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Contingent claims valuation of optional calling plan contracts in telephone industry
International Review of Financial Analysis 11 (2002) 433 – 448
Contingent claims valuation of optional calling plan contracts in telephone industry Hyun-Woo Choia,*, In Joon Kimb, Tong Suk Kimb a
POSCO Research Institute, POSCO Building, 147 Samsungdon, Kangnamgu, 135-090 Seoul, South Korea Graduate School of Management, Korea Advanced Institute of Science and Technology, Seoul, South Korea
b
Abstract This paper presents a valuation methodology for optional calling plan contracts (OCP contracts) on free phone calls in the telephone industry. Contingencies in these OCPs stem from the uncertainty in the accumulated call usage during a given time period. Using a financial contingent claims approach, we investigate the basic nature of the model and extend the model to popular variants of the OCP in the industry. Utilization of the model is not limited to valuation and consequent decision making for the subscribers and provides a useful guideline for telephone companies in designing calling plans and assessing subscribers’ behavior. D 2002 Elsevier Science Inc. All rights reserved. JEL classification: G13; D81 Keywords: Real option; Optional calling plan; Telecommunications pricing
1. Introduction Generally, an optional calling plan contract (hereafter OCP contract) is constituent of a price discount on telephone calls incurred during a billing period under the condition that a customer pays the option charge (i.e., up-front fixed fee) before the start of a billing period.1 This fixity, coupled with the uncertainty in accumulated usage, creates a contingency in the
* Corresponding author. E-mail address: [email protected] (H.-W. Choi). 1 The OCP is called various terms such as nonuniform pricing, optional tariffs, and/or nonlinear price in the economic field. 1057-5219/02/$ – see front matter D 2002 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 7 - 5 2 1 9 ( 0 2 ) 0 0 0 6 3 - 7
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Fig. 1. Outlay schedule of a BOT-type OCP contractor. The customers who choose the standard price (i.e., nonOCP contractor) pay the amount that is the product of standard unit price p and the accumulated call usage AT at the end of the billing period (solid line). The BOT-type OCP contractor pays option charge ocBOT at the beginning of period but no charge for the calls up to the level h. For the calls exceeding h, QT h, the customer pays a discount amount p(1 p)max[0,QT h] or (1 p)max[0,AT B], where 0 p 1 (thick dotted line). The thick solid lines represent his total outlay schedule at the end of the billing period.
payoff of the OCP. Of the other features, the customer’s voluntary subscription (that is, making a contract) and the levying of an option charge are the two key characteristics that distinguish the OCP from other pricing schemes in telephone services—namely, standard pricing, night time discount pricing, and promotional pricing. Because it is one of the characteristics of telephone service marketing that potential users might have difficulties in judging the service quality of one company over another, the premiums on telephone calls become a major deciding factor in a customer’s choice of a telephone company. Hence, in the early stages of competition since the liberalization of the 1980s, price reduction took place in almost all telephone markets. However, such a price war could not continue for long. Clearly, price wars have a danger of draining companies of their revenue. When companies realized that further price cutting would be financially disastrous, they began developing the OCP contract as an alternative strategy. The first OCP introduced in June 1984 by AT&T in the name of the ‘‘Reach-Out America’’ became popular and a marketing success. Other competitors followed suit with their own OCP plans. Developing an attractive OCP contract emerged as one of the key factors to succeed in the telephone business. Although there are more than 100 types of OCP contracts, they can be broadly classified into three distinct groups: Block of Time (BOT) types, Volume Discount Program (VDP) types, and Minimum Usage Guarantee (MUG) types. Under the BOT-type OCP contract,2 in
2
Notable examples of BOT-type OCP include Reach-Out America (AT&T), Primer Calling Plan (MCI) in the US, Dial Coupon 001 (KDD), and Telejoyoz (NTT) in Japan.
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Fig. 2. Outlay schedule of a VDP-type OCP contractor. The solid line represents an outlay of the non-OCP contractor. The VDP-type OCP contractor pays ocVDP as an option charge at the beginning of the billing period and the amount calculated using different discount rates depending on the predetermined usage ranges at the end of the billing period. That is, if his call usage QT falls between k1 and k2, he pays k1p(1 p0)+( QT k1)p(1 p1). When the call usage falls between k2 and k3, he pays k1p(1 p0)+(k2 k1)p(1 p1)+( QT k2)p(1 p2) and so on (thick dotted lines). The thick solid lines represent his total outlay schedule at the end of the billing period.
return for an option charge (ocBOT) paid up-front, calls up to a predetermined level (h) are provided free of charge and only the calls that exceed the predetermined level are charged, but at a lower rate than that of the standard price.3 Fig. 1 shows the payment schedule. Let the call charge paid by the non-OCP subscriber over a period from 0 to T be AT, which is the standard price p times the accumulated call usage over the period QT. If we denote the discount rate by p, the call charge SBOT paid by an OCP subscriber at time T can be represented by Eq. (1): S BOT ¼ max½0,ð1 pÞðAT BÞ,
ð1Þ
where B ( = ph) is the predetermined hurdle amount over which the discount rate p is to be applied. Hence, the OCP contract gives the subscriber, at time T, a profit (or loss) of [AT (ocBOTerT + SBOT)] where r is risk-free rate of interest. Fig. 2 shows call charges under the VDP-type OCP contract.4 This contract divides a call charge into several ranges. Different discount rates are applied to each range of the call charge (i.e., the discount rate pi is applied on a charge between Ki and Ki + 1 where Ki < Ki + 1). To
3
Economists often use the term ‘‘measured service’’, ‘‘linear price’’, or ‘‘uniform price’’ instead of the term ‘‘standard price.’’ It is a tariff that is simply a product of the unit price and the call usage during a billing period. 4 The most famous VDP-type OCPs are the Wide Area Telephone Service (WATS) and the 800 number services in the US, the Members-Net of NTT, and the Freefone 0800 of BT.
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enter this contract, the subscriber pays the option charge ocVDP up-front. The call charge paid by the subscriber SVDP at time T would be (see Eq. (2) below) S VDP ¼ pk1 ð1 p0 Þ þ pðk2 k1 Þð1 p1 Þ þ pðk3 k2 Þð1 p2 Þ þ : : : þpðAT " ki1 Þð1 pi1 Þ # i X ¼ pQT pQT pi pkj ðpj pj1 Þ " ¼ AT AT pi
j¼0 i X
#
Kj ðpj pj1 Þ
if Ki < AT < Kiþ1 ,
ð2Þ
j¼0
where k0 = 0, Ki = pki, kn + 1 = 1, pi < 0 = 0, and i = 0, 1, 2, 3,. . .n. The terms in bracket are discount amount from the OCP contract, whereas the first term represents outlay under standard price schedule and is very similar to the payoff This contract Pof generalized options. gives the subscriber a profit (or loss) of erT AT pi ij¼0 Kj ðpj pj1 Þ ocVDP . The MUG-type OCP contracts5 also create tiers on a call charge as in the VDP type but impose no option charge. Rather, it requires the customer’s guarantee that his call charge for a billing period will be greater than a certain amount of G ( G = pg), and a penalty payment of P will be imposed if the call charge falls short of amount G at the end of a billing period (see Fig. 3). That is, the call charge paid by the OCP subscriber is given in Eq. (3):
S MUG
" # 8 i P > > > > < At AT pi j¼1 Kj ðpj pj1 Þ þ P " # ¼ i > P > > > : AT AT pi Kj ðpj pj1 Þ
if Ki AT < G ð3Þ if G Ki AT ,
j¼1
Pi the subscriber a profit of A p where Kn G < Ki + 1. The contract gives T i j¼0 Kj ðpj pj1 Þ Pi if G AT or a loss of P AT pi j¼0 Kj ðpj pj1 Þ if AT < G at time T. OCP contracts are applied to either ordinary or free phone calls (i.e., the 800 number service in the US). With regards to the free phone call, the receiving party pays telephone charge once a caller makes a call to a designated ‘‘free phone’’ number, whereas the telephone charge is paid for by the caller in an ordinary call. The free phone call has become an important marketing tool in many service industries that wish to solicit customer inquiries and orders.
5
In the US, the Virtual Private Network (VPN) service is designed for high-volume customers, and most carriers require a minimum volume. One of the most famous example is AT&T’s Software Designed Network (SDN).
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Fig. 3. Outlay schedule of a MUG-type OCP contractor. The solid line represents outlay of the non-OCP contractor. The call charge on the MUG-type OCP contract is similar to that of the VDP-type OCP contracts having tiered discounts. The MUG-type OCP, however, requires a guarantee that his accumulated call charge during the billing period be greater than a certain amount G ( G = gp). If the usage falls below the predetermined level g, a penalty P is imposed. Instead of the up-front option charge, the contractor pays SVDP + P if his call usage falls short of amount G and SVDP otherwise. The thick solid line represents total outlay schedule at the end of the billing period.
