2,andS 3 were indeendent Y 1, Y 2, and Y 3 would not be, because the market rices deend on the oint-to-oint routings. Also from (2.1) we see that the robability distribution of Pfuture Y i is different from S i for all i 2f1; 2; 3g, because Y i ðtþ S j ðtþ and, therefore, the uer tail of Y i s robability j2f1;2;3gfig distribution is shorter than the S i s corresonding tail. However, all the uncertainties in the market sot rices are from the S-rocesses and, therefore, these rocesses are the risk factors of the bandwidth market. If we had more comlex network structure then in equation (2.1) we would have the minimum over all ossible routings. We consider a finite time horizon [0, s]. In describing the robabilistic structure of the market, we will refer to an underlying robability sace (W, F, P). Here W is a set, F is a r-algebra of subsets of W, and P is a robability measure on F. We assume that E½jS i ðtþjš < 1 for all t 2½0; sš and i 2f1; 2; 3g so that we can calculate the exected values and we denote the conditional exected value of the fixed routing rice S i by S i ðt; T Þ¼E½S i ðt ÞjF t Š for all t 2½0; T Š and T 2½0; sš. We model these exected rices and the following assumtion characterizes their stochastic rocess. Assumtion 2.1 The rocess of the exected fixed routing rice S i (t,t) is given by the following Itoˆ stochastic differential equation ds i ðt; T Þ¼S i ðt; T Þr i db i ðtþ for all t 2½0; T Š; i 2f1; 2; 3g; ð2:2þ where S i ðt; T Þ¼E½S i ðt ÞjF t Š; S i ð; T Þ : ½0; T Š!R þ ; T 2½0; sš; r i is bounded and constant, and B i (Æ) is a standard Brownian motion corresonding to the link i on the robability sace (W, F, P) along with the standard filtration ff t : t 2½0; sšg. We will denote by q i,j the correlation between the i th and j th Brownian motions. According to equation (2.2) the stochastic rocesses for the exected fixed routing rices follow geometric Brownian motion rocesses where S i ðt; T Þ 2 r 2 i is the rate of change of the conditional variance of S i (t,t). The boundedness of the volatility arameter r i guarantees the existence and uniqueness of the solution to (2.2). The solution to (2.2) can be written as S i ðt ; T Þ¼S i ðt Þ¼S i ðt; T Þ ex 1 2 r2 i ðt tþþr i½b i ðt ÞB i ðtþš : The extension to time deendent and deterministic volatility is straightforward since we can set r 2 i to equal the average of volatility square over the lifetime of the bandwidth contract under consideration. For instance, in the

7 Pricing of oint-to-oint bandwidth contracts T t R T t calculating qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of S i ðt Þ s distribution with time deendent volatility r i ðt; T Þ we use r i ¼ r i ðy; T Þ 2 dy: Because in Assumtion 2.1 we model the exected values, the rocess of the fixed routing rice S i ðtþ can be e.g. geometric Brownian motion or mean-reverting, and we can use, for instance, the electricity rice models resented in Deng, Johnson, and Sogomonian (2001), Keo and Ra sa nen (1999, 2000), Manoliu and Tomaidis (2000), and Oren (2001). Thus, equation (2.2) has already been used with a non-storable underlying asset. Also note that there can be cycles in the exected fixed routing rices. For instance, in our model we can have S i ð0; 1Þ ¼100 and S i ð0; 1:1Þ ¼1 The continuous-time rocess in (2.2) can also be seen as a limit of the corresonding discrete-time rocess. Since rices are always ositive let us model log-exected rices and define D m i ðkþ ¼logðS iðk T m; T ÞÞ logðs i ððk 1Þ T m; T ÞÞ for all k 2f1;...; mg; m 2f1; 2...g; where m is the number of discrete time intervals on ½0; T Š and k is the index for discrete times. This gives log Si m ðt Þ ¼ log ð Si ð0; T ÞÞþ P m k¼1 Dm i ðkþ. We assume that D m i ðkþ are mutually indeendent and identically distributed random variables with mean b i m and standard deviation r T qffiffiffi T i m. Then we get from the central limit theorem [see e.g. Billingsley (1995)] P lim m D m i ðkþb i T m ¼ ri B i ðt Þ, i.e., m!1 k¼1 lim log m!1 Sm i ðt Þ ¼ log ð Si ð0; T ÞÞþb i T þ r i B i ðt Þ; where according to (2.2) b i ¼ 1 2 r2 i and therefore the limit of our discretetime model is given by (2.2). Thus, if we assume that in discrete-time the differences of the log-exected rices are indeendent