A Comparative Fuzzy Real Options Valuation Model ...
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A Comparative Fuzzy Real Options Valuation Model using Trinomial Lattice and Black-Scholes Approaches: A Call Center Application A. ÇAGRITOLGA',CENGIZKAHRAMAN^ AND MURAT LÈVENTDEMIRCAN' ' Galaiasdray Universiry. Industrial Engineering Department. 34357 Ortakoy Istanbul Turkey F-mail: [email protected] Istanbul Technical Universiry. Industrial Engineering Depannienl. 34367 Macka Istanbul Turkey Received: March 25. 2009. Accepted: Juiy 29. 2009.
Valuation of Ihe investment projects is very serious in every dimen.sion. Convenlional discounted ca.shflowtechniques are usually insufficient since these techniques faii to account tor the (lexibility in business decisions and violations occur as a result of the existing uncertainty in projects. Real option valuation methods overcome this problem with its efficient and flexible nature. Financial option valuation methods are applied into real options area with little changes in variable definitions. When there is a lack of data or involuntary companies about giving their financial data, fuzzy numbers can be used to capture this vagueness. In this study, both fuzzy Black-Scholes and fuzzy trinomial lattice models are examined. These two fuzzy models are compared with each other for the first time in this paper. Differently from the previous works, tbe parameler dividend yield is added into tbe fuzzy trinomial lattice model. They botb are applied to a call center investment project. A comparison between these methods is made, and then a sensitivity analysis is discussed. Keywords: Real optiotis, Black-Scholes tnethod, Erinomiai lattice method, fuzzy.
I INTRODUCTION An optioti is defined as the right, but not the obligaiion, to buy (if a call) or sell (if a put) a nominated asset by paying a preset price on or before a specified date. Real options are based on financial options. However, the nature of real options involves permanent, fixed or immovable assets. In contrast to
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financial options, real options are not tradable e.g. the factory owner cannot sell the right to extend his factory to another party, he can only make this decision. The valuation of real options necessitates a real option analysis (ROA). The key advantage and value of real option analysis is to integrate managerial flexibility into the valuation process and thereby assist in making the best decisions [I]. Real options give a right but not an obligation to make or not to make an investment for a certain period. For instance, investing in the expansion of a firm's factory gives the company right to produce more but not the obligation. There are three valuation methods for real options; /. Partial differential equations (the most famous one is Black-Scholes formula), ii. Simulations (Monte Carlo), Hi. Lattices (binomial, trinomial etc.). Partial differential equation methods necessitate the solution of a partial differential equation with specified boundary conditions (i.e., option values at specified periods, type of option, etc.). Black-Scholes [2] offered their valuation method based on partial differential equations in 1973. Boyle [3] proposed Monte Carlo simulation approach for financial options built on the insight that whatever the distribution of stock value would be at the time the option expired, that distribution was determined by operations that driven the activities of the asset value between now and the expiration date. Cox et al. [4] developed a simplified option pricing model (the binomial options pricing model) based on a discrete-time approach. Black-Schoies approach is known as "black box" because of its mathematical complexity. Black-Scholes assumes a lognormal distribution of the underlying asset value, which may not be applicable for most real assets because of the cash flows" nature. During real option's life, there could be also more than one strike price but Black-Scholes assumes only one. Many complex models were developed against these deficiencies of Black-Schoies formula. In spite of complexity, easy to apply property is the advantageous and the most preferable side of Black-Schoies approach. Simulation approach is more easily applicable to European options. Nevertheless, when someone needs to deal with the sequential options, he/she has to run more than 1000 simulations. In the binomial lattice approach to real option valuations, input parameters such as the strike price and volatility can be changed easily over the option life. The binomial lattice model is easy to understand and elementary mathematics is sufficient. It uses probability distribution instead of volatility estimation. However, if multiple options and their interactions are needed to consider, it can be very cumbersome to construct the binomial asset tree. While Black-Scholes finds the most accurate option value, the binomial lattice method approximates to it. In daily life real circumstances are very often uncertain and vague in several ways. When there is a lack of information or no voluntary financial data out the company, a system might not be known completely. Zadeh [5] suggested a strict mathematical outline named fuzzy set theory that overcomes these inadequacies. Many engineering and decision prohlems implemented classes
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of groupings of data with boundaries, which are not explained exactly, can be simplified by the fuzzy set theory. The fuzzy approach to real option valuation (ROV) was first studied by Carlsson and Fuller [6]. Tbis work was based on Black-Scholes' real option valuation, but under fuzziness. Then, Wang and Hwang |7] offered fuzzy compound options for R&D project selection based on the Black-Scholes real option valuation. Tolga and Kahraman |7] offered a model that integrates this fuzzy Black-Scholes real option valuation and fuzzy AHP for selection among R&D projects. Another fuzzy real option valuation based on Cox et ¿i/.'s 14] method was investigated by Allenotor and Thulasiram |9]. They modeled pricing of grid/distributed computing resources as a problem of real option pricing. In this study, we will analyze the valuation of a call center with a fuzzy real option valuation model based on Black-Scholes (BS) and Trinomial Lattice (TL) individually, and then we will compare them. And at the last a sensitivity analysis for dividend yield and risk free interest rate will be given. The rest of this paper is organized as follows: At first, theoretical information on fuzzy Black-Scholes ROV and fuzzy trinomial lattice ROV will be given, respectively. Then, a call center application with real data in Turkey will be given. The following section will offer comparative analysis between fuzzy BlackScholes ROV and fuzzy trinomial lattice ROV with a sensitivity analysis. Conclusions and further study suggestions will come up at the last.
2 FUZZY BLACK-SCHOLES REAL OPTIONS VALUATION Merton [10] extended the Black-Scholes financial options valuation formula to dividends paying stocks but Carlsson and Fuller [6] applied that formula to real option valuation using Leslie and Michaels' 111 ] computation as follows:
ROV = Soe'^'^Nidi) -Xe~''^Nid2)
(1)
where
7 7 f •
^'^ (3)
where ROV denotes the real option value. So is the present value of expected cash flows, X is the (nominal) value of stationary costs, cr points out the uncertainly of expected cash flows, r quantifies the annualized continuously compounded rate on a safe asset, and 8 illustrates the value lost over the duration of the option.
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HGUREI A trapezoidal fuzzy number. À = (a\. ai. ay. i/4).