The purpose of this paper is to develop a methodology in evaluating a fair option charge for the OCP contract on free phone call charges by applying a contingent claims approach. The application of contingent claims approach to real-life problems has been currently received increasing attention and the analysis has been applied in a variety of contexts. Examples of these, although not exhaustive, include natural resource investments (Laughton & Jacoby, 1995), land development (Quigg, 1993), leasing (Trigeorgis, 1992), flexible manufacturing (Tannous, 1996), government subsidies and regulation (Teisberg, 1994), joint ventures (Kogut, 1991), accounting rule (Seow, 1995), and foreign investment (Kogut & Kulatilaka, 1994).6 By applying contingent claims approach to valuation of OCP contracts, the subscribers may get improved estimates of fair economic value of the contracts. It also provides a useful guideline for telephone companies in designing calling plans and assessing subscribers’ behavior. This paper is organized as follows. In Section 2, we investigate basic properties of a fair option charge in relation to existing option pricing researches. We then develop a valuation methodology that yields a fair option charge and extend that methodology to the three representative OCP contracts. In Section 3, we present some applications and discuss implications of the valuation model. Finally, Section 4 concludes the paper.
6 Chung (1993) applied the option pricing concept to sequential investment decisions. Smith and Nau (1995) compare option pricing analysis and decision analysis for valuing risky projects.
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2. Modeling OCP contracts 2.1. Distribution and preference free restrictions on the option charge To begin, let us consider a very simple OCP contract for free phone calls over a billing period from 0 to T (0 < T). Under this contract, a standard unit price p is applied up to a certain amount k and no charge is levied on the additional calls that exceed the level k in exchange for an option charge oc (see Fig. 4). It is called a three-part simple OCP in the telecommunications field, because three different prices (option charge oc, standard price p, and price 0) are applied according to the accumulated call usage level (0, from 0 to k, and more than k). If we denote the instantaneous call usage at time t (0 t T) by xt, then the instantaneous free phone call charge is qt = pxt and the call charge AT for a billing period from 0 to T is Z T AT ¼ qt dt: ð4Þ 0
Because there is no additional charge over the amount K (discount point, K = pk), the final call charge would be AT if AT < K and K otherwise; that is, min[K,AT] or AT max[0,AT K]. Notice that the second part of Eq. (4) is the same as the payoff of a call option with an exercise price K written on the uncertain call charge. Thus, subscribing to the OCP contract is identical to buying a call option from the telephone company. The option charge of the OCP contract, therefore, must be the same as the value of the call option. In a complete market, customers can duplicate the payoffs of OCPs by constructing a traded security (or a dynamic portfolio) that has the same risk characteristics (i.e., is perfectly correlated) as their free phone charge. Under the assumption that there is no arbitrage in the competitive OCP market (i.e., private agents can offer OCPs without restriction), the arbitrage relationships for European call options hold and the value of the fair option charge can be obtained by the use of the standard option pricing argument. 2.2. Valuing fair option charge of the three-part simple OCP contracts We will consider the problem of valuing the fair option charge of a three-part simple OCP contract for the free phone call. The interest rate is assumed to be a constant r and the instantaneous free phone call charge qt follows a geometric Brownian motion process generated by the following stochastic differential equation (Eq. (5)): dqt ¼ aqt dt þ qqt dZ;
qð0Þ ¼ q:
ð5Þ
The constant a reflects expected growth or decline in the level of free phone call charges. The constant q determines the instantaneous volatility of the process. The term dZ represents a standard Wiener process, with mean zero and variance dt. The free phone call charge process is correlated with the market portfolio of Merton (1973). The correlation coefficient is r.
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To this structure, we add the following assumptions for the telephone company and the economy as a whole: A1 The telephone company enjoys full observability of the underlying free phone call charge. The free phone call charge occurs exogenously, in the absence of any influence on the part of the customer.7 There are no moral hazard and no adverse selection problem. A2 The assumptions of the Cox, Ingersoll, and Ross (1985) model of intertemporal capital asset pricing hold. These include (1) existence of a single consumption/investment good, (2) existence of a set of linear production activities where changes in productivity follow exogenously specified the Wiener processes, (3) existence of a finite dimensional state variable vector whose components follow a Wiener process, (4) free entry and exit with competitive price-taking individuals and firms, (5) a common riskless borrowing and lending rate, (6) markets for contingent claims where values of claims follow Wiener processes with endogenously determined parameters, (7) a fixed number of identical individuals with homogeneous expectations and von Neumann–Morgenstern utility functions, and (8) a continuous and frictionless trading. A3 Investors have intertemporal utility functions that exhibit a constant relative risk aversion. The investment opportunity set (including the riskless rate of interest) is constant. Under the above assumptions, the fair option charge of an OCP contract O( q,A,t;K,T) must evolve according to the following partial differential equation: @O @O 1 2 2 @ 2 O @O þq þ qq rO ¼ 0, þ a*q 2 @t @A 2 @q @q
ð6Þ
subject to O( q,A,T;K,T) = max[AT K,0], O( q,1,t;K,T) = 1, and O(1,A,t;K,T) = 1, where a* = a lrq, l=(aM r)/sM. The term a* can be thought of as a certainty equivalent growth rate for call charge levels, as discussed by Constantinides (1978). The term l represents the market price of risk for the call charge level, and M is the market portfolio, which follows a geometric Brownian motion with a constant drift aM and volatility sM. A closed-form solution to Eq. (6) cannot be obtained, because AT is the sum of correlated lognormally distributed random variables and the distribution of the sum of lognormal components has no explicit representation. However, if we convert the payoff under the threepart simple OCP contract, that is, max[AT K,0], to
AT K Oðq, A, T ; K, TÞ ¼ T max ,0 , T T
ð7Þ
we can regard the problem as a continuously sampled, full period, unweighted arithmetic average rate Asian option with an exercise price K/T (Eq. (7)). Hence, we can utilize various methods developed for valuing Asian options. 7
Because the OCP contract provides substantial discount, customers may take strategic behavior (call stimulation effect) for ordinary calls. However, in the case of free phone calls, we can hardly expect that OCPs affect calling behavior.
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There are two ways to derive the average rate Asian options. Firstly, through numerical methods including the Monte Carlo approach (Kemna & Vorst, 1990), the Fourier transform approach (Carverhill & Clewlow, 1990), and the extended binomial approach (Hull & White, 1990), one can generate averages of the underlying assets and calculate the discounted average payoff of the Asian option. Although these methods provide an acceptable solution, they are time consuming. This is especially profound during an analysis of large portfolios. The second way to valuate Asian rate options is through approximations. Several authors have proposed approximations (see, for example, Levy, 1992; Ritchken, Sankarasubramanian, & Vijh, 1993; Rogers & Shi, 1995; Ruttiens, 1990; Turnbull & Wakeman, 1991), and the accuracy of which is sufficient for a practitioner’s purpose (Vorst, 1996). As we will see in Section 2.3, most OCP contracts form a portfolio of three-part simple OCPs, and the numerical procedure consumes much time with unnecessary complications. Hence, we present here an analytical approximation as a solution. This involves calculating the first two moments of the probability distribution of the sum exactly and then assuming that the distribution of the sum is lognormal with the same first two moments (Hull, 1997). Let us define (see Eqs. (8) and (9))
M1 ¼
ea*T 1 a*
ð8Þ
" # 2 2eð2a*þq ÞT 2 1 ea* : M2 ¼ þ ða* þ q2 Þð2a* þ q2 Þ a* ð2a* þ q2 Þ ða* þ q2 Þ
ð9Þ
Then, the first and second moments of the sum as seen at time 0 for a period of time T are qM1 and qM2. If we define m = 2ln(qM1) ln( q2M2)/2 and s2 = ln( q2M2) ln(qM1), then the analytical approximation of the fair option charge of an OCP contract can be expressed as follows:
Oðq, A, 0; K, T Þ ¼ erT bexpðm þ s2 =2ÞNðd1 Þ KN ðd2 Þc,
ð10Þ
where d1 = b ln(K) + m + s2c/s, d2 = d1 s, N( ) = standard cumulative normal distribution. From Eq. (10), we are able to conjecture that the option charge increases with the present call charge q, summing period T and volatility q, because they offer more chance to exercise. However, contrary to the stock option, an increase in interest rates decreases the option charge. The option charge also decreases as the discount point increases. Eq. (10) is an analytical approximation of the fair option charge for three-part simple OCP contracts when a subscriber receives a 100% discount on call charges above K. More generally, if the discount rate is expressed by p (where 0 p 1), because it is identical to a
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long position of p units of three-part simple OCP contracts, the valuation Eq. (10) can be adjusted as follows: Oðq, A, 0; K, T , pÞ ¼ perT bexpðm þ s2 =2ÞNðd1 Þ KN ðd2 Þc:
ð11Þ