and identically distributed then if we seed u the arrivals of the discrete-time events we get our aroximation, Assumtion 2.1. Because there are three risk factors they cause at most three indeendent sources of uncertainty in the market. That is, all the uncertainties in the market rices are driven by the risk factors and the correlation structure between the risk factors is free in our model. If the market caacity suly is constant then the uncertainties in the S-rices are generated from the caacity demand rocesses. In order to get analytical formulas for bandwidth oint-to-oint contracts, we will aroximate the sum of two geometric Brownian motions with a geometric Brownian motion. Thus, we make the following assumtion on the exected alternative routing rices and the same assumtion is done in the ricing of financial derivatives. Assumtion 2.2 The exected alternative routing rices are given by the following Itoˆ stochastic differential equations

8 198 J. Keo dx 1 ðt;t Þ¼dS 2 ðt;t ÞþdS 3 ðt;t Þ¼X 1 ðt;t Þ x 1;2 r 2 db 2 ðtþþx 1;3 r 3 db 3 ðtþ dx 2 ðt;t Þ¼dS 1 ðt;t ÞþdS 3 ðtt Þ¼X 2 ðt;t Þ x 2;1 r 1 db 1 ðtþþx 2;3 r 3 db 3 ðtþ dx 3 ðt;t Þ¼dS 1 ðt;t ÞþdS 2 ðt;t Þ¼X 3 ðt;t Þ x 3;1 r 1 db 1 ðtþþx 3;2 r 2 db 2 ðtþ ð2:3þ where X i ðt; T Þ¼ P k2f1;2;3gfig S kðt; T Þ and x i;j ¼ S jðt;t Þ X i ðt;t Þ is constant for all i; j 2f1; 2; 3g. In many ractical situations, equation (2.3) is accurate enough and, therefore, the same method is used, e.g., in the ricing of basket otions [see Gentle (1993)]. The correct rocesses of X 1 ðt; T Þ¼S 2 ðt; T ÞþS 3 ðt; T Þ, X 2 ðt; T Þ¼S 1 ðt; T ÞþS 3 ðt; T Þ, and X 3 ðt; T Þ¼S 1 ðt; T ÞþS 2 ðt; T Þ are given by dx 1 ðt; T Þ¼dS 2 ðt; T ÞþdS 3 ðt; T Þ¼S 2 ðt; T Þr 2 db 2 ðtþþs 3 ðt; T Þr 3 db 3 ðtþ dx 2 ðt; T Þ¼dS 1 ðt; T ÞþdS 3 ðtt Þ¼S 1 ðt; T Þr 1 db 1 ðtþþs 3 ðt; T Þr 3 db 3 ðtþ dx 3 ðt; T Þ¼dS 1 ðt; T ÞþdS 2 ðt; T Þ¼S 1 ðt; T Þr 1 db 1 ðtþþs 2 ðt; T Þr 2 db 2 ðtþ and, therefore, combining this with equation (2.3) we get x i;j ¼ S jðt;t Þ X i ðt;t Þ. In order to justify more the X-rocess of Assumtion 2.2, Aendix 1 analyzes the aroximation error by using two first moments and comares equation (2.3) with Monte Carlo Simulation. Note that equation (2.3) is for the exected values P k2f1;2;3gfig S kðt; T Þ and not for P k2f1;2;3gfig S kðtþ. We consider a market where telecommunications caacity and bandwidth instruments are bought and sold continuously. Because of the new exchanges (for instance Band-X and InterXion) and OTC-market this kind of international market already exists, but the market is not liquid and is in an early stage. Therefore, we assume the following general ricing function for bandwidth instruments and it is not based on continuous time hedging. Assumtion 2.3 The rice of T-maturity bandwidth contingent claim is given by ðt; T Þ¼exðrðT tþþe ½/ðT ÞjF t Š for all t 2 ½0; T Š; T 2 ½0; sš; ð2:4þ where r is a constant discount rate, /(T) is the ayoff at time T, and the exectation is with resect to robability measure P. Assumtion 2.3 imlies that the agents in the market rice the bandwidth instruments by using the discounted exected ayoff formula. The discount rate and the robability measure in equation (2.4) may deend on the agent s utility function. However, for simlicity we assume that the robability measure is equal to the objective measure P and the discount rate is constant. Note that if the market was liquid, arbitrage free, and comlete then the discount rate in (2.4) would be the risk-free rate and the exectation would be under the risk-neutral ricing measure Q [see e.g. Harrison and Kres (1979), Harrison and Pliska (1981), Kres (1981), and Bjo rk (1998)]. However, since in this aer we do not assume this kind of market we use directly