In case of incomplete data or when it is difficult to take out exact financial data out the company, fuzzy sets are very helpful. A fuzzy set is a special fuzzy set G — {.r e R\ßcix)], where x takes its values on the real line R\ : —00 < J: < +00 and fMcix) is a continuous mapping from R] to the close interval [0, 1 ]. A trapezoidal fuzzy number can be denoted as A = (íii.íi2. ^'3- "4)- Í'I £ "2 5 03 < «4 and traced as in Figure 1. Its membership f u n c l i o n ß ^ i x ) : R ^>^ [0, l ] \ s e q u a l t o X — a\ X € [ « I , «2
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In addition, the possibilistic mean value and variance of fuzzy number À are calculated from Equations (9) and (10) [61:
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- Xe''^'N{d2)
(11)
where
,
\niEiSo)/E(X))
(12)
and where So denotes the possible values of the present value of expected cash flows, in a similar manner X quantifies the possible values of investment cost., EiSo) is the possibilistic mean value of the present value of expected cash flows, EiX) stands for the possibilistic mean value of expected costs, a i So) denotes the possibilistic variance of the present value of the expected cash flows, and d2 is given by Equation (3). In options, dividend yield reduces the value of the option at the paid-out dividend times. The analogy between options and real options starts here. In real options, competitors could come into the market when applying a new project, and this event reduces the value of the real option as in financial options. Dividend yield rate is carried out from dividing the estimated cost of waiting to the present value of cash flows as below (in fuzzy type problem, the expected value (£(So)) should take the place of (5o)): S = thecost of waiting/£(So) (13) Sensitivity analysis to the dividend yield rate will be made in the fifth section.
3 FUZZY REAL OPTIONS VALUATION BY TRINOMIAL TREE METHOD Binomial tree method was developed by Cox et al. [4] for pricing the options simply. Later trinomial tree method was developed for more complex situations.
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\ FIGURE 2 Trinomial tree model with dividend yield.
Trinomial tree method provides much better approximation to the continuous time process than the binomial lattice for the same number of steps. The trinomial tree is easier to work with because of its more regular grid and is more flexible, allowing relatively easy extension to time-varying drift and volatility parameters [12]. In this section, we first present general form of the trinomial tree model as illu.strated in Clewlow and Strickland [12]. Then, the fuzzy forms of these representations will he developed for the case of incomplete data or vagueness. The trinomial tree approach is equivalent to the explicit finite difference method (introduced by [ 13]) and it is illustrated in Figure 2. For being intelligibility, notations in Section 2 is used in this section without changing anything. Suppose that pu, p^, and p¿ are the probabilities of up, middle, and down movements at each node and Ar is the length of the time step. Dividend paying stocks" real option valuation techniques for binomial lattices are evaluated hy Brandao and Dyer [14]. For a dividend paying stock in trinomial lattice, parameter values that match the mean and standard deviation of price changes
FUZZY REAL OPTIONS VALUATION MODEL
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are as follows: I
, (14) (15) (16) (17)
Ax / a^At + li-^Ar
2 V
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/
Computations for a trinomial tree lattice are similar to those for a binomial tree. At time zero, the present value of expected cash flows, 5o is known. At time A/, there are three possible 5» values, SQU. SQ. and S{)d\ at time 2Ai, there are five possible 5o values, SQU^. SOU, SO, Sod, and Sod^; and so on. In general, at time i At, 2i + 15o values are considered. These are: S¡j=So{\-8)u^.j^-i....,i
(21)
Notice that the u = \ -=r d relationship is used in computing the So value at each node of the tree in Figure 2. For example, Sou~^ — 5ÜÍ/". Real option valuations by lattice methods are evaluated by starting at the end of the tree {time T) and working backward. The value of the real option is known at time T. For being intelligibility, notations in Section 2 is used in this section without changing anything. Let us express the approach algebraical ly. Because the value of the real option at its expiration date is max(O, S¡j — X), we know that CT,j = max(O, STJ - X), j =-T
0
T
(22)
After computing the crj values, (no early exercise) Cjj values at each node could be calculated by the formula below: Ci,i = e^''^'[puCi+\j+i -\- p,„Ci+]j -fprfc-í+i.j-i]
(23)
for O < í < 7" — I and —/ < j < i. When early exercise is considered, this value for c, j must be compared with the option's intrinsic value, and we obtain: = max ^ [ p u i + \ j + \
i PrnQ+ij + P d i + \ j i l
Q
(24)
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In case of incomplete data or vagueness, all the calculations are same, but So (the present value of expected cash fiows). and X (the nominal value of stationary costs) will be joined to these formulae in a fuzzy form ÍSQ, X). In this way, the Equation 21 turns into Equation 25: j 5 ) u J , j
= -i,...,i
(25)
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0
T
(26)
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(27)
for 0 < / < r — 1 and —/ < j < i. And in early exercise: c, J = m a x
., „ . , ..^
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, _ . , ..J
(28)
4 A CALL CENTER APPLICATION The objective ofthe project is the design, development, deployment and operation of a top-up system for GSM operators that will enable their subscribers to recharge their credits via USSD, SMS, IVR. web and call center channels. When a registered subscriber needs to top-up credit, s^e will be able to reach any of the 'top-up with pin' channels such as USSD, SMS, IVR, web and call center. The top-up process is completed instantly upon request. The application given will aim to design, implement and manage the call center process perspective only. In this process, the subscriber registers to the system via call center. During the registration process, the credit card data as well as necessary personal information is gathered from the subscriber and later controlled and validated for fraud protection purposes. The subscriber is then ready to use the top-up system. The call center will bring the following benefits to GSM operators and their subscribers, the followings: • Customer satisfaction will be ensured by easy and fast access to the product. • Subscribers will be able to charge their credits virtually, therefore there will be no physical card preparation and distribution cost for GSM operators.
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• Convenient sales channels and a well trained sales team will increase the ARPU (Average Revenue Per Unit) for GSM operators. • GSM Operators will obtain the ability to interact with the subscribers and manage the customer experience. • Subscribers will be able to reach the product 24 hours a day, 7 days a week. The call center has two financial aspects. The first one is initial investment which includes capital expenditures (CAPEX); the second one is generated cash flows which include operating expenditures (OPEX). labor costs and revenues. In Table I, you will find the capital expenditures of the overall call center establishment costs which we call initial investment. The costs are extracted from different proposals from different outsourcing companies. For instance for Phone Infrastructure and GSM gateway we received 6 different proposals from 4 different providers. Then the financial figures are derived from the proposals as their two minimums (a\ and«?) and two maximums {CIT, andíí4}. The total capital expenditure is calculated {205155, 210443. 212558, 217845) YTL as a trapezoidal fuzzy number. In order to calculate the expected cash flows, we, first, calculated the cost aspect which includes the labor cost and operating expenditures (OPEX). Operating expenditures and labor costs are shown in Table 2 and in Table 3, respectively. Fuzziness of both financial costs is explained as the changes and variations of bonus, augmentation, personnel turn over, etc. Since we have a 5 year investment planning horizon with 25 working agents, we calculated the present value of costs (331217072, 3397535.94, 3431682.03, 3517047.25) YTL. We assume that over 5 years, the call center will generate 5500 transactions per month by 18 YTL gross revenue per transaction. The present value of gross revenue as calculated as (3587141.40, 3679593.51, 3716574.35, 3809026.45) YTL. In order to calculate the present value figures we used discrete monthly discount rate as % 1.708. The final financial table is shown in Table 4.