2.3. Valuing the fair option charge of three representative OCP contracts To this point, we have been exclusively concerned with a very simple OCP contract, which can be used as a building block for constructing more general OCP contracts. If there are no arbitrage opportunities, and if a payoff function of the OCP contract provided by telephone companies at the end of the billing period is equal to that of the portfolio constructed by threepart simple OCPs, then the two fair option charges must be equal. This means that the valuation methodology for the three-part simple OCP contract in Section 2.2 could be used for the valuation of the more general OCP contracts, which can be found in a telephone market. Here, we extend the methodology in Section 2.2 to three popular variants of the OCPs. 2.3.1. The BOT-type OCP contracts As shown in Section 1, the call charge paid by the subscriber of the BOT-type OCP contract is SBOT = max[0,(1 p)(AT B)]. Because the call charge of the nonsubscriber is AT, the total amount of discounts that the contract subscriber is entitled to is DB ¼ AT ð1 pÞmax½AT B,0:
ð12Þ
This payoff from the contract DB can be exactly duplicated by an appropriately chosen portfolio of three-part simple OCP contracts. It is equal to the payoff from a portfolio composed of one unit of three-part simple OCP contracts with a discount point of 0 and (p 1) units of contracts with a discount point of B. As a result, the fair option charge of BOT-type OCPs can be expressed as OB ¼ Oðq, A, 0; 0, T Þ þ ðp 1ÞOðq, A, 0; B, TÞ:
ð13Þ
2.3.2. The VDP-type OCP contracts The OCP contract that is most frequently used for the free phone call is the VDP-type OCP. As shown in Fig. 3, this contract cuts off the price with a different discount rate for each range of call charges under the condition that the customer pays the option charge. In other words, under the VDP-type OCP contract, the discount rate pi is applied to the free phone call charge between Ki and Ki + 1 (where Ki + 1 > Ki, i = 0, 1, 2, 3,. . .,n, n < 1) and pn is applied to all charges above Kn. Hence, a telephone company’s reimbursement function of the OCP contract is expressed by DV ¼ p0 AT þ ðp1 p0 Þmax½AT K1 ,0 þ : : : þ ðpn pn1 Þmax½AT Kn ,0:
ð14Þ
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Eq. (14) is exactly duplicated by three-part simple OCPs, which comprise (pi + 1 pi) units of OCPs that have discount point Ki. Hence, the fair option charge of the VDP-type OCP contract OV is OV ¼ p0 Oðq, A, 0; 0, T Þ þ ðp1 p0 ÞOðq, A, 0; K1 , T Þ þ : : : þ ðpn pn1 ÞOðq, A, 0; Kn , TÞ:
ð15Þ
Eq. (15) indicates that the BOT-type OCP contracts can be considered as a special case of VDP-type OCPs with p0 = 1, p1 = p and K0 = 0, K1 = B. 2.3.3. The MUG-type OCP contracts The MUG-type OCP contract does not levy an option charge to a customer. Rather, it requires a customer’s guarantee that his call charge during a billing period will not be less than G. When a customer violates his promise, he should pay a predetermined penalty P at the end of the billing period.8 Here, we deal with the problem of determining the fair penalty P when the customer guarantees that his free phone call charge will be more than G, and the discount structure (pi, Ki, i = 0, 1, 2, 3,. . .,n) is determined. The MUG-type OCP contract can be regarded as a portfolio constituent by buying a VDPtype OCP contract and selling a continuously sampled, full period, unweighted arithmetic average rate Asian cash-or-nothing put with an exercise price G/T and an underlying asset AT. If there is no arbitrage in the competitive OCP contract market, the cost of constructing this portfolio has to be zero. The cash-or-nothing put would be like selling a three-part simple OCP except that, although the holder receives the penalty amount, he is under no obligation to give the underlying asset. From Eq. (13), such a partial put option CP would be worth CP ¼ erT PN ðd2 Þ,
ð16Þ
where d2=[ ln( G) + m]/s, to the holder. Therefore, the fair penalty P for a MUG-type OCP contract is given by P¼
8
OV : erT ½Nðd2 Þ
ð17Þ
The MUG-type OCP has been offered as a mixture of VDP-type OCPs on 800 or WATS in the US. Under the OCP, especially under the MUG-type OCPs, the subscribers reveal some information about their demands before the purchase decision. Some telephone companies argue that this information can be valuable to improve their production decision, and OCP’s price discounts include this effect. However, as noted by Mitchell and Vogelsang (1991), ‘‘This information would be potentially valuable when the supplier has to commit capacity before the consumer makes her actual purchasing decision. Because it would be inappropriate to bind a subscriber to a tariff for too long, it is not clear that it plays an important role for telephone carriers.’’ Therefore, in this paper, we do not consider such an information effect.
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Fig. 4. Outlay schedule and profit function of three-part simple OCP contractor. (a) The outlay of a three-part simple OCP contract where K is the hurdle amount over which a flat fee is charged. (b) The profit from the threepart simple OCP contract (AT K).
We have evaluated the fair option charges of three representative types of OCP contracts (Fig. 4), which we developed in Section 2.2. The methodology also can be used to determine the fair option charge of many other types of OCP contracts.
3. Applications and implications When the discount mechanism, the customer’s instantaneous free phone charge, and the volatility are given, applying the methodology developed in Section 2 to the valuation of actual OCP contracts is straightforward.9 However, there are few OCPs in the telephone market where the discount structure and the option charge vary for their subscribers. This lack of customer-tailored OCP contracts stems from the fact that telephone prices are usually regulated by regulatory bodies, most of which prohibit providing such OCPs.10 As a result, telephone companies have no choice but to design several kinds of OCP contracts. Customers are faced with the problem of choosing the most valuable OCP among a multitude of others. For telephone companies, forecasting the take rates has emerged as one of their main tasks. The standard theoretical approach employed by the regulatory economists has been based on the premise that customers will subscribe to the OCP services when their customer’s surplus exceeds the subscription price, as noted by Faulhaber and Panzar (1977). However, demand estimation and welfare analysis for telecommunications services have often been plagued by apparent inconsistencies between actual consumer behavior and standard 9
It is easy to estimate the variables a, q, and qt, because customers as well as telephone companies usually keep call records for several months for the purpose of preparing for charge-related conflicts. 10 An exception is that in the US, prices and accounting procedures can be tailored to customer preferences under the ‘‘competitive necessity’’ doctrine, Tariff nos. 12 and 15.
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Table 1 Price schedule for Freefone 0800 of British Telecoma Price and discount rates for accumulated usageb Types of option
Monthly rental (£)
Cumulative telephone usage
Pricec (discount rate, %)
Option 100 Option 105 Option 110
Nothing 33.33 333.33
Option 115
2000.00
Option 120
3333.33
– – 0 – 30,000 min 30,000 + 0 – 30,000 30,001 – 100,000 100,000 + 0 – 100,000 100,000 – 1,000,000 1,000,000 +
0.1055 0.0900 0.0792 0.0684 0.0792 0.0684 0.0576 0.0684 0.0576 0.0540
(0.00) (14.69) (24.93) (35.17) (24.93) (35.17) (45.30) (35.17) (45.40) (48.82)
British Telecom put five price schedules (OCPs) for their Freefone 0800 customers: Options 100, 105, 110, 115, and 120. In Option 100, which is the standard price of Freefone 0800, the customers pay no monthly rental (i.e., option charge) but pays 10.55 pence for each of usage minutes (blanket in right-hand side). For the rest of the price schedules, different monthly rental rates are levied at the beginning of the month and different discount rates are applied as the call usage increases. For example, the Option 115 levies £2000 a month as a monthly rental. However, customers benefit a 24.93% discount (0.0792 per minute) up to 30,000 monthly accumulate usage minutes, a 35.17% discount (0.0684 per minute) for the additional 70,000 minutes and a 45.30% discount (0.0576 per minute) for the call usage over 100,000 minutes. Source: Thairs (1997). a Tariffs valid from 1 April 1997. We modified the text of the original price table. First, each option in the original table discounts every monthly call charge for 3 months under the condition that the customer pays the option charge. For example, in Option 115, the monthly call charge is discounted by 24.93%, 35.17%, and 45.4% for ranges £0 – 3165, £3165 – 10,550, and over £10,550 for 3 months in return for an additional 6000 option charge. The fair option charge in these cases could also be easily derived from the result of Section 2. To secure simplicity, however, we assumed OCP as monthly OCP with the option charge £2000. Second, we adjust one minor fact. In Option 100, telephone price is 9.0 pence and it is charged on a per-minute basis, but in the another option, it is charged on a per-second basis. It means that the telephone company charges 9.0 pence under Option 100 and (9.0/60) 10 pence under Option 105 when a user made a 10-second call. If we assume that the conversation time per call follows an exponential distribution and the average conversation time per call is 180 seconds, then the average charge per call at 9.0 pence/minute under the per-minute charging system is 1{1/ [1 exp( 60/180)]}, and under the per-second charging system, the average charge is 1{1/[1 exp( 1/180)]}. That is, the price 9.0 pence/minute under the per-minute charging system equals 10.58 in per-second charging system (the price in the per-minute charging system is 14.72% higher than in the per-second charging system). To account for such a difference in charging systems among options, we treat the price in Option 100 as 10.58 pence instead of 9.0 pence. b Daytime call price per minute. c Number in blanket is per minute.
economic theory.11 Train, Ben-Akiva, and Atherton (1989) and subsequent researches have shown that a substantial fraction of customers does not correctly anticipate their own demands, yield ex post choices that do not minimize their expenditures for the quantities 11
For example, ‘‘Reach-Out America’’ shows that 45% of customers who choose an OCP have low monthly calling volumes and incur a somewhat higher average monthly bill than they would pay for the same use under the standard price.