9 Pricing of oint-to-oint bandwidth contracts 199 Assumtion 2.3. The ayoff /(T) in equation (2.4) can be viewed as a money metric utility and, therefore, Assumtion 2.3 can also be seen as a utility based ricing aroach. For the general utility based ricing models see, e.g., Davis (1998, 2001) and Cochrane (2002) and for alications in telecommunications see, e.g., Stoenescu and Taneketzis (2002) and Courcoubetis, Kelly, Siris, and Weber (2000). In bandwidth markets there are leasing contracts traded between telecommunications comanies. As mentioned earlier, these leasing contracts can be modeled as forward contracts since they oblige the buyer of the contract to acquire the underlying bandwidth connection at a certain future date (maturity) for a certain ayment (forward rice). Therefore, in this aer we call these contracts as forwards. When the forward contract is agreed uon, no ayments are made. Instead, at maturity the seller of the contract receives the forward rice from the buyer. Thus, if the claim in Assumtion 2.3 is a forward contract then we get from equation (2.4) that the T-maturity forward rice at time t with instantaneous duration is given by Y i ðt; T Þ¼EY ½ i ðt ÞjF t Š ð2:5þ because, as mentioned above, in this case by definition of the forward contract /ðt Þ¼Y i ðt ÞY i ðt; T Þ and (t,t) = 0, where Y i (T) is the corresonding bandwidth sot rice at time T. Note that according to Assumtion 2.3 the forward rice with duration D can be calculated as follows Y i ðt; T ; DÞ ¼ r R T þd exðrðy tþþy T i ðt; yþdy exðrðt þ D tþþexðrðt tþþ ; ð2:6þ where the contract s start date is T and the end date is T+D. 3. Forward ricing From now on we only consider the ricing of contracts on Y 1 and the contract rices on Y 2 and Y 3 can be derived in the same way. Unlike in other commodity markets, the bandwidth forward contract includes otionality. That is, the seller can rovide either the direct routing or the alternative one. According to Section 2, we assume that the exectation hyothesis for bandwidth forward rices holds under P and, therefore, the forward rices are equal to the exected sot rices: Y 1 ðt; T Þ¼Eðmin½S 1 ðt Þ; X 1 ðt ÞŠjF t Þ for all t 2 ½0; T Š; T 2 ½0; sš; ð3:1þ where Y 1 (t,t) is the T-maturity forward rice at time t and X 1 ðtþ ¼ S 2 ðtþ þs 3 ðtþ. Equation (3.1) imlies that the forward contracts are a kind of combined otions where the forward rice is equal to the minimum value of an asset and a ortfolio. Note that (3.1) is indeendent of the stochastic rocesses of S 1 and X 1. Further, because the sot rices [equation (2.1)] are free of network arbitrage, according to (3.1) also the forward rices satisfy the no-arbitrage condition. Equation (3.1) can also be written in the following way Y 1 ðt; T Þ¼ES ð 1 ðt ÞjF t ÞEðmax½0; S 1 ðt ÞX 1 ðt ÞŠjF t Þ ð3:2þ

10 200 J. Keo Thus, the market forward rice is equal to the rice with fixed routing minus the otion to change the oint-to-oint routing. This otion is due to the network arbitrage condition and it is a kind of exchange otion [for the ricing of exchange otions see e.g. Margrabe (1978)]. By allowing T vary from t to s we get the whole bandwidth forward curve Y 1 ðt; Þ : ½t; sš! R þ at time t. Using the ricing formula for exchange otions and the martingale measure for X 1 we get the following roosition. Proosition 3.1 The forward rice is given by Y 1 ðt; T Þ¼S 1 ðt; T ÞHðt; T ; S 1 ; X 1 Þ for all t 2 ½0; T Š; T 2 ½0; sš; ð3:3þ where the exchange otion Ht; ð T ; S 1 ; X 1 Þ ¼ S 1 ðt; T ÞNðz t ÞX 1 ðt; T ÞN z t r ffiffiffiffiffiffiffiffiffiffiffi T t ; ð3:4þ and ln S 1ðt;T Þ X 1 ðt;t Þ þ 1 2 r2 ðt tþ z t ¼ r ffiffiffiffiffiffiffiffiffi T t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r¼ r 2 1 þx2 1;2 r2 2 þx2 1;3 r2 3 þ2x 1;2x 1;3 q 2;3 r 2 r 3 2r 1 x 1;2 q 1;2 r 2 þx 1;3 q 1;3 r 3 and N(Æ) is the cumulative standard normal distribution. Proof. From equation (3.2) we get Ht; ð T ; S 1 ; X 1 Þ ¼ E X 1 ðt Þ max S 1ðT Þ 1; 0 jf t X 1 ðt Þ ¼ X 1 ðt; T ÞE X 1ðT Þ X 1 ðt; T Þ max S 1ðT Þ 1; 0 jf t X 1 ðt Þ ¼ X 1 ðt; T ÞE Q 1 max S 1ðT ; T Þ 1; 0 jf t ; ð3:5þ X 1 ðt ; T Þ where S 1 ðt; T Þ¼ES ½ 1 ðt ÞjF t Š, X 1 ðt; T Þ¼EX ½ 1 ðt ÞjF t Š, E Q 1 ðþ in the last line of (3.5) is the exectation with resect to the martingale measure Q 1 for the numeraire rocess X 1 (Æ,T) [see e.g. Geman (1989) and Bjo rk (1998, chater 19)], and the Radon-Nikodym derivative is given by dq 1 dp ¼ X 1ðT ; T Þ on F T : X 1 ðt; T Þ Equation (3.5) imlies that H(t,T,S 1,X 1 ) is equal to X 1 (t,t) numbers of Black- Scholes tye T-maturity call otions on S 1ðt;T Þ X 1 ðt;t Þ with zero discount rate and unit strike rice. The rocess of S 1ðt;T Þ X 1 ðt;t Þ is a martingale under the martingale measure Q 1. Therefore, it has zero exected change and the rocess is given by [see e.g. Bjo rk (1998, chater 19)] d S 1ðt; T Þ ¼ S 1ðt; T Þ X 1 ðt; T Þ X 1 ðt; T Þ r 1d ^B 1 ðtþx 1;2 r 2 d ^B 2 ðtþx 1;3 r 3 d ^B 3 ðtþ ; ð3:6þ