5 COMPARATIVE ANALYSIS In this part, fuzzy real option valuation of the call center with each BlackScholes and Trinomial method separately will be made. Then benchmarking is going to be analyzed. After these, sensitivity analysis of the annualized continuously compounded rate on a safe asset, and the value lost over the duration of the option will be framed. For the annualized continuously compounded rate on a safe asset, five year Treasury bond interest rate belonging Central Bank of the Republic of Turkey
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Communication Infrastructure
Cost values (in YTL)
Phone Infrastructure GSM Gateway Voice Recording Infrastructure
(48298. 49304, 51317, 52323) (20736, 21168. 22032. 22464) (19008. 19404,20196.20592)
User Hardware Desktop PC Handheld Device Headphone Wired System
(17280, 17640, 18360, 18720) (3456. 3528, 3672, 3744) (1728, 1764, 1836, 1872) (3456,3528, 3672, 3744)
System Hardware Voice Recording Backup Unit CRM Server Admin Server Firewall System Cabling
(2592. 2646, 2754, 2808) (2592, 2646, 2754, 2808) (2592, 2646. 2754, 2808) (5184,5292,5508,5616) (8640,8820,9180,9360)
Furnishings Agent Tables Management Tables Seats Other Furnishings
(4320,4410,4590.4680) (173, 177, 184, 188) (2592, 2646, 2754, 2808) (3370,3440,3581,3651)
Software & Other CRM Software HR Cost
(51840, 52920, 55080, 56160) (5184,5292,5508,5616)
Total
(205155, 210443, 212558, 217845)
TABLE I
Capital Expenditure (Initial Investment) figures for the Call Center E.stablishment (Turkish Lira. YTL) as trapezoidal fuzzy numbers
will be utilized which is 20.50%, the average of July 2008. We have to retain that this value contains deep volatility. For the value lost over the duration of the option, the estimated i5(= 10.50%) value will be used. The value lost over the duration of the option means the entrance of the competitors to the market, so at this time in Turkey, there are many competitors in the market of call center. That is why we take the value in this manner, however in the latest days; there may be bankruptcies or new foundations. We assume S is the total dividend yield associated with all ex-dividend dates between time zero and time iSt. So because of these reasons, sensitivity analyses on competitor entrance and on risk free rate have to be made.
FUZZY REAL Oprttws VALUATION MODEL
General Management Call Center Manager Team Uader Back Office Manager Back Office Staff Call Quality Specialist Helpdesk Staff Stationary HR Staff Finance & Accounting Staff Office Rent Energy and Other Costs System and Technical Maintenance Software Updates and Upgrades Other Total
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(1964.56, 2015.20, 2035.45, 2086.08) (1414.49. 1450.94, 1465.52, 1501.98) (2946.85, 3022.80, 3053.18. 3129.13) (1414.49. 1450.94. 1465.52, 1501.98) (10372.90, 10640.25, 10747.18, 11014.53) (1728.82, 1773.37, 1791.20, 1835.75) (1728.82, 1773.37, 1791.20. 1835.75) (1100.16, 1128.51, 1139.85, 1168.21) (589.37. 604.56. 610.64, 625.83) (589,37, 604.56, 610.64, 625.83) (2750.39, 2821.28, 2849.63, 2920.52) (1178.74. 1209.12, 1221.27, 1251.65) (982.28. 1007.60, 1017.73. 1043.04) (982.28, 1007,60. 1017.73. 1043.04) (216.10,221.67,223.90.229.47) (29959.61. 30731.77. 31040.63. 31812.79)
TABLE 2 Motiihly Operating Expenditure figures (Turkish Lira YTL) as trapezoidal fuzzy numbers
Gross Salary Social Insurance (Employer Fee) Social Insurance (Employer Fee) Revenue Tax Premium Revenue Tax Fee Other Tax Premium Base Salary Bonus Other Total NetSalaiy Total Gross Salary
(1800.36. 1846.76. 1865.32, 1911.73) (387.63, 397.62, 401.62. 411.61) (269.97, 276.93, 279.72, 286.67) (1530.39, 1569.83, 1585.61, 1625.05) (229.68, 235.60, 237.97, 243.89) (11.28, 11.57, 11.69. 11.98) (483.54. 496.00. 500.98, 513.44) (805.89, 826.66, 834.97, 855.74) (161.18, 165.33, 166.99, 171.15) (1289.43, 1322.66. 1335.95, 1369.19) (2348.37, 2408.90, 2433.11, 2493.63)
TABLE 3 Monthly Labor Cost per Agentfigures(YTL) as trapezoidal fuzzy numbers
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Gross Revenue Cost {So) Initial Investment {X) Net Cash Flow (SQ - X)
(3587141.4 L 3679593.51. 3716574.35. 3809026.45) (3312170.72. 3397535.94, 3431682.03, 3517047.25) (70088.30,247890.76,319011.74,496814.20) (205155. 210.443, 212558, 217845) (-147756.70, 35333.26, 108569.24. 291659.20)
TABLE 4 Present value offinancialfigures(YTL) as trapezoidal fuzzy numbers
As given in Section 4 and above, initial values are listed below: So = (70088.30, 247890.76, 319011.74,496814.20) YTL, X = (205155, 210443, 212558, 217845) YTL, T =5 years, r = 20.50%, S = 10.50%. 5.1 Fuzzy Black-Scholes computation For FROV by Black-Scholes method, one can easily compute £(5o), E{X) and (T values from Equations (8) and (9), i.e. 283451 YTL; 211500 YTL; 36.6% successively. Also, from Equations (11) and (12), d\ and d2 values could be found 1.2658 and 0.4480 respectively. At the last, F R O V B S = (-10400.19, 79870.59, 115978.90, 206249.67) value can be computed from Equation ( 10). 5.2 Fuzzy Trinomial lattice computation For FROV by Trinomial lattice method, at first step, i», AJ:, U, and d values have to be calculated from Equations (14-17); and those could be found 0.033, 0.634, 1.884.0.53! respectively. Then, the probabilities of up. middle, and down movements for each node are calculated by Equations (18-20) at the second step: Pu - 0 . 1 9 4 ,
p„, =0.664,
p,¡ ^0.142.