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purchased. As a result, economists realized the expected customer’s surplus is generally not an adequate basis for subscription decision. Kridel, Lehman, and Weisman (1993) emphasized that customers exhibit substantial risk aversion when faced with bill uncertainty and pay attention to the customer’s attitudes toward risks and means available for reducing risks. The other economists protest the risk aversion arguments (Clay, Sibley, & Srinagesh, 1992). Our research provides one solution of the debates. It is clear that the uncertainty of future call charges generates option values and the financial contingent claims approach provide an option value (i.e., fair option charge), which is independent of customers risk preference. Consideration of option values requires modification of the existing framework for analyzing customer subscription decisions. Let us examine an actual OCP. Table 1 illustrates the ‘‘Freefone 0800’’ of British Telecom in the United Kingdom. If we assume Option 100 is a standard price, Options 105, 110, 115, and 120 can be considered VDP-type OCP contracts with expiration date 1/12 (1 month). That is, in the case of Option Table 2 Value of Option 115 of Freefone 0800 Fair option charge
Call charge ( q)
Expected call charge
CCA (A)
60,000 62,400 64,800 67,200 69,600 72,000 74,400 76,800 79,200 81,600 84,000 86,400 88,800 91,200 93,600 96,000 98,400 100,800
5030.3 5231.5 5432.8 5634.0 5835.2 6036.4 6237.6 6438.8 6640.0 6841.2 7042.5 7243.7 7444.9 7646.1 7847.3 8048.5 8249.7 8451.0
1446.6 1517.4 1588.1 1658.8 1729.5 1800.2 1870.9 1941.7 2012.4 2083.1 2153.8 2224.5 2295.3 2366.0 2436.7 2507.5 2578.2 2649.0
Profit from subscription
STA (B)
Option charge (C)
(A – C)
(B – C)
1433.0 1503.1 1573.3 1643.5 1713.7 1783.8 1854.0 1924.2 1994.3 2064.5 2134.7 2204.9 2275.0 2345.2 2415.4 2485.5 2555.7 2625.9
2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000
553.4 482.6 411.9 341.2 270.5 199.8 129.1 58.3 12.4 83.1 153.8 224.5 295.3 366.0 436.7 507.5 578.2 649.0
567.0 496.9 426.7 356.5 286.3 216.2 146.0 75.8 5.8 64.5 134.7 204.9 275.0 345.2 415.4 485.5 555.7 625.9
The first column represents charge per year. Expected call charge during a month (second column R ð30=360Þthe current call of left-hand side) is 0 qtdt ¼ ½qað30=360Þ 1=a under the assumption that the instantaneous free phone call charge follows a geometric Brownian motion process. The fair option charges are calculated using the contingent claims approach (CCA), the Eq. (12) in the main text, and the standard theoretical approach (STA), which is based on the expected call charge during a month. For example, if a customer’s present call charge is 79,200, his fair option charge under CCA is £2012.4 but is £1994.3 under the standard theoretical approach (1994.3 = 3165 0.2493+(6640 3165) 0.3517). The telephone company, British Telecom, sets £2000 as an option charge. The balance of subscription of Option 115 is £12.4 under CCA but is £ 5.8 under STA. The default parameter values are expected growth rate = 0.07, risk-free interest rate = 0.1, volatility = 0.5, lr = 0.15, and maturity of contract = 30 days.
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115, by paying £2000 of monthly option charges, the subscriber gets a discount of 24.93% for a call charge ranging from £0 to £3165, 35.17% for charges ranging from £3165 to £10,550, and 45.40% for amounts over £10,550. The fair option charge of each option could easily be derived from Eqs. (11) and (12) in Section 2. Customers will choose the most valuable option for them by comparing the option charges that the telephone company offers with the fair value, which is based on their present call charge and volatility. That is, customers will subscribe to the OCP when the profit from the subscription (that is, the fair option charge less the option charge imposed by the telephone company) is positive. Table 2 shows the profits of the subscription of Option 115 in a selected range of call charges when other parameters are given. As we can see in the table, if a customer’s present call charge is £79,200 a year, and he has to choose one between Option 100 and Option 115, subscribing to Option 115 is profitable. However, based on the standard theoretical approach, it could be regarded as an incorrect choice. This conflicting result for a customer’s subscription behavior stems from option values in the OCP contract. A contingent claims approach quantifies the value of option, whereas the standard theoretical approach could not. Table 2, which focuses on Option 115, also shows a lack of consideration for risks in taking the OCP contract leads to a consistent undervaluation of the customer’s benefit from subscribing to the OCP contract and to an underestimation of subscription demand.
4. Concluding remarks The application of the option pricing model to real-life problems is currently receiving increased attention, and the model has been applied in a variety of contexts. We applied the option pricing methodology to the valuation problem of OCP contracts in the telephone industry. We show that the payoff of a three-part simple OCP contract is identical to that of a call option, and option pricing theory can provide useful guidelines for OCP designers. Using the three-part simple OCP contracts as a building block, we developed a valuation model for the fair option charge of general OCPs found in the actual market. When customers choose a tariff among OCPs, because regulatory bodies prohibit providing custom tailored OCPs, the valuation model can be used as a tool in making subscription decisions by potential customers and in predicting take rates by a telephone company. We also showed that the standard theoretical approach consistently undervalues the OCP contract. The valuation results explored in this paper could be extended to ordinary call OCP contracts as well as to optional pricing schemes in other industries, such as gas or electricity, where the customer’s price elasticity is very small. The option concept used in this industry can be utilized in other competitive environment as well. Its use, however, has been limited to goods and services that are not dissimilar such as the telephone services, gas, and electricity. The goods and services that can be differentiated easily by other means have not been given much attention. The concept and the methodology explored in this paper would be quite useful where there is an uncertainty and
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the need for packaging and transferring the risk from customers to suppliers or vice versa. The option pricing scheme is an effective means to alter the exposure and get the fair values of such a design.
Acknowledgments We thank Jae Chul Kim, Thomas A. Fetherston, and an anomymous referee for helpful comments, discussions, and insights.
References Carverhill, A. R., & Chewlow, L. J. (1990). Valuing average rate (Asian) options. RISK, 3(4), 25 – 29. Chung, K. H. (1993). The contingent claims approach to investment decisions. Decision Science, 24, 1215 – 1221. Clay, K. B., Sibley, D. S., & Srinagesh, P. (1992). Ex post vs. ex ante pricing: optional calling plan and tapered tariffs. Journal of Regulatory Economics, 4, 115 – 138. Constantinides, G. (1978). Market risk adjustment in project valuation. Journal of Finance, 33, 603 – 616. Cox, J., Ingersoll, J., & Ross, S. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 3, 363 – 384. Faulhaber, G., & Panzar, J. (1977). Optimal two-part tariffs with self-selection. Bell Laboratories Economics, Discussion Paper, No. 74. Hull, J., & White, A. (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573 – 592. Hull, J. C. (1997). Options, futures and other derivatives. (3rd ed.). Englewood Cliffs, NJ: Prentice-Hall. Kemna, A. G. Z., & Vorst, A. C. F. (1990). A pricing method for options based on average asset values. Journal of Banking and Finance, 14, 113 – 129. Kogut, B. (1991). Joint ventures and the option to expand and acquire. Management Science, 37, 19 – 33. Kogut, B., & Kulatilaka, N. (1994, January). Evaluating the effects of time and risk on investment choice: a comparison of finance theory and decision analysis. Electric Power Research Institute (EPRI), P-5028, Project 2379-4. Kridel, D. K., Lehman, D. E., & Weisman, D. L. (1993). Option value, telecommunications demand, and policy. Information Economics and Policy, 5, 125 – 144. Laughton, D. G., & Jacoby, H. D. (1995). The effects of reversion on commodity projects of different length. Lenos trigeorgis, real options in capital investment: models, strategies, and applications ( pp. 185 – 206) 88 Post Road West, CT: Praeger Publisher. Levy, E. (1992). Pricing European average rate currency options. Journal of International Money and Finance, 11, 474 – 491. Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica, 41, 867 – 887. Mitchell, B. M., & Vogelsang, I. (1991). Telecommunications pricing: theory and practice. New York: Cambridge University Press. Quigg, L. (1993). Empirical testing of real option-pricing models. Journal of Finance, 48, 621 – 640. Ritchken, P., Sankarasubramanian, L., & Vijh, A. M. (1993). The valuation of path dependent contracts on the average. Management Science, 39, 1202 – 1213. Rogers, L., & Shi, Z. (1995). The value of an Asian option. Journal of Applied Probability, 32, 1077 – 1088. Ruttiens, A. (1990). Classical replica. RISK, 3(2), 33 – 36. Seow, G. S. (1995). A contingent claims model of corporate pension obligations. Decision Science, 26, 145 – 173. Smith, J. E., & Nau, R. F. (1995). Valuing risky projects: option pricing theory and decision analysis. Management Science, 41, 795 – 816.