11 Pricing of oint-to-oint bandwidth contracts 201 where ^B 1 ; ^B 2,and^B 3 are Brownian motions under Q 1. Note that the correlation structure between the Q 1 -Brownian motions is the same as the correlation structure between the P-Brownian motions of Assumtion 2.1. Using equations (3.5) and (3.6) and Black-Scholes formula we get (3.3) and (3.4). u As, e.g., in Davis (1998) we end u Black-Scholes tye ricing equation even though we started with the general ricing function of Assumtion 2.3. Note that the change of measure in the roof of Proosition 3.1 is just an efficient technique to calculate the exectation of Assumtion 2.3. By using equations (3.3) and (3.4) the forward rice can also be written as follows Y 1 ðt; T Þ¼S 1 ðt; T Þð1 Nðz t ÞÞþX 1 ðt; T ÞN z t r ffiffiffiffiffiffiffiffiffiffiffi T t ¼ S 1 ðt; T ÞNðz t ÞþX 1 ðt; T ÞN z t r ffiffiffiffiffiffiffiffiffiffiffi T t ; ð3:7þ where z t ¼ ln X1ðt;T Þ 1 S 1 ðt;t Þ ffiffiffiffiffiffi 2 r2 ðtþ and z r T t t r ffiffiffiffiffiffiffiffiffiffiffi T t ¼ ln S1ðt;T Þ 1 X 1 ðt;t Þ 2 r2 ðt tþ ffiffiffiffiffiffi. r T t Thus, equation (3.7) is just the exected marker rice because if S 1 (T) < X 1 (T) then the market selects the S 1 -rice and direct routing and, otherwise, the rice X 1 and the longer routing are used. Because Y 1 (t,t) is a nonlinear function of S-rices, the instantaneous correlations between the forward rices (Y-rices) are not constant and the correlations deend on time and the S-rices. That is, from equation (3.7) and Itoˆ s lemma we get dy 1 ðt; T ðt; S 1; X 1 þ 1 2 Y ðt; S 1 ; X 1 1 ðt; T Þ 2 dt ðt; S 1; X 1 1 ðt; T Þ ðds 1 ðt; T ÞÞ 2 þ 1 2 ds 1 ðt; T ðt; S 1; X 1 Þ dx 1 ðt; T 1 ðt; T 2 Y ðt; S 1 ; X 1 1 ðt; T Þ 2 ðdx 1 ðt; T ÞÞ 2 Y ðt; S 1 ; X 1 1 ðt; T Þ@X 1 ðt; T Þ dx 1ðt; T ÞdS 1 ðt; T Þ; ð3:8þ where Y(t,S,X) is the forward rice given by the right-hand side of equation (3.7) and it is a function of time and direct and alternative routing rices. Because the artial derivatives in (3.8) are not constant, the arameters of the forward rice dynamics are neither constant even though the S-rocess arameters are constant. That is, the drift and diffusion terms of Y 1 are changing all the time. According to equation (3.1) the forward rices (Y-rices) are martingale under P. Therefore, the drift term in (3.8) is zero and we get by using (2.2), (2.3), and (3.7) dy 1 ðt; T Þ¼m 1;1 ðt; T ÞdB 1 ðtþþm 1;2 ðt; T ÞdB 2 ðtþþm 1;3 ðt; T ÞdB 3 ðtþ; ð3:9þ where

12 202 J. Keo m 1;1 ðt; T Þ¼Nðz t ÞS 1 ðt; T Þr 1 m 1;2 ðt; T Þ¼N z t r ffiffiffiffiffiffiffiffiffiffiffi T t S 2 ðt; T Þr 2 m 1;3 ðt; T Þ¼N z t r ffiffiffiffiffiffiffiffiffiffiffi T t S 3 ðt; T Þr 3 : According to equation (3.9) the forward rice uncertainty is equal to the robability of direct routing multilied by the direct routing s uncertainty lus the robability of alternative routing multilied by the alternative routing s uncertainty. Thus, we have 0 m 1;i ðt; T ÞS i ðt; T Þr i for all i 2f1; 2; 3g. Further, equation (3.9) imlies the facts that are already mentioned earlier: the market forward rices are martingales and the correlation structure of the forward rices is not constant even though the correlation between S-rices is constant (see Assumtion 2.1). Note that even if S 1, S 2,andS 3 were indeendent, i.e., even if the Brownian motions were indeendent the forward rices would deend on each other because of the routing otions. In order to see this, let us use equation (3.9) and assume momentarily that the S-rocesses are indeendent. Then the instantaneous covariance between Y 1 (t,t) and Y 2 (t,t) isgivenby cov½y 1 ; Y 2 Š ¼ m 1;1 ðt; T Þm 2;1 ðt; T Þþm 1;2 ðt; T Þm 2;2 ðt; T Þþm 1;3 ðt; T Þm 2;3 ðt; T Þ > 0; ð3:10þ where the rocess of Y 2 (t,t) follows dy 2 ðt; T Þ¼m 2;1 ðt; T ÞdB 1 ðtþþm 2;2 ðt; T ÞdB 2 ðtþþm 2;3 ðt; T ÞdB 3 ðtþ and according to (3.9) all the diffusion terms of Y 2 (t,t) are ositive. Because these terms are ositive and because the correlations between the Brownian motions were assumed to be zero, we get (3.10). That is, the instantaneous covariance between Y 1 (t,t) andy 2 (t,t) is ositive and if there are no routing otions then the artial derivatives in equations (3.8) and (3.9) are zero and we get cov½y 1 ; Y 2 Š ¼ 0. Thus, the network arbitrage creates ositive correlation between caacity rices. This is natural since network routing smoothens the oint-tooint demands over the whole network and therefore rices move together. From now on we again assume the general correlation structure between the S-rices. 