Aii calculated Si_j values at each node are illustrated in Appendix A. For example, 55.5 and S5._5 could be found as below: 55.5 = (1489831.11,5269287.16,6781069.58, 10560525.63) 55,-5 - (2641.19.9341.46. 12021.57. 18721.84) After finding these eleven ends of the period values. Equation (26) have to be computed to find the values of the fuzzy real option values at its expiration
FUZZY REAL OPTIONS VALUATION MODEL
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date. Application of Equation (26) is not easy in fuzzy terms, so we have to defuzzify all values by centroid method to compare these values by zero. At the following rows, there are some calculated FROVs at its expiration date: C5.5 = (1271986.11,5056729.66,6570627.08, 10355370.63), C5,o = (-155115.97.9304.73.75073.01.239493.71), C5.-5 = (0.000,0.000, 0.000. 0.000). At the next step, all ¿-¡j values at each node should be calculated by the Equation (27) with backward induction. Finally, ro.o value - shown below will be carried out which means FROV by Trinomial lattice method: FROV-TL ^ (-22602.70,65922.50, 101332.58, 189857.78) The defuzzified values for FROVßs and F R O V J L are calculated as 97924.74 and 83627.54, respectively. Trinomial lattice model gives a lower value than Black-Scholes model in every combination of parameters, as seen in Tables Bl, B2, and B3. Trinomial lattice model can be said to be a more pessimistic approach relative to Black-Scholes model. If we survey to the NPV as computed in section 4. this value underestimates to both F R O V B S and F R O V J L - We can make this comparison by Lee and Li's method [15], 5.3 Sensitivity analysis As we mentioned before, volatility in the annualized continuously compounded rate on a safe asset and competitor entrance overextend us to make sensitivity analyses on these subjects. The experts guessed that in five year period there may be one entrance to the market. Estimation of the waiting cost is very difficult; but as a result of consultancy with the experts in the firm, 5952.47 YTL value lost per five year period is reached. So from Equation ( 13), one can easily obtain dividend yield rate as: ¿ = (5) X 5952.47/283451.25 = 10.50% It is thought thatfiveyear Treasury bond interest rate belonging Central Bank of the Republic of Turkey could change from 15.5% to 25.00 percent spectrum by 50 base points. Though there are few competitors in the market, S is estimated 10.50% as shown above. Let us assume tbat this value varies between 8% and 12.75% increasing 25 points at each step. All these changing variables and their F R O V B S and FROVXL corresponding values are listed in Appendix B. Exchange of expected real options values calculated by Black-Scholes method according to the riskless rate and dividend yield rate variations is illustrated in Figure 3.
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FROVÏBSJ
80000 0.08
0.17
0.15
0.25 0.23 FIGURE 3 Surface Plol of FROVßs vs. á; r.
FROV|TL)
0.09
0,10 0,11
0.15 0,12
0,25
0,23
FIGURE 4 Surface Plot of F R O V J L VS. S\ r.
Exchange of expected real options values calculated by Trinomial lattice method according to the riskless rate and dividend yield rate variations is illustrated in Figure 4. White obtaining Figures 3 and 4, we defuzzify the FROVs of each data set separately. In sensitivity analysis, we have to find 20 x 20 = 400 values for each FROV because of riskless rate-dividend yield changes. So that means 800 number data has to be found. We wrote a VBA code in © MSExcel to save time. Then, graphs seen in Figures 3 ^ are traced by © Miniiab statistical software. As seen from the graphs both FROVs (computed by Black-Scholes and Trinomial lattice method) are so close to each other. When both the riskless rate and dividend yield rate increase, the FROVs ( F R O V B S and F R O V J L )
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climb down in each figure. When the riskless rate increases and dividend yield rate decreases, FROVs also increase. This means some competitors in the market pack up and the gaining of the company will increase.
6 CONCLUSIONS Recently, real option valuation methods have taken the place of conventional net present value methods. Because of the option valuation methods give the decision maker too flexibility, much more experts prefer real option valuation methods. Many application areas could be found to put into practice. As in our real case subject a call center project could be one of these areas. In this paper's application, classical net present value method underestimates the real option valuation methods both FROVßs and FROVTL by 36%. Net revenue computed with classical methods may not be sufficient to the decision maker; however with efficiency proven method called real option valuation method one can take the decision maker's mind easily. The decision maker will consider in that way: risk free interest rate is 20.50%, but real option valuation methods offer nearly double the amount. In addition, there is always option to expand or contract the decision maker can hold. That means the decision maker may execute the project, however the future conditions may be ambiguous, so if the state of affairs goes well, the company can increase the capacity, and vice versa. Many corporations act so consecutive in giving the financial data outside. However, if someone offers a possibilistic approach expressions, they could be willing to devote their special data. Sometimes theircost estimates could be in a fuzzy manner also. According to these reasons, fuzzy real option valuation methods are used. A comparison between fuzzy real option valuation methods is made in this work. One can see there is 2.04% difference between FROVBS and FROVXL methods. This is not a meaningful difference; however the trinomial lattice method offers more flexibility. So, this flexibility reflects this little difference. Black-Schotes method computes the exact result, but in trinomial lattice method, analyzer can make computation every year point to revise the project's efficiency. Estimating the dividend yield rate is the serious problem in these types of works. In future studies, researchers can follow a way to find a mediod for estimating 5 or making some strategically decisions about the dividend yield rate.
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REFERENCES 11 ] Brach. M. A. Real Options in Praclice, Hoboken, New Jersey. John Wiley & Sons, Inc., 2003. [2j Black. F.. Schoies M. The pricing of opiions and corporate lîabililies. Journal of Political Economy 81 ( I973|. 637-654. [3| Boyle, P. P. Options: A Monte Carlo approach. Journal of Financial Economies 4 (1977). 323-338. [4) Cox, J. C . Ross. S. A . Rubinslein, M., Option pricing: a simplified approach. Journal of Financial Economie- 7 ( 1979), 229-263. |5| Zadeh L. Fuzzy sets, fn/orm Control 8 (1965). 338-353. |6| Carlsson. C , Fuller, R. A fuzzy approach to real option valuation. Fuzzv Sets and Systems 139(2003), 297-312. I7| Wang, J., Hwang, W. L. A fuzzy set approach for R&D portfolio selection using a real options valuation model. Omega 35 (2007), 247-257. [8] Tolga, A. C . Kahraman, C. Fuzzy multiattribute evaluation of R&D projects using a real options valuation model. Imernatianal Journal of Intelligent Svstems 1M\ I) (2008), 11531176. [9] Allenotor. D., Thuksirani, R. K. A grid resources valuation model using fuzzy real option. Lecture notes in Computer Science, 4742 (2007), 622-632. 110) Merton, R. Theory of rational option pricing. Bell Journal of Economics and Management
Science 4 {\97^l 141-183. [Ill Leslie. K.J., Michaeis, M. P. The real power of real options. The McKinsey Quart 3 I ]991). 5-22. [121 Clewlow, L., Strickland, C. Implementing derivative.s models. Chichester: John Wiley & son.s. Inc., 1998. [13] Schwartz, E. S. The valuation of warrants: implementing a new approach. Journal of Financial Economics 4 (1977), 79-94. [14] Brandao, L. E., Dyer, J . S. Decision analysis and real options: A discrete time approach to real option va]\iaûon. Anruils of Operations Research 135 (2005), 21-39. [151 Lee, E.S., Li. R.L. Comparison of fuzzy numbers based on the probability measure of fuzzy events. Computer and Mathematics with Applications 15 (1988), 887-896.