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Tannous, G. F. (1996). Capital budgeting for volume flexible equipment. Decision Sciences, 27, 157 – 184. Teisberg, E. O. (1994). An option valuation analysis of investment choices by a regulated firm. Management Science, 40, 535 – 548. Thairs, S. (1997). T-guide tariffs. Teligen, Brentford TW8 OBB, UK: Eurodata Foundation. Train, K. E., Ben-Akiva, M., & Atherton, T. (1989). Consumption patterns and self-selecting tariffs. Review of Economics and Statistics, 71, 62 – 73. Trigeorgis, L. (1992). Evaluating lease with a variety of operating options. Working Paper, Boston University. Turnbull, M. S., & Wakeman, L. M. (1991). A quick algorithm for pricing European average options. Journal of Financial and Quantitative Analysis, 26, 377 – 389. Vorst, A. C. F. (1996). Prices and hedge ratios of average exchange rate options. International Review of Financial Analysis, 1, 179 – 193.
Contingent claims valuation of optional calling plan contracts in telephone industry Hyun-Woo Choia,*, In Joon Kimb, Tong Suk Kimb a
POSCO Research Institute, POSCO Building, 147 Samsungdon, Kangnamgu, 135-090 Seoul, South Korea Graduate School of Management, Korea Advanced Institute of Science and Technology, Seoul, South Korea
b
Abstract This paper presents a valuation methodology for optional calling plan contracts (OCP contracts) on free phone calls in the telephone industry. Contingencies in these OCPs stem from the uncertainty in the accumulated call usage during a given time period. Using a financial contingent claims approach, we investigate the basic nature of the model and extend the model to popular variants of the OCP in the industry. Utilization of the model is not limited to valuation and consequent decision making for the subscribers and provides a useful guideline for telephone companies in designing calling plans and assessing subscribers’ behavior. D 2002 Elsevier Science Inc. All rights reserved. JEL classification: G13; D81 Keywords: Real option; Optional calling plan; Telecommunications pricing
1. Introduction Generally, an optional calling plan contract (hereafter OCP contract) is constituent of a price discount on telephone calls incurred during a billing period under the condition that a customer pays the option charge (i.e., up-front fixed fee) before the start of a billing period.1 This fixity, coupled with the uncertainty in accumulated usage, creates a contingency in the
* Corresponding author. E-mail address: [email protected] (H.-W. Choi). 1 The OCP is called various terms such as nonuniform pricing, optional tariffs, and/or nonlinear price in the economic field. 1057-5219/02/$ – see front matter D 2002 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 7 - 5 2 1 9 ( 0 2 ) 0 0 0 6 3 - 7
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Fig. 1. Outlay schedule of a BOT-type OCP contractor. The customers who choose the standard price (i.e., nonOCP contractor) pay the amount that is the product of standard unit price p and the accumulated call usage AT at the end of the billing period (solid line). The BOT-type OCP contractor pays option charge ocBOT at the beginning of period but no charge for the calls up to the level h. For the calls exceeding h, QT h, the customer pays a discount amount p(1 p)max[0,QT h] or (1 p)max[0,AT B], where 0 p 1 (thick dotted line). The thick solid lines represent his total outlay schedule at the end of the billing period.
payoff of the OCP. Of the other features, the customer’s voluntary subscription (that is, making a contract) and the levying of an option charge are the two key characteristics that distinguish the OCP from other pricing schemes in telephone services—namely, standard pricing, night time discount pricing, and promotional pricing. Because it is one of the characteristics of telephone service marketing that potential users might have difficulties in judging the service quality of one company over another, the premiums on telephone calls become a major deciding factor in a customer’s choice of a telephone company. Hence, in the early stages of competition since the liberalization of the 1980s, price reduction took place in almost all telephone markets. However, such a price war could not continue for long. Clearly, price wars have a danger of draining companies of their revenue. When companies realized that further price cutting would be financially disastrous, they began developing the OCP contract as an alternative strategy. The first OCP introduced in June 1984 by AT&T in the name of the ‘‘Reach-Out America’’ became popular and a marketing success. Other competitors followed suit with their own OCP plans. Developing an attractive OCP contract emerged as one of the key factors to succeed in the telephone business. Although there are more than 100 types of OCP contracts, they can be broadly classified into three distinct groups: Block of Time (BOT) types, Volume Discount Program (VDP) types, and Minimum Usage Guarantee (MUG) types. Under the BOT-type OCP contract,2 in
2
Notable examples of BOT-type OCP include Reach-Out America (AT&T), Primer Calling Plan (MCI) in the US, Dial Coupon 001 (KDD), and Telejoyoz (NTT) in Japan.
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Fig. 2. Outlay schedule of a VDP-type OCP contractor. The solid line represents an outlay of the non-OCP contractor. The VDP-type OCP contractor pays ocVDP as an option charge at the beginning of the billing period and the amount calculated using different discount rates depending on the predetermined usage ranges at the end of the billing period. That is, if his call usage QT falls between k1 and k2, he pays k1p(1 p0)+( QT k1)p(1 p1). When the call usage falls between k2 and k3, he pays k1p(1 p0)+(k2 k1)p(1 p1)+( QT k2)p(1 p2) and so on (thick dotted lines). The thick solid lines represent his total outlay schedule at the end of the billing period.
return for an option charge (ocBOT) paid up-front, calls up to a predetermined level (h) are provided free of charge and only the calls that exceed the predetermined level are charged, but at a lower rate than that of the standard price.3 Fig. 1 shows the payment schedule. Let the call charge paid by the non-OCP subscriber over a period from 0 to T be AT, which is the standard price p times the accumulated call usage over the period QT. If we denote the discount rate by p, the call charge SBOT paid by an OCP subscriber at time T can be represented by Eq. (1): S BOT ¼ max½0,ð1 pÞðAT BÞ,
ð1Þ
where B ( = ph) is the predetermined hurdle amount over which the discount rate p is to be applied. Hence, the OCP contract gives the subscriber, at time T, a profit (or loss) of [AT (ocBOTerT + SBOT)] where r is risk-free rate of interest. Fig. 2 shows call charges under the VDP-type OCP contract.4 This contract divides a call charge into several ranges. Different discount rates are applied to each range of the call charge (i.e., the discount rate pi is applied on a charge between Ki and Ki + 1 where Ki < Ki + 1). To
3
Economists often use the term ‘‘measured service’’, ‘‘linear price’’, or ‘‘uniform price’’ instead of the term ‘‘standard price.’’ It is a tariff that is simply a product of the unit price and the call usage during a billing period. 4 The most famous VDP-type OCPs are the Wide Area Telephone Service (WATS) and the 800 number services in the US, the Members-Net of NTT, and the Freefone 0800 of BT.