4. Otion ricing In addition to the routing otions described in section 2 and 3 some bandwidth service contracts include also other otion tye characteristics. For instance, the seller can have a right to disconnect the service for a redefined enalty ayment. These rights can be modeled as bandwidth otions and, therefore, the understanding of otion ricing is imortant in the bandwidth markets even though there do not exist traded otion contracts. Bandwidth otions are modeled as otions on forwards. Therefore, a Euroean bandwidth call otion s ayoff on the T-maturity forward at the otion s exiration date T C 2½0; T Š is max½y 1 ðt c ; T ÞK; 0Š, where K is the strike rice. Because Y 1 can be viewed as an exchange otion, the call is a kind of comound otion on S 1 and, in the ricing of bandwidth otions we utilize the ricing theory of comound otions [see Geske (1979) and Rubinstein (1992)]. In the same way, bandwidth ut otions are comound ut otions

13 Pricing of oint-to-oint bandwidth contracts 203 and the terminal ayoff is max½k Y 1 ðt c ; T Þ; 0Š. We concentrate on the Euroean bandwidth call otion ricing because the corresonding ut otion rice can be solved by using the following ut-call arity Cðt; T c ; S 1 ; X 1 ; KÞPðt; T c ; S 1 ; X 1 ; KÞ¼½Y 1 ðt; T ÞKŠexðrðT c tþþ; ð4:1þ where Cðt; T C ; S 1 ; X 1 ; KÞ and Pðt; T c ; S 1 ; X 1 ; KÞ are the bandwidth call and ut rices at time t 2½0; T C Š and 0 t T c T s. At the otions exiration date T c the left hand side of (4.1) is Y 1 (T c,t) ) K. Using Assumtion 2.3 we get EY ½ 1 ðt c ; T ÞKF j t ŠexðrðT c tþ Þ ¼ ½Y 1 ðt; T ÞK ŠexðrðT c tþþ: This is the same as the right hand side of (4.1) and, therefore, equation (4.1) holds. Thus, this ut-call arity is got from Assumtion 2.3. In the same way, according to Assumtion 2.3 before exiration date T c the bandwidth call otion rice has to be given by the discounted exected ayoff: Cðt; T c ; S 1 ; X 1 ; KÞ ¼exðrðT c tþþe max S 1 ðt c ; T ÞNðz Tc Þ þ X 1 ðt c ; T ÞN z Tc r ffiffiffiffiffiffiffiffiffiffiffiffiffi T T c K; 0 F t : ð4:2þ Equation (4.2) is solved in two stes. Firstly, we calculate the conditional otion rice on the value of X 1 (T c,t) and, secondly, we integrate the conditional otion rice over the X 1 s robability distribution. The first ste, i.e., the calculation of the conditional call otion rice means integrating the otion s ayoff over the S 1 s robability distribution and because the underlying forward rice formula includes cumulative normal distributions, this conditional otion rice is described in terms of bivariate normal distributions. This gives the fact that the final otion rice formula includes trivariate normal distributions because we have to consider also the X 1 s distribution. The derivation of Euroean bandwidth call otion formula is resented in Aendix 2. The result of the first ste, i.e., the conditional call rice is given in the following lemma. Lemma 4.1 The conditional call otion rice is given by Cðt; T c ; S 1 ; X 1 ; KÞ j X1 ðt c ;T Þ¼x ¼ ex ð r ð T c tþþ½s 1 ðt; T ÞðNq ð t ðxþþ M q t ðxþ; z t ; q q;z Þ ffiffiffiffiffiffiffiffiffiffiffiffi þ xm q t ðxþr 1 T c t; z t r ffiffiffiffiffiffiffiffiffiffiffi T t; q q;z ffiffiffiffiffiffiffiffiffiffiffiffii KN q t ðxþr 1 T c t ; ð4:3þ S 1 ðt;t Þ h S ðxþ þ 1 2 r2 1 ðt ctþ ffiffiffiffiffiffiffi r 1 T c t where q t ðxþ ¼ ln ; h S ðxþ solves 1 h S ðxþn r ffiffiffiffiffiffiffiffiffiffiffiffiffi ln h SðxÞ þ ½r 2 ðt T c Þ T T c x 1 þ xn r ffiffiffiffiffiffiffiffiffiffiffiffiffi ln h SðxÞ ½r 2 ðt T c Þ ¼ K; T T c x

