A real options approach to the baseball game betting

Wang, Shin-yun Article A real options approach to the baseball game betting Economics & Finance Research Provided in Cooperation with: Taylor & Francis Group Suggested Citation: Wang, Shin-yun (2013) : A real options approach to the baseball game betting, Economics & Finance Research, ISSN 2164-9499, Taylor & Francis, Abingdon, Vol. 1, Iss. 1, pp. 75-81, https://doi.org/10.1080/21649480.2013.870489 This Version is available at: https://hdl.handle.net/10419/147693 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/3.0/ Economics & Finance Research, 2013 Vol. 1, 75–81, http://dx.doi.org/10.1080/21649480.2013.870489 A real options approach to the baseball game betting Shin-Yun Wang Department of Finance, National Dong Hwa University, Shou-Feng, Hualien, Taiwan. E-mail: gr[email protected] Applications of real options can be found in real assets, but not in sports. This article proposes the real options model and applies it to baseball games, allowing the option to switch between the two teams during the game. We believe this is the first attempt to apply the switching options to the baseball game scheme. To do so, we first design some rules in our assumptions and define variables that are relevant to the baseball options.We combine features of the wager volatility and options to determine the optimal time of team switching, and we find that with low switching cost, the expected investment costs fall and so does risk. The application of switching options to baseball game also presents a feasible method to other sport games. I. Introduction Baseballhasbeenoneofthemostpopularsportsintheworld;many countries regard baseball as the national sport. Because of the popularity, not only the number of professional teams has increased, but also the baseball fans. To mitigate the illegal game-cheating problem, which often involves gang threats and violence, regulation and legalizing baseball game gambling such as using public welfare lottery has been proposed. Since baseball games have large number of fans, and the trading of derivatives has gained an important footing in the financial markets, the proposed combination of baseball games and switching options is indeed a new endeavour to our best knowledge. Legalizing the trading of such a new product mayincreasethetrading volumeof options, reduce illegalgambling and enhance government tax revenues. To shed light on such a proposal, we propose and design the use of switching option to add to the flexibility of baseball game gambling.To be sure, application of the switching option is also feasible for other sport games. A bettor owns an option and has the opportunity to switch from one team to another team as soon as future wager volatility favours suchswitch.Thisopportunityiscomparabletoacalloption,sincein effectthebettorpaysaprice(theoptionpremium)fortherighttopay a fixed exercise price (the investment cost) and obtain the switching option. With some costs, a bettor owns the switching option to exploit wager volatility movements but can switch back to the holding strategy when losses occur. The result is that, just like the holder of a financial option, the bettor can in principle benefit from unlimited profit opportunities, while cap their potential loss. Therefore, uncertainty of wager volatility is not only a source of concern to the bettors; it also creates the potential for abnormal payoffs. In the ideal situation, winning takes place wherever the bettors have more information or take advantage of wager volatility movements. The more uncertain the wager volatility and the lower the switching cost, the more valuable is the bettor’s ability to respond to new information swiftly. We combine features of the wager volatility and the bettor’s ability to switch between teams in this article. We start out with the premise that bettors can choose team or combination of teams to bet, and have the options to switch between teams by purchasing an option after paying a switching cost. The relative cost or return for the holding versus the switching strategy is determined by the team-specific wager volatility and is thus variable over time. The rest of this article is organized as follows: Section II presents the literature review about baseball option and models of sport researches.InSectionIII,wediscusstheconceptofbaseballoptions where we present the application of switching options to the valuation of baseball game betting. Some rules in the assumptions and variables to estimate the variance in the baseball options are proposed.SectionIVgivesnumericalexamplestocompareandanalyse theresultsofbaseballoptions.WeuseStochasticDynamicProgramming (SDP) procedure to discuss risk implications and flexibility. Conclusions are presented in Section V. II. Literature Review Early researchers sought evidence of inefficiencies in the form of systematic biases in bookmakers’ prices, such as the homeaway team or favourite-longshot biases. More recently, forecasting models have been used to establish whether historical information available in previous match results can be extrapolated to formulate profitable betting strategies. Gandar et al. (2001) are sceptical over the existence of systematic biases favouring bets on home teams in baseball and basketball. Moroney (1956) and Reep et al. (1971) use the Poisson and negative binomial distributions to model at an aggregate level the © 2013 TheAuthor(s). Published by Taylor & Francis. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The moral rights of the named author(s) have been asserted. 76 S.-Y. Wang 1/2 1.03 Path 1: (1.0)-(1.01)-(1.02)-(1.03) 1/2 1.02 Path 2: (1.0)-(1.01)-(1.02)-(1.01) 1/2 1.01 1/2 1.01 Path 3: (1.0)-(1.01)-(1.0)-(1.01) 1.0 1/2 1.0 Path 4: (1.0)-(1.01)-(1.0)-(0.99) 1/2 0.99 1/2 0.99 Path 5: (1.0)-(0.99)-(1.0)-(1.01) 1/2 0.98 Path 6: (1.0)-(0.99)-(1.0)-(0.99) 1/2 0.97 Path 7: (1.0)-(0.99)-(0.98)-(0.99) Path 8: (1.0)-(0.99)-(0.98)-(0.97) 1:00 1:10 1:20 1:30 Fig.1. Numericalexample–pathsofwagerinthefirstfourinnings distributions of the numbers of goals scored per game, but this approach precludes the generation of specific forecasts for individual matches based on information about the respective strengths of the two teams. Hill (1974) demonstrates that individual match results do have a predictable element, and are not determined solely by chance. Maher (1982) develops a model in which the home and away team scores follow independent Poisson distributions, with means reflecting the attacking and defensive capabilities of the two teams. A full set of attacking parameters and a set of defensive parameters for each team are estimated ex post, but the model does not predict scores or results ex ante. A tendency to underestimate the proportion of draws is attributed to interdependence between the home and away scores, and corrected using the bivariate Poisson distribution to model scores. Dixon and Coles (1997) develop a forecasting model capable of generating ex ante probabilities for scores and match outcomes. The home and away team scores follow independent Poisson distributions, but for low-scoring matches an ad hoc adjustment allows for interdependence. Using a similar framework, Rue and Salvesen (2000) assume that the time-varying attacking and defensive parameters of all teams vary randomly over time. In the real world of sport, once the bettor bets on a team, he/she seldom can switch bet between teams, which greatly limits the flexibility of betting. In fact, the best team in baseball has the lowest probability to win an average game compare with other sports such as football and basketball. Switching options thus add decision dynamics to the betting. We explain in more details about the switching between teams in Fig. 1, Tables 4and 5. With the introduction of switching options in the baseball game, it serves three purposes.First,theabilitytoswitchteamsoffersthebettorcapability to ‘hedge’during the game with little cost; second, the introduction of the switching options to the baseball game betting enhances the liquidity of the wager; and finally, introducing options to the baseball game betting offers the bettors more time to observe and to make decisions, hence delay his/her ‘investment decisions’ until more information becomes available. Most of the game models use the Poisson distribution, bivariate Poisson distribution and negative binomial distributions model (e.g. Dixon and Coles, 1997;Rue and Salvesen, 2000;Crowder et al., 2002). This article thus uses the bivariate Poisson dynamic stochastic model outlined above to identify potentially profitable betting strategies. III. An Application of Switching Options to Baseball Game Betting Although applications of option pricing models to financial assets are abundant, applications to real assets are less popular. To be sure, there are applications of option model to subject matters such as R&D project evaluation and selection (e.g. Newton and Pearson, 1994;Robert, 1998); nevertheless, application to sports is absent. We buildourschemebaseduponthe conceptof consumerswitching costs (Grossman, 1981;Klemperer, 1995), and the notion of the value of an option to exchange one asset for another (Margrabe, 1978). The money line and switching options In the traditional baseball games, sportsbooks use a money line for betting