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enter this contract, the subscriber pays the option charge ocVDP up-front. The call charge paid by the subscriber SVDP at time T would be (see Eq. (2) below) S VDP ¼ pk1 ð1 p0 Þ þ pðk2 k1 Þð1 p1 Þ þ pðk3 k2 Þð1 p2 Þ þ : : : þpðAT " ki1 Þð1 pi1 Þ # i X ¼ pQT pQT pi pkj ðpj pj1 Þ " ¼ AT AT pi
j¼0 i X
#
Kj ðpj pj1 Þ
if Ki < AT < Kiþ1 ,
ð2Þ
j¼0
where k0 = 0, Ki = pki, kn + 1 = 1, pi < 0 = 0, and i = 0, 1, 2, 3,. . .n. The terms in bracket are discount amount from the OCP contract, whereas the first term represents outlay under standard price schedule and is very similar to the payoff This contract Pof generalized options. gives the subscriber a profit (or loss) of erT AT pi ij¼0 Kj ðpj pj1 Þ ocVDP . The MUG-type OCP contracts5 also create tiers on a call charge as in the VDP type but impose no option charge. Rather, it requires the customer’s guarantee that his call charge for a billing period will be greater than a certain amount of G ( G = pg), and a penalty payment of P will be imposed if the call charge falls short of amount G at the end of a billing period (see Fig. 3). That is, the call charge paid by the OCP subscriber is given in Eq. (3):
S MUG
" # 8 i P > > > > < At AT pi j¼1 Kj ðpj pj1 Þ þ P " # ¼ i > P > > > : AT AT pi Kj ðpj pj1 Þ
if Ki AT < G ð3Þ if G Ki AT ,
j¼1
Pi the subscriber a profit of A p where Kn G < Ki + 1. The contract gives T i j¼0 Kj ðpj pj1 Þ Pi if G AT or a loss of P AT pi j¼0 Kj ðpj pj1 Þ if AT < G at time T. OCP contracts are applied to either ordinary or free phone calls (i.e., the 800 number service in the US). With regards to the free phone call, the receiving party pays telephone charge once a caller makes a call to a designated ‘‘free phone’’ number, whereas the telephone charge is paid for by the caller in an ordinary call. The free phone call has become an important marketing tool in many service industries that wish to solicit customer inquiries and orders.
5
In the US, the Virtual Private Network (VPN) service is designed for high-volume customers, and most carriers require a minimum volume. One of the most famous example is AT&T’s Software Designed Network (SDN).
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Fig. 3. Outlay schedule of a MUG-type OCP contractor. The solid line represents outlay of the non-OCP contractor. The call charge on the MUG-type OCP contract is similar to that of the VDP-type OCP contracts having tiered discounts. The MUG-type OCP, however, requires a guarantee that his accumulated call charge during the billing period be greater than a certain amount G ( G = gp). If the usage falls below the predetermined level g, a penalty P is imposed. Instead of the up-front option charge, the contractor pays SVDP + P if his call usage falls short of amount G and SVDP otherwise. The thick solid line represents total outlay schedule at the end of the billing period.
The purpose of this paper is to develop a methodology in evaluating a fair option charge for the OCP contract on free phone call charges by applying a contingent claims approach. The application of contingent claims approach to real-life problems has been currently received increasing attention and the analysis has been applied in a variety of contexts. Examples of these, although not exhaustive, include natural resource investments (Laughton & Jacoby, 1995), land development (Quigg, 1993), leasing (Trigeorgis, 1992), flexible manufacturing (Tannous, 1996), government subsidies and regulation (Teisberg, 1994), joint ventures (Kogut, 1991), accounting rule (Seow, 1995), and foreign investment (Kogut & Kulatilaka, 1994).6 By applying contingent claims approach to valuation of OCP contracts, the subscribers may get improved estimates of fair economic value of the contracts. It also provides a useful guideline for telephone companies in designing calling plans and assessing subscribers’ behavior. This paper is organized as follows. In Section 2, we investigate basic properties of a fair option charge in relation to existing option pricing researches. We then develop a valuation methodology that yields a fair option charge and extend that methodology to the three representative OCP contracts. In Section 3, we present some applications and discuss implications of the valuation model. Finally, Section 4 concludes the paper.
6 Chung (1993) applied the option pricing concept to sequential investment decisions. Smith and Nau (1995) compare option pricing analysis and decision analysis for valuing risky projects.
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2. Modeling OCP contracts 2.1. Distribution and preference free restrictions on the option charge To begin, let us consider a very simple OCP contract for free phone calls over a billing period from 0 to T (0 < T). Under this contract, a standard unit price p is applied up to a certain amount k and no charge is levied on the additional calls that exceed the level k in exchange for an option charge oc (see Fig. 4). It is called a three-part simple OCP in the telecommunications field, because three different prices (option charge oc, standard price p, and price 0) are applied according to the accumulated call usage level (0, from 0 to k, and more than k). If we denote the instantaneous call usage at time t (0 t T) by xt, then the instantaneous free phone call charge is qt = pxt and the call charge AT for a billing period from 0 to T is Z T AT ¼ qt dt: ð4Þ 0
Because there is no additional charge over the amount K (discount point, K = pk), the final call charge would be AT if AT < K and K otherwise; that is, min[K,AT] or AT max[0,AT K]. Notice that the second part of Eq. (4) is the same as the payoff of a call option with an exercise price K written on the uncertain call charge. Thus, subscribing to the OCP contract is identical to buying a call option from the telephone company. The option charge of the OCP contract, therefore, must be the same as the value of the call option. In a complete market, customers can duplicate the payoffs of OCPs by constructing a traded security (or a dynamic portfolio) that has the same risk characteristics (i.e., is perfectly correlated) as their free phone charge. Under the assumption that there is no arbitrage in the competitive OCP market (i.e., private agents can offer OCPs without restriction), the arbitrage relationships for European call options hold and the value of the fair option charge can be obtained by the use of the standard option pricing argument. 2.2. Valuing fair option charge of the three-part simple OCP contracts We will consider the problem of valuing the fair option charge of a three-part simple OCP contract for the free phone call. The interest rate is assumed to be a constant r and the instantaneous free phone call charge qt follows a geometric Brownian motion process generated by the following stochastic differential equation (Eq. (5)): dqt ¼ aqt dt þ qqt dZ;
qð0Þ ¼ q:
ð5Þ
The constant a reflects expected growth or decline in the level of free phone call charges. The constant q determines the instantaneous volatility of the process. The term dZ represents a standard Wiener process, with mean zero and variance dt. The free phone call charge process is correlated with the market portfolio of Merton (1973). The correlation coefficient is r.
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To this structure, we add the following assumptions for the telephone company and the economy as a whole: A1 The telephone company enjoys full observability of the underlying free phone call charge. The free phone call charge occurs exogenously, in the absence of any influence on the part of the customer.7 There are no moral hazard and no adverse selection problem. A2 The assumptions of the Cox, Ingersoll, and Ross (1985) model of intertemporal capital asset pricing hold. These include (1) existence of a single consumption/investment good, (2) existence of a set of linear production activities where changes in productivity follow exogenously specified the Wiener processes, (3) existence of a finite dimensional state variable vector whose components follow a Wiener process, (4) free entry and exit with competitive price-taking individuals and firms, (5) a common riskless borrowing and lending rate, (6) markets for contingent claims where values of claims follow Wiener processes with endogenously determined parameters, (7) a fixed number of identical individuals with homogeneous expectations and von Neumann–Morgenstern utility functions, and (8) a continuous and frictionless trading. A3 Investors have intertemporal utility functions that exhibit a constant relative risk aversion. The investment opportunity set (including the riskless rate of interest) is constant. Under the above assumptions, the fair option charge of an OCP contract O( q,A,t;K,T) must evolve according to the following partial differential equation: @O @O 1 2 2 @ 2 O @O þq þ qq rO ¼ 0, þ a*q 2 @t @A 2 @q @q
ð6Þ
subject to O( q,A,T;K,T) = max[AT K,0], O( q,1,t;K,T) = 1, and O(1,A,t;K,T) = 1, where a* = a lrq, l=(aM r)/sM. The term a* can be thought of as a certainty equivalent growth rate for call charge levels, as discussed by Constantinides (1978). The term l represents the market price of risk for the call charge level, and M is the market portfolio, which follows a geometric Brownian motion with a constant drift aM and volatility sM. A closed-form solution to Eq. (6) cannot be obtained, because AT is the sum of correlated lognormally distributed random variables and the distribution of the sum of lognormal components has no explicit representation. However, if we convert the payoff under the threepart simple OCP contract, that is, max[AT K,0], to
AT K Oðq, A, T ; K, TÞ ¼ T max ,0 , T T
ð7Þ
we can regard the problem as a continuously sampled, full period, unweighted arithmetic average rate Asian option with an exercise price K/T (Eq. (7)). Hence, we can utilize various methods developed for valuing Asian options. 7
Because the OCP contract provides substantial discount, customers may take strategic behavior (call stimulation effect) for ordinary calls. However, in the case of free phone calls, we can hardly expect that OCPs affect calling behavior.