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A Comparative Fuzzy Real Options Valuation Model using Trinomial Lattice and Black-Scholes Approaches: A Call Center Application A. ÇAGRITOLGA',CENGIZKAHRAMAN^ AND MURAT LÈVENTDEMIRCAN' ' Galaiasdray Universiry. Industrial Engineering Department. 34357 Ortakoy Istanbul Turkey F-mail: [email protected] Istanbul Technical Universiry. Industrial Engineering Depannienl. 34367 Macka Istanbul Turkey Received: March 25. 2009. Accepted: Juiy 29. 2009.
Valuation of Ihe investment projects is very serious in every dimen.sion. Convenlional discounted ca.shflowtechniques are usually insufficient since these techniques faii to account tor the (lexibility in business decisions and violations occur as a result of the existing uncertainty in projects. Real option valuation methods overcome this problem with its efficient and flexible nature. Financial option valuation methods are applied into real options area with little changes in variable definitions. When there is a lack of data or involuntary companies about giving their financial data, fuzzy numbers can be used to capture this vagueness. In this study, both fuzzy Black-Scholes and fuzzy trinomial lattice models are examined. These two fuzzy models are compared with each other for the first time in this paper. Differently from the previous works, tbe parameler dividend yield is added into tbe fuzzy trinomial lattice model. They botb are applied to a call center investment project. A comparison between these methods is made, and then a sensitivity analysis is discussed. Keywords: Real optiotis, Black-Scholes tnethod, Erinomiai lattice method, fuzzy.
I INTRODUCTION An optioti is defined as the right, but not the obligaiion, to buy (if a call) or sell (if a put) a nominated asset by paying a preset price on or before a specified date. Real options are based on financial options. However, the nature of real options involves permanent, fixed or immovable assets. In contrast to
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financial options, real options are not tradable e.g. the factory owner cannot sell the right to extend his factory to another party, he can only make this decision. The valuation of real options necessitates a real option analysis (ROA). The key advantage and value of real option analysis is to integrate managerial flexibility into the valuation process and thereby assist in making the best decisions [I]. Real options give a right but not an obligation to make or not to make an investment for a certain period. For instance, investing in the expansion of a firm's factory gives the company right to produce more but not the obligation. There are three valuation methods for real options; /. Partial differential equations (the most famous one is Black-Scholes formula), ii. Simulations (Monte Carlo), Hi. Lattices (binomial, trinomial etc.). Partial differential equation methods necessitate the solution of a partial differential equation with specified boundary conditions (i.e., option values at specified periods, type of option, etc.). Black-Scholes [2] offered their valuation method based on partial differential equations in 1973. Boyle [3] proposed Monte Carlo simulation approach for financial options built on the insight that whatever the distribution of stock value would be at the time the option expired, that distribution was determined by operations that driven the activities of the asset value between now and the expiration date. Cox et al. [4] developed a simplified option pricing model (the binomial options pricing model) based on a discrete-time approach. Black-Schoies approach is known as "black box" because of its mathematical complexity. Black-Scholes assumes a lognormal distribution of the underlying asset value, which may not be applicable for most real assets because of the cash flows" nature. During real option's life, there could be also more than one strike price but Black-Scholes assumes only one. Many complex models were developed against these deficiencies of Black-Schoies formula. In spite of complexity, easy to apply property is the advantageous and the most preferable side of Black-Schoies approach. Simulation approach is more easily applicable to European options. Nevertheless, when someone needs to deal with the sequential options, he/she has to run more than 1000 simulations. In the binomial lattice approach to real option valuations, input parameters such as the strike price and volatility can be changed easily over the option life. The binomial lattice model is easy to understand and elementary mathematics is sufficient. It uses probability distribution instead of volatility estimation. However, if multiple options and their interactions are needed to consider, it can be very cumbersome to construct the binomial asset tree. While Black-Scholes finds the most accurate option value, the binomial lattice method approximates to it. In daily life real circumstances are very often uncertain and vague in several ways. When there is a lack of information or no voluntary financial data out the company, a system might not be known completely. Zadeh [5] suggested a strict mathematical outline named fuzzy set theory that overcomes these inadequacies. Many engineering and decision prohlems implemented classes
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of groupings of data with boundaries, which are not explained exactly, can be simplified by the fuzzy set theory. The fuzzy approach to real option valuation (ROV) was first studied by Carlsson and Fuller [6]. Tbis work was based on Black-Scholes' real option valuation, but under fuzziness. Then, Wang and Hwang |7] offered fuzzy compound options for R&D project selection based on the Black-Scholes real option valuation. Tolga and Kahraman |7] offered a model that integrates this fuzzy Black-Scholes real option valuation and fuzzy AHP for selection among R&D projects. Another fuzzy real option valuation based on Cox et ¿i/.'s 14] method was investigated by Allenotor and Thulasiram |9]. They modeled pricing of grid/distributed computing resources as a problem of real option pricing. In this study, we will analyze the valuation of a call center with a fuzzy real option valuation model based on Black-Scholes (BS) and Trinomial Lattice (TL) individually, and then we will compare them. And at the last a sensitivity analysis for dividend yield and risk free interest rate will be given. The rest of this paper is organized as follows: At first, theoretical information on fuzzy Black-Scholes ROV and fuzzy trinomial lattice ROV will be given, respectively. Then, a call center application with real data in Turkey will be given. The following section will offer comparative analysis between fuzzy BlackScholes ROV and fuzzy trinomial lattice ROV with a sensitivity analysis. Conclusions and further study suggestions will come up at the last.