baseball. The money line determines the amount of money laidandmoneywon.The moneyline,however, does not providethe flexibility of allowing the bettor to switch bet between teams during the game. The bettor cannot change his/her original bet during the game when condition changes. Therefore, this article proposes a switching options approach to baseball games, allowing bettor the option to switch bets between the two teams during the game. Hence it is a combination of traditional money line with the real option model. The proposed baseball option relies on the concept of switching option–avariation of real options. To do so, we first design the rules for the model; then we demonstrate how to measure primary variables and calculate the value of these variables related to the baseball options. Finally, we use these variables to price switching baseball options. The rules of baseball options This article presents a framework for baseball options in a manner similar to stock options. We assume that there is a home team competingagainsttheguestteaminasinglegameinwhichtherearenine innings. To facilitate the discussions, the following assumptions are made without losing generality: (1) There are no more than nine innings in a single game; tie is permitted. (2) The quality of teams is very competitive through a fair scheme of public selection and a free market for trading bettors. (3) Game continues until the end of the ninth inning. (4) The duration of the game is assigned a value of 1.5 hours, hence for each inning is 10 minutes. (5) Standard deviation is estimated based upon the team’s past performance. (6) Thebettorincursa switching cost forchangingbetbetween teams. (7) Switching is no longer permitted when the difference in scores exceeds ‘y’points in a single game. (8) Bettingisnolongerallowedafterthe nth inning (e.g. fourth inning). The value of the switching options Switching options allow the bettors to switch bet from one team to anotherduringthegame at a fixedcost.This option is moreversatile than the Black–Scholes options pricing model (1973), and is more adapted to the real-world situation. In the real options framework, the bettor holds the real option of switching between the two teams. It is therefore interesting to determine the value of this switching option, i.e. what additional value the bettor receives from having the option to switch bet between teams. We define the value of switching options as follows: vAB(p0)−max(vA(p0),vB(p0)), (1) where p0>1 is the initial present value of wagering on team A. In Equation 1,vAB(p0)is the maximum value to the bettor from The real options approach to the baseball game betting 77 having the option of switching bets between teams A and B and max(vA(p0),vB(p0)) is the maximum value from betting only on one of the two teams.We assume optimal policies exist within each of the game determined by the boundary conditions at the switching points at which bettor switches between teams (Dixit, 1989). Assume that a bettor has already betted on A. Due to the seesaw game, the bettor is now considering the following options: continue the bet with team A; or buy a flexible switching option that allows him/her to switch from team A to B for a cost of $CAB (or from B toA for $CBA if the initial bet is on B)We then use the Real Options Analysis (ROA) method to value the flexibility feature of the option (CopelandandAntikarov,2001).Sincethevalueofswitchingoption is the marginal value of such flexibility, the first question we need to answer is: given the choice of only either team A or team B, which one should be wagered on?A simple decision can be made based on the Net Present Value (NPV) of the expected cash flows each team provides. Following the binomial model, we can compute the value of a real option as (Cox et al., 1979) ROV =V0B(n≥a|T,p)−K(1+r)−TB(n≥a|T,p),(2) where pis the up-movement probability such that p=(1+r− d)/(u−d),p=u/(1+r)p. ROV denotes the current real option value; V0is the value of expected cash flow; B(n≥a|T,p)is the cumulative probability of having an in-the-money option (i.e. n≥a), where the probabilities arethecertainty-equivalentprobabilitiesdeterminedbytherisk-free hedge portfolio; and K(1+r)−Tis the discounted initial cash outflow.Assuming uniform through the game, the up-movement factor for each inning is defined as u=eσ√t, and the down-movement factor is defined as d=e−σ√t, where σquantifies the uncertainty of expected cash flows, rdenotes the risk-free interest rate and T=1.5 hours so that tis 10 minutes (nine innings in a game). Giving the risk-adjusted discount rate per period (k), the risk-free rate of return and the objective probability