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There are two ways to derive the average rate Asian options. Firstly, through numerical methods including the Monte Carlo approach (Kemna & Vorst, 1990), the Fourier transform approach (Carverhill & Clewlow, 1990), and the extended binomial approach (Hull & White, 1990), one can generate averages of the underlying assets and calculate the discounted average payoff of the Asian option. Although these methods provide an acceptable solution, they are time consuming. This is especially profound during an analysis of large portfolios. The second way to valuate Asian rate options is through approximations. Several authors have proposed approximations (see, for example, Levy, 1992; Ritchken, Sankarasubramanian, & Vijh, 1993; Rogers & Shi, 1995; Ruttiens, 1990; Turnbull & Wakeman, 1991), and the accuracy of which is sufficient for a practitioner’s purpose (Vorst, 1996). As we will see in Section 2.3, most OCP contracts form a portfolio of three-part simple OCPs, and the numerical procedure consumes much time with unnecessary complications. Hence, we present here an analytical approximation as a solution. This involves calculating the first two moments of the probability distribution of the sum exactly and then assuming that the distribution of the sum is lognormal with the same first two moments (Hull, 1997). Let us define (see Eqs. (8) and (9))
M1 ¼
ea*T 1 a*
ð8Þ
" # 2 2eð2a*þq ÞT 2 1 ea* : M2 ¼ þ ða* þ q2 Þð2a* þ q2 Þ a* ð2a* þ q2 Þ ða* þ q2 Þ
ð9Þ
Then, the first and second moments of the sum as seen at time 0 for a period of time T are qM1 and qM2. If we define m = 2ln(qM1) ln( q2M2)/2 and s2 = ln( q2M2) ln(qM1), then the analytical approximation of the fair option charge of an OCP contract can be expressed as follows:
Oðq, A, 0; K, T Þ ¼ erT bexpðm þ s2 =2ÞNðd1 Þ KN ðd2 Þc,
ð10Þ
where d1 = b ln(K) + m + s2c/s, d2 = d1 s, N( ) = standard cumulative normal distribution. From Eq. (10), we are able to conjecture that the option charge increases with the present call charge q, summing period T and volatility q, because they offer more chance to exercise. However, contrary to the stock option, an increase in interest rates decreases the option charge. The option charge also decreases as the discount point increases. Eq. (10) is an analytical approximation of the fair option charge for three-part simple OCP contracts when a subscriber receives a 100% discount on call charges above K. More generally, if the discount rate is expressed by p (where 0 p 1), because it is identical to a
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long position of p units of three-part simple OCP contracts, the valuation Eq. (10) can be adjusted as follows: Oðq, A, 0; K, T , pÞ ¼ perT bexpðm þ s2 =2ÞNðd1 Þ KN ðd2 Þc:
ð11Þ
2.3. Valuing the fair option charge of three representative OCP contracts To this point, we have been exclusively concerned with a very simple OCP contract, which can be used as a building block for constructing more general OCP contracts. If there are no arbitrage opportunities, and if a payoff function of the OCP contract provided by telephone companies at the end of the billing period is equal to that of the portfolio constructed by threepart simple OCPs, then the two fair option charges must be equal. This means that the valuation methodology for the three-part simple OCP contract in Section 2.2 could be used for the valuation of the more general OCP contracts, which can be found in a telephone market. Here, we extend the methodology in Section 2.2 to three popular variants of the OCPs. 2.3.1. The BOT-type OCP contracts As shown in Section 1, the call charge paid by the subscriber of the BOT-type OCP contract is SBOT = max[0,(1 p)(AT B)]. Because the call charge of the nonsubscriber is AT, the total amount of discounts that the contract subscriber is entitled to is DB ¼ AT ð1 pÞmax½AT B,0:
ð12Þ
This payoff from the contract DB can be exactly duplicated by an appropriately chosen portfolio of three-part simple OCP contracts. It is equal to the payoff from a portfolio composed of one unit of three-part simple OCP contracts with a discount point of 0 and (p 1) units of contracts with a discount point of B. As a result, the fair option charge of BOT-type OCPs can be expressed as OB ¼ Oðq, A, 0; 0, T Þ þ ðp 1ÞOðq, A, 0; B, TÞ:
ð13Þ
2.3.2. The VDP-type OCP contracts The OCP contract that is most frequently used for the free phone call is the VDP-type OCP. As shown in Fig. 3, this contract cuts off the price with a different discount rate for each range of call charges under the condition that the customer pays the option charge. In other words, under the VDP-type OCP contract, the discount rate pi is applied to the free phone call charge between Ki and Ki + 1 (where Ki + 1 > Ki, i = 0, 1, 2, 3,. . .,n, n < 1) and pn is applied to all charges above Kn. Hence, a telephone company’s reimbursement function of the OCP contract is expressed by DV ¼ p0 AT þ ðp1 p0 Þmax½AT K1 ,0 þ : : : þ ðpn pn1 Þmax½AT Kn ,0:
ð14Þ
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Eq. (14) is exactly duplicated by three-part simple OCPs, which comprise (pi + 1 pi) units of OCPs that have discount point Ki. Hence, the fair option charge of the VDP-type OCP contract OV is OV ¼ p0 Oðq, A, 0; 0, T Þ þ ðp1 p0 ÞOðq, A, 0; K1 , T Þ þ : : : þ ðpn pn1 ÞOðq, A, 0; Kn , TÞ:
ð15Þ
Eq. (15) indicates that the BOT-type OCP contracts can be considered as a special case of VDP-type OCPs with p0 = 1, p1 = p and K0 = 0, K1 = B. 2.3.3. The MUG-type OCP contracts The MUG-type OCP contract does not levy an option charge to a customer. Rather, it requires a customer’s guarantee that his call charge during a billing period will not be less than G. When a customer violates his promise, he should pay a predetermined penalty P at the end of the billing period.8 Here, we deal with the problem of determining the fair penalty P when the customer guarantees that his free phone call charge will be more than G, and the discount structure (pi, Ki, i = 0, 1, 2, 3,. . .,n) is determined. The MUG-type OCP contract can be regarded as a portfolio constituent by buying a VDPtype OCP contract and selling a continuously sampled, full period, unweighted arithmetic average rate Asian cash-or-nothing put with an exercise price G/T and an underlying asset AT. If there is no arbitrage in the competitive OCP contract market, the cost of constructing this portfolio has to be zero. The cash-or-nothing put would be like selling a three-part simple OCP except that, although the holder receives the penalty amount, he is under no obligation to give the underlying asset. From Eq. (13), such a partial put option CP would be worth CP ¼ erT PN ðd2 Þ,
ð16Þ
where d2=[ ln( G) + m]/s, to the holder. Therefore, the fair penalty P for a MUG-type OCP contract is given by P¼
8
OV : erT ½Nðd2 Þ
ð17Þ
The MUG-type OCP has been offered as a mixture of VDP-type OCPs on 800 or WATS in the US. Under the OCP, especially under the MUG-type OCPs, the subscribers reveal some information about their demands before the purchase decision. Some telephone companies argue that this information can be valuable to improve their production decision, and OCP’s price discounts include this effect. However, as noted by Mitchell and Vogelsang (1991), ‘‘This information would be potentially valuable when the supplier has to commit capacity before the consumer makes her actual purchasing decision. Because it would be inappropriate to bind a subscriber to a tariff for too long, it is not clear that it plays an important role for telephone carriers.’’ Therefore, in this paper, we do not consider such an information effect.
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Fig. 4. Outlay schedule and profit function of three-part simple OCP contractor. (a) The outlay of a three-part simple OCP contract where K is the hurdle amount over which a flat fee is charged. (b) The profit from the threepart simple OCP contract (AT K).
We have evaluated the fair option charges of three representative types of OCP contracts (Fig. 4), which we developed in Section 2.2. The methodology also can be used to determine the fair option charge of many other types of OCP contracts.
3. Applications and implications When the discount mechanism, the customer’s instantaneous free phone charge, and the volatility are given, applying the methodology developed in Section 2 to the valuation of actual OCP contracts is straightforward.9 However, there are few OCPs in the telephone market where the discount structure and the option charge vary for their subscribers. This lack of customer-tailored OCP contracts stems from the fact that telephone prices are usually regulated by regulatory bodies, most of which prohibit providing such OCPs.10 As a result, telephone companies have no choice but to design several kinds of OCP contracts. Customers are faced with the problem of choosing the most valuable OCP among a multitude of others. For telephone companies, forecasting the take rates has emerged as one of their main tasks. The standard theoretical approach employed by the regulatory economists has been based on the premise that customers will subscribe to the OCP services when their customer’s surplus exceeds the subscription price, as noted by Faulhaber and Panzar (1977). However, demand estimation and welfare analysis for telecommunications services have often been plagued by apparent inconsistencies between actual consumer behavior and standard 9
It is easy to estimate the variables a, q, and qt, because customers as well as telephone companies usually keep call records for several months for the purpose of preparing for charge-related conflicts. 10 An exception is that in the US, prices and accounting procedures can be tailored to customer preferences under the ‘‘competitive necessity’’ doctrine, Tariff nos. 12 and 15.