2 FUZZY BLACK-SCHOLES REAL OPTIONS VALUATION Merton [10] extended the Black-Scholes financial options valuation formula to dividends paying stocks but Carlsson and Fuller [6] applied that formula to real option valuation using Leslie and Michaels' 111 ] computation as follows:
ROV = Soe'^'^Nidi) -Xe~''^Nid2)
(1)
where
7 7 f •
^'^ (3)
where ROV denotes the real option value. So is the present value of expected cash flows, X is the (nominal) value of stationary costs, cr points out the uncertainly of expected cash flows, r quantifies the annualized continuously compounded rate on a safe asset, and 8 illustrates the value lost over the duration of the option.
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HGUREI A trapezoidal fuzzy number. À = (a\. ai. ay. i/4).
In case of incomplete data or when it is difficult to take out exact financial data out the company, fuzzy sets are very helpful. A fuzzy set is a special fuzzy set G — {.r e R\ßcix)], where x takes its values on the real line R\ : —00 < J: < +00 and fMcix) is a continuous mapping from R] to the close interval [0, 1 ]. A trapezoidal fuzzy number can be denoted as A = (íii.íi2. ^'3- "4)- Í'I £ "2 5 03 < «4 and traced as in Figure 1. Its membership f u n c l i o n ß ^ i x ) : R ^>^ [0, l ] \ s e q u a l t o X — a\ X € [ « I , «2
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FUZZY REAL OmoNS VALUATION MODEL
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In addition, the possibilistic mean value and variance of fuzzy number À are calculated from Equations (9) and (10) [61:
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- Xe''^'N{d2)
(11)
where
,
\niEiSo)/E(X))
(12)
and where So denotes the possible values of the present value of expected cash flows, in a similar manner X quantifies the possible values of investment cost., EiSo) is the possibilistic mean value of the present value of expected cash flows, EiX) stands for the possibilistic mean value of expected costs, a i So) denotes the possibilistic variance of the present value of the expected cash flows, and d2 is given by Equation (3). In options, dividend yield reduces the value of the option at the paid-out dividend times. The analogy between options and real options starts here. In real options, competitors could come into the market when applying a new project, and this event reduces the value of the real option as in financial options. Dividend yield rate is carried out from dividing the estimated cost of waiting to the present value of cash flows as below (in fuzzy type problem, the expected value (£(So)) should take the place of (5o)): S = thecost of waiting/£(So) (13) Sensitivity analysis to the dividend yield rate will be made in the fifth section.
3 FUZZY REAL OPTIONS VALUATION BY TRINOMIAL TREE METHOD Binomial tree method was developed by Cox et al. [4] for pricing the options simply. Later trinomial tree method was developed for more complex situations.
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\ FIGURE 2 Trinomial tree model with dividend yield.
Trinomial tree method provides much better approximation to the continuous time process than the binomial lattice for the same number of steps. The trinomial tree is easier to work with because of its more regular grid and is more flexible, allowing relatively easy extension to time-varying drift and volatility parameters [12]. In this section, we first present general form of the trinomial tree model as illu.strated in Clewlow and Strickland [12]. Then, the fuzzy forms of these representations will he developed for the case of incomplete data or vagueness. The trinomial tree approach is equivalent to the explicit finite difference method (introduced by [ 13]) and it is illustrated in Figure 2. For being intelligibility, notations in Section 2 is used in this section without changing anything. Suppose that pu, p^, and p¿ are the probabilities of up, middle, and down movements at each node and Ar is the length of the time step. Dividend paying stocks" real option valuation techniques for binomial lattices are evaluated hy Brandao and Dyer [14]. For a dividend paying stock in trinomial lattice, parameter values that match the mean and standard deviation of price changes
FUZZY REAL OPTIONS VALUATION MODEL
141
are as follows: I
, (14) (15) (16) (17)
Ax / a^At + li-^Ar
2 V
AA--
AA-
/
Computations for a trinomial tree lattice are similar to those for a binomial tree. At time zero, the present value of expected cash flows, 5o is known. At time A/, there are three possible 5» values, SQU. SQ. and S{)d\ at time 2Ai, there are five possible 5o values, SQU^. SOU, SO, Sod, and Sod^; and so on. In general, at time i At, 2i + 15o values are considered. These are: S¡j=So{\-8)u^.j^-i....,i
(21)
Notice that the u = \ -=r d relationship is used in computing the So value at each node of the tree in Figure 2. For example, Sou~^ — 5ÜÍ/". Real option valuations by lattice methods are evaluated by starting at the end of the tree {time T) and working backward. The value of the real option is known at time T. For being intelligibility, notations in Section 2 is used in this section without changing anything. Let us express the approach algebraical ly. Because the value of the real option at its expiration date is max(O, S¡j — X), we know that CT,j = max(O, STJ - X), j =-T
0
T
(22)
After computing the crj values, (no early exercise) Cjj values at each node could be calculated by the formula below: Ci,i = e^''^'[puCi+\j+i -\- p,„Ci+]j -fprfc-í+i.j-i]
(23)
for O < í < 7" — I and —/ < j < i. When early exercise is considered, this value for c, j must be compared with the option's intrinsic value, and we obtain: = max ^ [ p u i + \ j + \
i PrnQ+ij + P d i + \ j i l
Q
(24)
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In case of incomplete data or vagueness, all the calculations are same, but So (the present value of expected cash fiows). and X (the nominal value of stationary costs) will be joined to these formulae in a fuzzy form ÍSQ, X). In this way, the Equation 21 turns into Equation 25: j 5 ) u J , j
= -i,...,i
(25)
Then, Equation 22 turns into Equation 26: CTJ = max(O, ST.J - X), j = -T
0
T
(26)
To select the maximum term in Equation (26). a defuzzification method must be used. In this study, the centroid method is used for this aim. The Equations 23 and 24 turns into the Equations 27 and 28 consecutively: Ci.j = e~''^'[puCi+\j+\ + p,,,Ci+\,j + PdQ+ij-i]
(27)
for 0 < / < r — 1 and —/ < j < i. And in early exercise: c, J = m a x
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J
, _ . , ..J
(28)
4 A CALL CENTER APPLICATION The objective ofthe project is the design, development, deployment and operation of a top-up system for GSM operators that will enable their subscribers to recharge their credits via USSD, SMS, IVR. web and call center channels. When a registered subscriber needs to top-up credit, s^e will be able to reach any of the 'top-up with pin' channels such as USSD, SMS, IVR, web and call center. The top-up process is completed instantly upon request. The application given will aim to design, implement and manage the call center process perspective only. In this process, the subscriber registers to the system via call center. During the registration process, the credit card data as well as necessary personal information is gathered from the subscriber and later controlled and validated for fraud protection purposes. The subscriber is then ready to use the top-up system. The call center will bring the following benefits to GSM operators and their subscribers, the followings: • Customer satisfaction will be ensured by easy and fast access to the product. • Subscribers will be able to charge their credits virtually, therefore there will be no physical card preparation and distribution cost for GSM operators.