14 204 J. Keo z t ¼ ln S1ðt;T Þ x þ 1 2 r2 ðt tþ ffiffiffiffiffiffi ; Mðq r T t t ; z t ; q q;z Þ is the area under a standard bivariate normal distribution function covering the region from ) to q t and qffiffiffiffiffiffiffi ) to z t, and the T two random variables have correlation q q;z ¼ c t T tq S, and q S ¼ 1 r r 1 x 1;2 q 1;2 r 2 x 1;3 q 1;3 r 3 Þ is the instantaneous correlation between S 1 and S 1 X 1 : Proof. See Aendix 2. Q.E.D. The first term after the discount factor S 1 ðt; T Þ Nq ð t ðxþþm q t ðxþ; z t ; q q;z ¼ ES1 ½ ðt; T ÞIfY 1 ðt c ; T ÞKg IfX 1 ðt ÞS 1 ðt ÞgjX 1 ðt c ; T Þ¼xŠ; where I is the indicator function, i.e. IfxPS 1 ðt Þg ¼ 1 if x S 1ðT Þ : 0 if x < S 1 ðt Þ Thus, IfY 1 ðt c ; T ÞKgIfX 1 ðt ÞS 1 ðt Þg ¼ 1 if the bandwidth otion exires in the money and if at the exiration of the forward contract X 1 (T) S 1 (T). In this case the bandwidth market rice at time T is S 1 (T). The second term ffiffiffiffiffiffiffiffiffiffiffiffi xm q t ðxþr 1 T c t; z t r ffiffiffiffiffiffiffiffiffiffiffi T t; q q;z is x multilied by the robability that the bandwidth otion exires in the money and that at the exiration of the forward contract X 1 (T) <S 1 (T). In this case at ffiffiffiffiffiffiffiffiffiffiffiffi time T the bandwidth market rice is X 1 (T). The last term KN q t ðxþr 1 T c t is K multilied by the robability that the bandwidth otion exires in the money. Thus, equation (4.3) is the discounted exected value of the bandwidth call otion given that X 1 (T c,t) = x. Note that if x K then always Y 1 (T c,t) K and, therefore, the call otion exires worthless. The conditional otion rice, equation (4.3), is similar to the comound otion ricing formula in Geske (1979) and Rubinstein (1992). However, we have to consider also the uncertainties in the alternative routing rice X 1 and, therefore, we integrate the conditional otion rice over the X 1 s robability distribution. In order to get analytical otion ricing formula in this integration we assume that the uer boundaries of the cumulative normal distributions in (4.3) are indeendent of X 1 (T c,t) s outcome. The derivation of the otion ricing formula is resented in Aendix 2 and the result is given by the following roosition. Proosition 4.1 Bandwidth call otion rice is given by h ffiffiffiffiffiffiffiffiffiffi Cðt;T c ;S 1 ;X 1 ;KÞ¼exðrðT c tþþ S 1 ðt;t Þ M q t ðx q Þ;w t r X T c t ;q q;w ffiffiffiffiffiffiffiffiffiffiffiffi G q t ðx q Þ; z t ; w t r X T c t; q q;z ; q q;w ; q z;w ffiffiffiffiffiffiffiffiffiffiffiffi þ X 1 ðt; T ÞG q t ðx q Þr 1 T c t; z t r ffiffiffiffiffiffiffiffiffiffiffi T t; w t ; q q;z ; q q;w ; q z;w ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi i KM q t ðx q Þr 1 T c t; w t r X T c t; q q;w ; ð4:4þ

15 Pricing of oint-to-oint bandwidth contracts 205 where x q ¼ X 1ðt;T ÞNðw t Þ ffiffiffiffiffiffiffi Nðw t r X T c tþ ; w t ¼ ln X 1 ðt;t Þ K þ 1 2 r2 X ð T ctþ ffiffiffiffiffiffiffi ; Gq r X T c t t ; z t ; w t ; q q;z ; q q;w ; q z;w is the area under a standard trivariate normal distribution function covering the region from ) to q t, ) q to ffiffiffiffiffiffiffi z t, and ) to w t and the three qffiffiffiffiffiffiffi random variables T have correlations q q;z ¼ c t T T tq S ; q q;w ¼ q S;X ; and q z;w ¼ c t T tq X ; q S;X ¼ 1 x 1;2 q r X r 1;2 r 1 r 2 þ x 1;3 q 1;3 r 1 r 3 1 is the correlation between S 1 and X 1, q X ¼ 1 r X r r 1 x 1;2 q 1;2 r 2 þ x 1;3 q 1;3 r 3 r 2 X is the correlation between X 1 and S 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r X ¼ x 2 1;2 r2 2 þ x2 1;3 r2 3 þ 2x 1;2x 1;3 q 2;3 r 2 r 3 is the volatility of X 1,and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ r 2 1 þ r2 X 2r 1 x 1;2 q 1;2 r 2 þ x 1;3 q 1;3 r 3 X 1, is the volatility of S 1 X 1. Proof. See Aendix 2. u Equation (4.4) is the discounted exected value of the bandwidth call otion. The first term after the discount factor ffiffiffiffiffiffiffiffiffiffiffiffi S 1 ðt; T Þ M q t ðx q Þ; w t r X T c t; q q;w ffiffiffiffiffiffiffiffiffiffiffiffi G q t ðx q Þ; z t ; w t r X T c t; q q;z ; q q;w ; q z;w ¼ ES ½ 1 ðt; T ÞIfY 1 ðt c ; T ÞKgIfX 1 ðt ÞS 1 ðt ÞgjF t Š is now the exected value of S 1 ðt; T ÞIfY 1 ðt c ; T ÞKgIfX 1 ðt ÞS 1 ðt Þg: Thus, IfY 1 ðt c ; T ÞKgIfX 1 ðt ÞS 1 ðt Þg ¼ 1 if the bandwidth otion exires in the money and if at the exiration of the forward contract X 1 (T) S 1 (T). The second term ffiffiffiffiffiffiffiffiffiffiffiffi X 1 ðt; T ÞG q t ðx q Þr 1 T c t; z t r ffiffiffiffiffiffiffiffiffiffiffi T t; w t ; q q;z ; q q;w ; q