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Table 1 Price schedule for Freefone 0800 of British Telecoma Price and discount rates for accumulated usageb Types of option
Monthly rental (£)
Cumulative telephone usage
Pricec (discount rate, %)
Option 100 Option 105 Option 110
Nothing 33.33 333.33
Option 115
2000.00
Option 120
3333.33
– – 0 – 30,000 min 30,000 + 0 – 30,000 30,001 – 100,000 100,000 + 0 – 100,000 100,000 – 1,000,000 1,000,000 +
0.1055 0.0900 0.0792 0.0684 0.0792 0.0684 0.0576 0.0684 0.0576 0.0540
(0.00) (14.69) (24.93) (35.17) (24.93) (35.17) (45.30) (35.17) (45.40) (48.82)
British Telecom put five price schedules (OCPs) for their Freefone 0800 customers: Options 100, 105, 110, 115, and 120. In Option 100, which is the standard price of Freefone 0800, the customers pay no monthly rental (i.e., option charge) but pays 10.55 pence for each of usage minutes (blanket in right-hand side). For the rest of the price schedules, different monthly rental rates are levied at the beginning of the month and different discount rates are applied as the call usage increases. For example, the Option 115 levies £2000 a month as a monthly rental. However, customers benefit a 24.93% discount (0.0792 per minute) up to 30,000 monthly accumulate usage minutes, a 35.17% discount (0.0684 per minute) for the additional 70,000 minutes and a 45.30% discount (0.0576 per minute) for the call usage over 100,000 minutes. Source: Thairs (1997). a Tariffs valid from 1 April 1997. We modified the text of the original price table. First, each option in the original table discounts every monthly call charge for 3 months under the condition that the customer pays the option charge. For example, in Option 115, the monthly call charge is discounted by 24.93%, 35.17%, and 45.4% for ranges £0 – 3165, £3165 – 10,550, and over £10,550 for 3 months in return for an additional 6000 option charge. The fair option charge in these cases could also be easily derived from the result of Section 2. To secure simplicity, however, we assumed OCP as monthly OCP with the option charge £2000. Second, we adjust one minor fact. In Option 100, telephone price is 9.0 pence and it is charged on a per-minute basis, but in the another option, it is charged on a per-second basis. It means that the telephone company charges 9.0 pence under Option 100 and (9.0/60) 10 pence under Option 105 when a user made a 10-second call. If we assume that the conversation time per call follows an exponential distribution and the average conversation time per call is 180 seconds, then the average charge per call at 9.0 pence/minute under the per-minute charging system is 1{1/ [1 exp( 60/180)]}, and under the per-second charging system, the average charge is 1{1/[1 exp( 1/180)]}. That is, the price 9.0 pence/minute under the per-minute charging system equals 10.58 in per-second charging system (the price in the per-minute charging system is 14.72% higher than in the per-second charging system). To account for such a difference in charging systems among options, we treat the price in Option 100 as 10.58 pence instead of 9.0 pence. b Daytime call price per minute. c Number in blanket is per minute.
economic theory.11 Train, Ben-Akiva, and Atherton (1989) and subsequent researches have shown that a substantial fraction of customers does not correctly anticipate their own demands, yield ex post choices that do not minimize their expenditures for the quantities 11
For example, ‘‘Reach-Out America’’ shows that 45% of customers who choose an OCP have low monthly calling volumes and incur a somewhat higher average monthly bill than they would pay for the same use under the standard price.
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purchased. As a result, economists realized the expected customer’s surplus is generally not an adequate basis for subscription decision. Kridel, Lehman, and Weisman (1993) emphasized that customers exhibit substantial risk aversion when faced with bill uncertainty and pay attention to the customer’s attitudes toward risks and means available for reducing risks. The other economists protest the risk aversion arguments (Clay, Sibley, & Srinagesh, 1992). Our research provides one solution of the debates. It is clear that the uncertainty of future call charges generates option values and the financial contingent claims approach provide an option value (i.e., fair option charge), which is independent of customers risk preference. Consideration of option values requires modification of the existing framework for analyzing customer subscription decisions. Let us examine an actual OCP. Table 1 illustrates the ‘‘Freefone 0800’’ of British Telecom in the United Kingdom. If we assume Option 100 is a standard price, Options 105, 110, 115, and 120 can be considered VDP-type OCP contracts with expiration date 1/12 (1 month). That is, in the case of Option Table 2 Value of Option 115 of Freefone 0800 Fair option charge
Call charge ( q)
Expected call charge
CCA (A)
60,000 62,400 64,800 67,200 69,600 72,000 74,400 76,800 79,200 81,600 84,000 86,400 88,800 91,200 93,600 96,000 98,400 100,800
5030.3 5231.5 5432.8 5634.0 5835.2 6036.4 6237.6 6438.8 6640.0 6841.2 7042.5 7243.7 7444.9 7646.1 7847.3 8048.5 8249.7 8451.0
1446.6 1517.4 1588.1 1658.8 1729.5 1800.2 1870.9 1941.7 2012.4 2083.1 2153.8 2224.5 2295.3 2366.0 2436.7 2507.5 2578.2 2649.0
Profit from subscription
STA (B)
Option charge (C)
(A – C)
(B – C)
1433.0 1503.1 1573.3 1643.5 1713.7 1783.8 1854.0 1924.2 1994.3 2064.5 2134.7 2204.9 2275.0 2345.2 2415.4 2485.5 2555.7 2625.9
2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000
553.4 482.6 411.9 341.2 270.5 199.8 129.1 58.3 12.4 83.1 153.8 224.5 295.3 366.0 436.7 507.5 578.2 649.0
567.0 496.9 426.7 356.5 286.3 216.2 146.0 75.8 5.8 64.5 134.7 204.9 275.0 345.2 415.4 485.5 555.7 625.9
The first column represents charge per year. Expected call charge during a month (second column R ð30=360Þthe current call of left-hand side) is 0 qtdt ¼ ½qað30=360Þ 1=a under the assumption that the instantaneous free phone call charge follows a geometric Brownian motion process. The fair option charges are calculated using the contingent claims approach (CCA), the Eq. (12) in the main text, and the standard theoretical approach (STA), which is based on the expected call charge during a month. For example, if a customer’s present call charge is 79,200, his fair option charge under CCA is £2012.4 but is £1994.3 under the standard theoretical approach (1994.3 = 3165 0.2493+(6640 3165) 0.3517). The telephone company, British Telecom, sets £2000 as an option charge. The balance of subscription of Option 115 is £12.4 under CCA but is £ 5.8 under STA. The default parameter values are expected growth rate = 0.07, risk-free interest rate = 0.1, volatility = 0.5, lr = 0.15, and maturity of contract = 30 days.
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115, by paying £2000 of monthly option charges, the subscriber gets a discount of 24.93% for a call charge ranging from £0 to £3165, 35.17% for charges ranging from £3165 to £10,550, and 45.40% for amounts over £10,550. The fair option charge of each option could easily be derived from Eqs. (11) and (12) in Section 2. Customers will choose the most valuable option for them by comparing the option charges that the telephone company offers with the fair value, which is based on their present call charge and volatility. That is, customers will subscribe to the OCP when the profit from the subscription (that is, the fair option charge less the option charge imposed by the telephone company) is positive. Table 2 shows the profits of the subscription of Option 115 in a selected range of call charges when other parameters are given. As we can see in the table, if a customer’s present call charge is £79,200 a year, and he has to choose one between Option 100 and Option 115, subscribing to Option 115 is profitable. However, based on the standard theoretical approach, it could be regarded as an incorrect choice. This conflicting result for a customer’s subscription behavior stems from option values in the OCP contract. A contingent claims approach quantifies the value of option, whereas the standard theoretical approach could not. Table 2, which focuses on Option 115, also shows a lack of consideration for risks in taking the OCP contract leads to a consistent undervaluation of the customer’s benefit from subscribing to the OCP contract and to an underestimation of subscription demand.
4. Concluding remarks The application of the option pricing model to real-life problems is currently receiving increased attention, and the model has been applied in a variety of contexts. We applied the option pricing methodology to the valuation problem of OCP contracts in the telephone industry. We show that the payoff of a three-part simple OCP contract is identical to that of a call option, and option pricing theory can provide useful guidelines for OCP designers. Using the three-part simple OCP contracts as a building block, we developed a valuation model for the fair option charge of general OCPs found in the actual market. When customers choose a tariff among OCPs, because regulatory bodies prohibit providing custom tailored OCPs, the valuation model can be used as a tool in making subscription decisions by potential customers and in predicting take rates by a telephone company. We also showed that the standard theoretical approach consistently undervalues the OCP contract. The valuation results explored in this paper could be extended to ordinary call OCP contracts as well as to optional pricing schemes in other industries, such as gas or electricity, where the customer’s price elasticity is very small. The option concept used in this industry can be utilized in other competitive environment as well. Its use, however, has been limited to goods and services that are not dissimilar such as the telephone services, gas, and electricity. The goods and services that can be differentiated easily by other means have not been given much attention. The concept and the methodology explored in this paper would be quite useful where there is an uncertainty and
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the need for packaging and transferring the risk from customers to suppliers or vice versa. The option pricing scheme is an effective means to alter the exposure and get the fair values of such a design.
Acknowledgments We thank Jae Chul Kim, Thomas A. Fetherston, and an anomymous referee for helpful comments, discussions, and insights.
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