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• Convenient sales channels and a well trained sales team will increase the ARPU (Average Revenue Per Unit) for GSM operators. • GSM Operators will obtain the ability to interact with the subscribers and manage the customer experience. • Subscribers will be able to reach the product 24 hours a day, 7 days a week. The call center has two financial aspects. The first one is initial investment which includes capital expenditures (CAPEX); the second one is generated cash flows which include operating expenditures (OPEX). labor costs and revenues. In Table I, you will find the capital expenditures of the overall call center establishment costs which we call initial investment. The costs are extracted from different proposals from different outsourcing companies. For instance for Phone Infrastructure and GSM gateway we received 6 different proposals from 4 different providers. Then the financial figures are derived from the proposals as their two minimums (a\ and«?) and two maximums {CIT, andíí4}. The total capital expenditure is calculated {205155, 210443. 212558, 217845) YTL as a trapezoidal fuzzy number. In order to calculate the expected cash flows, we, first, calculated the cost aspect which includes the labor cost and operating expenditures (OPEX). Operating expenditures and labor costs are shown in Table 2 and in Table 3, respectively. Fuzziness of both financial costs is explained as the changes and variations of bonus, augmentation, personnel turn over, etc. Since we have a 5 year investment planning horizon with 25 working agents, we calculated the present value of costs (331217072, 3397535.94, 3431682.03, 3517047.25) YTL. We assume that over 5 years, the call center will generate 5500 transactions per month by 18 YTL gross revenue per transaction. The present value of gross revenue as calculated as (3587141.40, 3679593.51, 3716574.35, 3809026.45) YTL. In order to calculate the present value figures we used discrete monthly discount rate as % 1.708. The final financial table is shown in Table 4.
5 COMPARATIVE ANALYSIS In this part, fuzzy real option valuation of the call center with each BlackScholes and Trinomial method separately will be made. Then benchmarking is going to be analyzed. After these, sensitivity analysis of the annualized continuously compounded rate on a safe asset, and the value lost over the duration of the option will be framed. For the annualized continuously compounded rate on a safe asset, five year Treasury bond interest rate belonging Central Bank of the Republic of Turkey
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Communication Infrastructure
Cost values (in YTL)
Phone Infrastructure GSM Gateway Voice Recording Infrastructure
(48298. 49304, 51317, 52323) (20736, 21168. 22032. 22464) (19008. 19404,20196.20592)
User Hardware Desktop PC Handheld Device Headphone Wired System
(17280, 17640, 18360, 18720) (3456. 3528, 3672, 3744) (1728, 1764, 1836, 1872) (3456,3528, 3672, 3744)
System Hardware Voice Recording Backup Unit CRM Server Admin Server Firewall System Cabling
(2592. 2646, 2754, 2808) (2592, 2646, 2754, 2808) (2592, 2646. 2754, 2808) (5184,5292,5508,5616) (8640,8820,9180,9360)
Furnishings Agent Tables Management Tables Seats Other Furnishings
(4320,4410,4590.4680) (173, 177, 184, 188) (2592, 2646, 2754, 2808) (3370,3440,3581,3651)
Software & Other CRM Software HR Cost
(51840, 52920, 55080, 56160) (5184,5292,5508,5616)
Total
(205155, 210443, 212558, 217845)
TABLE I
Capital Expenditure (Initial Investment) figures for the Call Center E.stablishment (Turkish Lira. YTL) as trapezoidal fuzzy numbers
will be utilized which is 20.50%, the average of July 2008. We have to retain that this value contains deep volatility. For the value lost over the duration of the option, the estimated i5(= 10.50%) value will be used. The value lost over the duration of the option means the entrance of the competitors to the market, so at this time in Turkey, there are many competitors in the market of call center. That is why we take the value in this manner, however in the latest days; there may be bankruptcies or new foundations. We assume S is the total dividend yield associated with all ex-dividend dates between time zero and time iSt. So because of these reasons, sensitivity analyses on competitor entrance and on risk free rate have to be made.
FUZZY REAL Oprttws VALUATION MODEL
General Management Call Center Manager Team Uader Back Office Manager Back Office Staff Call Quality Specialist Helpdesk Staff Stationary HR Staff Finance & Accounting Staff Office Rent Energy and Other Costs System and Technical Maintenance Software Updates and Upgrades Other Total
145
(1964.56, 2015.20, 2035.45, 2086.08) (1414.49. 1450.94, 1465.52, 1501.98) (2946.85, 3022.80, 3053.18. 3129.13) (1414.49. 1450.94. 1465.52, 1501.98) (10372.90, 10640.25, 10747.18, 11014.53) (1728.82, 1773.37, 1791.20, 1835.75) (1728.82, 1773.37, 1791.20. 1835.75) (1100.16, 1128.51, 1139.85, 1168.21) (589.37. 604.56. 610.64, 625.83) (589,37, 604.56, 610.64, 625.83) (2750.39, 2821.28, 2849.63, 2920.52) (1178.74. 1209.12, 1221.27, 1251.65) (982.28. 1007.60, 1017.73. 1043.04) (982.28, 1007,60. 1017.73. 1043.04) (216.10,221.67,223.90.229.47) (29959.61. 30731.77. 31040.63. 31812.79)
TABLE 2 Motiihly Operating Expenditure figures (Turkish Lira YTL) as trapezoidal fuzzy numbers
Gross Salary Social Insurance (Employer Fee) Social Insurance (Employer Fee) Revenue Tax Premium Revenue Tax Fee Other Tax Premium Base Salary Bonus Other Total NetSalaiy Total Gross Salary
(1800.36. 1846.76. 1865.32, 1911.73) (387.63, 397.62, 401.62. 411.61) (269.97, 276.93, 279.72, 286.67) (1530.39, 1569.83, 1585.61, 1625.05) (229.68, 235.60, 237.97, 243.89) (11.28, 11.57, 11.69. 11.98) (483.54. 496.00. 500.98, 513.44) (805.89, 826.66, 834.97, 855.74) (161.18, 165.33, 166.99, 171.15) (1289.43, 1322.66. 1335.95, 1369.19) (2348.37, 2408.90, 2433.11, 2493.63)
TABLE 3 Monthly Labor Cost per Agentfigures(YTL) as trapezoidal fuzzy numbers
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Gross Revenue Cost {So) Initial Investment {X) Net Cash Flow (SQ - X)
(3587141.4 L 3679593.51. 3716574.35. 3809026.45) (3312170.72. 3397535.94, 3431682.03, 3517047.25) (70088.30,247890.76,319011.74,496814.20) (205155. 210.443, 212558, 217845) (-147756.70, 35333.26, 108569.24. 291659.20)
TABLE 4 Present value offinancialfigures(YTL) as trapezoidal fuzzy numbers
As given in Section 4 and above, initial values are listed below: So = (70088.30, 247890.76, 319011.74,496814.20) YTL, X = (205155, 210443, 212558, 217845) YTL, T =5 years, r = 20.50%, S = 10.50%. 5.1 Fuzzy Black-Scholes computation For FROV by Black-Scholes method, one can easily compute £(5o), E{X) and (T values from Equations (8) and (9), i.e. 283451 YTL; 211500 YTL; 36.6% successively. Also, from Equations (11) and (12), d\ and d2 values could be found 1.2658 and 0.4480 respectively. At the last, F R O V B S = (-10400.19, 79870.59, 115978.90, 206249.67) value can be computed from Equation ( 10). 5.2 Fuzzy Trinomial lattice computation For FROV by Trinomial lattice method, at first step, i», AJ:, U, and d values have to be calculated from Equations (14-17); and those could be found 0.033, 0.634, 1.884.0.53! respectively. Then, the probabilities of up. middle, and down movements for each node are calculated by Equations (18-20) at the second step: Pu - 0 . 1 9 4 ,
p„, =0.664,
p,¡ ^0.142.