z;w ¼ ES ½ 1 ðt; T ÞIfY 1 ðt c ; T ÞPKgIfX 1 ðt ÞS 1 ðt ÞgjF t Š and IfY 1 ðt c ; T ÞKgIfX 1 ðt ÞS 1 ðt Þg ¼ 1 if the otion exires in the money andif at the exiration ffiffiffiffiffiffiffiffiffiffiffiffi of the forward ffiffiffiffiffiffiffiffiffiffiffiffi contract X 1 ðt ÞS 1 ðt Þ. The last term KM q t ðx q Þr 1 T c t; w t r X T c t; q q;z is again K multilied by the robability that the bandwidth otion exires in the money. The selection of q t s variable according to x q ¼ X 1ðt;T ÞNðw t Þ ffiffiffiffiffiffiffi is an Nðw t r X T c tþ aroximation and this way the uer boundaries of the integrals in equation (4.4) are indeendent of X 1 (T c,t) s outcome. This selection imlies that x q EI ½ fx 1 ðt c ; T ÞKgjF t Š ¼ EX ½ 1 ðt c ; T ÞIfX 1 ðt c ; T ÞKgjF t Š;

16 206 J. Keo i.e., x q is the constant corresonding to X 1 (T c,t) in the sense of the above equation. Aendix 3 illustrates the error from x q by comaring our analytical otion ricing formula with Monte Carlo simulation with 200,000 outcomes. According to the results, the higher the volatility of X 1 and the smaller the X 1 (t,t)/s 1 (t,t) ratio the higher the aroximation error, and the error is zero if X 1 is deterministic. In the deriving of the otion ricing formula (4.4) we have made two aroximations: the rocess of X 1 ðt; T Þ¼S 2 ðt; T ÞþS 3 ðt; T Þ was assumed to follow a geometric Brownian motion [equation (2.3)] and the normal integrals uer boundary q t was assumed to be deterministic [equation (4.4)]. Both these aroximations are due to the stochastic alternative routing rice X 1 and in the case of deterministic X 1 equation (4.3) gives the correct call otion rice. 5. Examle In this section we illustrate our ricing models with numerical examles. Firstly, we analyze bandwidth forward rice as a function of the S 1 s volatility r 1. Secondly, we study how the volatility affects bandwidth call otion rice and comare our otion ricing model with Black-76 formula, which is a traditional commodity otion ricing method. Let us assume the following arameter values. Current time t ¼ 0, forward maturity T ¼ 2 years, bandwidth call otion maturity T c ¼ 1 year, otion strike rice K ¼ 2.8, and, for simlicity, discount rate r ¼ 0. Price S 1 (0,1) ¼ S 1 (0,2)= 2.8, S 2 (0,1) ¼ S 2 (0,2) ¼ 1, S 3 (0,1) ¼ S 3 (0,2) ¼ 2, r 1 ¼ 0.2, and r 2, r 3 ¼ 0. That is, the exected direct routing rices are 2.8, 1, and 2, and they satisfy the network arbitrage condition. For simlicity, we assume that S 2 and S 3 are constant and, therefore, X 1 (t) ¼ 3 for all t. The annual volatility of S 1 is 20%. Because X 1 is constant we do not have to use the aroximation of equations (2.3) and (4.4). Therefore, Proosition 3.1 and Proosition 4.1 give the correct forward and call otion rices. Using the above arameter values and the bandwidth forward and call ricing functions we get for the forward rice and for the call rice. Figure 2 illustrates the bandwidth forward rice as a function of volatility r 1. Figure 2 indicates that the volatility affects the forward rice and the function is decreasing. The volatility widens the robability distribution of future direct routing rice S 1. However, due to the existence of the alternative routing the future market rices Y 1 are bounded above and, therefore, the volatility mainly lengthens the lower tail of Y 1 s robability distribution. This imlies that the exected future sot rice and the forward rice decrease when the volatility increases. Thus, the bandwidth forward ricing is different than in other commodity markets. For instance, if the underlying asset is a storable commodity then the forward rices are given by the cost-of-carry model and the volatility does not affect the commodity forward rices [see e.g. Hull (1997)]. Next we analyze how the volatility affects the bandwidth call otion rices. Figure 3 illustrates the situation. The solid line is the call otion rice based on our model and the broken line is the corresonding otion rice given by the Black-76 formula, which does not consider the uer boundary

17 Pricing of oint-to-oint bandwidth contracts 207 Fig. 2. Bandwidth forward rice S 1 (t ¼ 0,T ¼ 2) as a function of annual volatility r 1 of the underlying forward rice. The increased volatility has two effects on bandwidth call rice. Firstly, it widens the robability distribution of future sot rices. This increases the otion rice. Secondly, according to Figure 2 the increased volatility lowers the forward rice and, therefore, the volatility also lowers the call otion rice. This is a natural otion character as the underlying asset decreases also the call otion value decreases. Thus, the total effect of volatility can be ositive or negative. As can be seen from Figure 3, with the given arameters our call otion rice is first increasing and then decreasing function of the volatility. With low volatility values the first effect is greater, but with higher than 5% volatility the second effect starts to lower the otion rice. This is because the S 1 -robability distribution is bounded above and, therefore, increasing volatility does not any more lengthen the uer robability tail. With the Black-76 model the first effect is greater all the time, because the model does not consider the alternative routing, i.e., the robability distribution is not bounded above. According to figures 2 and 3, the error term of Black-76 is significant in the situations where the second effect is strong, i.e., in the situations where there is a high robability that the alternative routing is used. Fig. 3. Bandwidth call otion rice Cðt ¼ 0; T c ¼ 1; S 1 ¼ 2:8; X 1 ¼ 3; K ¼ 2:8Þ as a function of annual volatility r 1 (solid line) and the corresonding call otion rice by using Black-76 model (broken line)

18 208 J. Keo 6. Conclusions In this aer we have modeled oint-to-oint bandwidth contracts under the network arbitrage condition. In order to understand the effects from this condition we used the fixed routing bandwidth rices as the underlying assets for these contracts. Due to the network arbitrage condition bandwidth forwards include routing exchange otions and, therefore, they are nonlinear instruments of the fixed routing rices. According to our bandwidth forward model, the network arbitrage creates ositive correlation between forward rices and shortens the uer tails of bandwidth market rice distributions. The more the uer tails are truncated the greater the correlation between the market rices. In the numerical examles, we have illustrated that the underlying bandwidth volatility affects the forward rices and the bandwidth forward rices are decreasing functions of the underlying volatility. Because a bandwidth forward contract can be viewed as an exchange otion, a bandwidth otion is a kind of comound otion. The analytical otion rice aroximation is described in terms of bivariate and trivariate normal distributions and the aroximation error is zero if the alternative routing rice is deterministic. Because the forward rice is a decreasing function of the underlying volatility and because the network arbitrage condition shortens the uer tail of the bandwidth rice distribution, a bandwidth call otion can be a decreasing function of the volatility. In the numerical examles, we have illustrated the difference between our otion ricing model and a traditional commodity otion ricing formula. The difference of these aroaches is significant in the situations when there is a high robability that the underlying oint-to-oint routing is changed. Aendix 1: Geometric Brownian motion aroximation for alternative routing In this aendix we analyze the geometric Brownian motion aroximation for alternative routing rice X 1. First, the aroximation error s two first moments are calculated. Second, we will comare the geometric Brownian motion assumtion with Monte Carlo Simulation with 20,000 outcomes. The geometric Brownian motion assumtion for X 1 can be written as X 1 ðt Þ¼½S 2 ðt; T ÞþS 3 ðt; T ÞŠex 1 2 r2 X ðt tþþr ffiffiffiffiffiffiffiffiffiffiffi X T tz X ; ða1:1þ where S 2 ðt; T Þ¼ES ½ 2 ðt ÞjF t Š, X 1 ðt; T Þ¼ES ½ 2 ðt ÞþS 3 ðt ÞjF t Š, Z X is a standard normal variable, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r X ¼ x 2 1;2 r2 2 þ x2 1;3 r2 3 þ 2x 1;2 x 1;3q 2;3 r 2 r 3 ; S 2 ðt; T Þ x 1;2 ¼ S 2 ðt; T ÞþS 3 ðt; T Þ ; and x S 3 ðt; T Þ 1;3 ¼ S 2 ðt; T ÞþS 3 ðt; T Þ : The distribution of (A1.1) is a lognormal with mean S 2 (t,t) +S 3 (t,t) and variance ½S 2 ðt; T ÞþS 3 ðt; T ÞŠ 2 ex r 2 X ðt tþ 1 : ða1:2þ