Aii calculated Si_j values at each node are illustrated in Appendix A. For example, 55.5 and S5._5 could be found as below: 55.5 = (1489831.11,5269287.16,6781069.58, 10560525.63) 55,-5 - (2641.19.9341.46. 12021.57. 18721.84) After finding these eleven ends of the period values. Equation (26) have to be computed to find the values of the fuzzy real option values at its expiration
FUZZY REAL OPTIONS VALUATION MODEL
147
date. Application of Equation (26) is not easy in fuzzy terms, so we have to defuzzify all values by centroid method to compare these values by zero. At the following rows, there are some calculated FROVs at its expiration date: C5.5 = (1271986.11,5056729.66,6570627.08, 10355370.63), C5,o = (-155115.97.9304.73.75073.01.239493.71), C5.-5 = (0.000,0.000, 0.000. 0.000). At the next step, all ¿-¡j values at each node should be calculated by the Equation (27) with backward induction. Finally, ro.o value - shown below will be carried out which means FROV by Trinomial lattice method: FROV-TL ^ (-22602.70,65922.50, 101332.58, 189857.78) The defuzzified values for FROVßs and F R O V J L are calculated as 97924.74 and 83627.54, respectively. Trinomial lattice model gives a lower value than Black-Scholes model in every combination of parameters, as seen in Tables Bl, B2, and B3. Trinomial lattice model can be said to be a more pessimistic approach relative to Black-Scholes model. If we survey to the NPV as computed in section 4. this value underestimates to both F R O V B S and F R O V J L - We can make this comparison by Lee and Li's method [15], 5.3 Sensitivity analysis As we mentioned before, volatility in the annualized continuously compounded rate on a safe asset and competitor entrance overextend us to make sensitivity analyses on these subjects. The experts guessed that in five year period there may be one entrance to the market. Estimation of the waiting cost is very difficult; but as a result of consultancy with the experts in the firm, 5952.47 YTL value lost per five year period is reached. So from Equation ( 13), one can easily obtain dividend yield rate as: ¿ = (5) X 5952.47/283451.25 = 10.50% It is thought thatfiveyear Treasury bond interest rate belonging Central Bank of the Republic of Turkey could change from 15.5% to 25.00 percent spectrum by 50 base points. Though there are few competitors in the market, S is estimated 10.50% as shown above. Let us assume tbat this value varies between 8% and 12.75% increasing 25 points at each step. All these changing variables and their F R O V B S and FROVXL corresponding values are listed in Appendix B. Exchange of expected real options values calculated by Black-Scholes method according to the riskless rate and dividend yield rate variations is illustrated in Figure 3.
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FROVÏBSJ
80000 0.08
0.17
0.15
0.25 0.23 FIGURE 3 Surface Plol of FROVßs vs. á; r.
FROV|TL)
0.09
0,10 0,11
0.15 0,12
0,25
0,23
FIGURE 4 Surface Plot of F R O V J L VS. S\ r.
Exchange of expected real options values calculated by Trinomial lattice method according to the riskless rate and dividend yield rate variations is illustrated in Figure 4. White obtaining Figures 3 and 4, we defuzzify the FROVs of each data set separately. In sensitivity analysis, we have to find 20 x 20 = 400 values for each FROV because of riskless rate-dividend yield changes. So that means 800 number data has to be found. We wrote a VBA code in © MSExcel to save time. Then, graphs seen in Figures 3 ^ are traced by © Miniiab statistical software. As seen from the graphs both FROVs (computed by Black-Scholes and Trinomial lattice method) are so close to each other. When both the riskless rate and dividend yield rate increase, the FROVs ( F R O V B S and F R O V J L )
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climb down in each figure. When the riskless rate increases and dividend yield rate decreases, FROVs also increase. This means some competitors in the market pack up and the gaining of the company will increase.
6 CONCLUSIONS Recently, real option valuation methods have taken the place of conventional net present value methods. Because of the option valuation methods give the decision maker too flexibility, much more experts prefer real option valuation methods. Many application areas could be found to put into practice. As in our real case subject a call center project could be one of these areas. In this paper's application, classical net present value method underestimates the real option valuation methods both FROVßs and FROVTL by 36%. Net revenue computed with classical methods may not be sufficient to the decision maker; however with efficiency proven method called real option valuation method one can take the decision maker's mind easily. The decision maker will consider in that way: risk free interest rate is 20.50%, but real option valuation methods offer nearly double the amount. In addition, there is always option to expand or contract the decision maker can hold. That means the decision maker may execute the project, however the future conditions may be ambiguous, so if the state of affairs goes well, the company can increase the capacity, and vice versa. Many corporations act so consecutive in giving the financial data outside. However, if someone offers a possibilistic approach expressions, they could be willing to devote their special data. Sometimes theircost estimates could be in a fuzzy manner also. According to these reasons, fuzzy real option valuation methods are used. A comparison between fuzzy real option valuation methods is made in this work. One can see there is 2.04% difference between FROVBS and FROVXL methods. This is not a meaningful difference; however the trinomial lattice method offers more flexibility. So, this flexibility reflects this little difference. Black-Schotes method computes the exact result, but in trinomial lattice method, analyzer can make computation every year point to revise the project's efficiency. Estimating the dividend yield rate is the serious problem in these types of works. In future studies, researchers can follow a way to find a mediod for estimating 5 or making some strategically decisions about the dividend yield rate.
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