for up- (down-) movement on every step, we can discount the expected cash flows along the event tree for team A and obtain a corresponding tree with the present values. The present value for team A at state ycan thus be expressed in the following: PVAy=p×PVAw+(1−p)PVAz (1+k)(3) Equation 3indicates that the present value of betting on team A at each possible stage of nature (inning) is equal to the expected value the team offers in the future discounted at a risk-adjusted discount rate. Next, let us assume that the bettor betted on team A at the previous state, he/she can stay with teamA or switch to team B and pay the switching cost CAB. The optimal decision rules at stage y are: SAy=MAX(PVAy,PVBy−CAB),if PVAy>PVBy−CAB,stay with A PVAy<PVBy−CAB,switch to B (4) After we have identified an optimal switching for the ending period state in both decision trees, we work backward to analyse the states before the ending period state, and find the present value forthetwodecisiontrees.We thenchecktoseewhether it isoptimal tostaywiththeircurrentteamortoswitchtoadifferentteam.Repeat the same process and work backward for each state all the way to the beginning period state.The final result provides an optimal contingent plan for executing the available options. We now can price the options-to-switch between the two teams. The marginal value of real options A (B) with the flexibility to switch to team B (A) is thus the difference between the real options value with switching option (ROVA,B) and the NPV of betting on one team without the switching feature (NPVA,B), that is VA,B =(ROVA,B)−(NPVA,B)(5) The estimation of variance in the baseball options Since our idea of baseball options evolves from the concept of real options, it is necessary to find the underlying asset standard deviation-equivalent in the baseball option. Therefore, one of the important concepts of this article is to objectively determine a feasible standard deviation for computing the baseball options price.We propose to estimate the expected variance of the underlying asset value,σ2,baseduponthreeimportantfactorsthatdeterminetheoutcomeofabaseballgame.Sincetheunderlyingassetofrealoptionsis often not traded, a proxy for volatility has to be found. (Copeland) This proxy can be a Monte Carlo simulation of the value of the project, another traded asset, or a synthetic portfolio of volatility that affect the outcome of a baseball game. We choose a synthetic portfolio of volatility that affects the outcome of a baseball game andthesefactorsare the past performance of baseball professionals. To create a synthetic portfolio of volatility that affect the outcome of a baseball game, we use the following three factors, which are generally considered the most influential factors by baseball professionals (seeYang and Swartz, 2004 for a very similar argument). These three factors are: (1) Earned RunAverage (ERA), (2) line-up players’ batting average (AVG) and (3) fielding percentage (FLP). They are defined as: ERA = n  i=1 (innings pitched by pitcher i/9∗ ×pitcher i’s ERA),where ERA =(earned runs ∗9)/(innings pitched) AVG =Hit At Batswhere At Bats =(Plate appearances −walks −sacrifice hits) FLP =(Total Chances −Errors) Total Chances Without the loss of generality, we assume that these three factors share equal weights; i.e. one-third each, in determining the variability of the payoffs. Moreover, since there are nine innings and two types of pitchers – starting and relief, we use two weights for the ERA, X/9 and (9−X)/9, where Xis the number of innings pitched by the starting pitcher. Therefore, proxy of the variability of the payoffs can be written as: σ2=1 3σ2 (AVG of line−up players−AVG of all players in league) +1 3σ2 (FLP of line−up players−FLP of all players in league) +1 3X 9σ2 (ERA of starting pitcher−ERA of all players in league) +9−X 9σ2 (ERA of relief pitcher−ERA of all players in league)(6) where Xis the number of innings pitched by the starting pitcher. 78 S.-Y. Wang Table 1. Team A line-up players’ERA, AVG, FLD Line-up AVG (batting FLD (fielding (player) ERA average) percentage) 3B 0.256 0.960 RF 0.161 0.955 SS 0.421 0.957 DH 0.500 1 1B 0.444 0.974 CF 0.462 1 LF 0.400 0.929 2B 0.476 0.969 C 0.310 1 SP 5.290 RP 0 CP 7.360 Average 5.170 0.381 0.972 This estimated σ2then can be used to calculate the node values of the decision tree, the present value event tree and finally the real options value of betting on each baseball teams in the game. IncorporatingtheROA,bettorsareallowedtoswitchbetweenteams withaswitchingcost.Thisflexibilityoffersbettorsthebestdecisionmaking tool during each innings, and the value of switching right can also be calculated using ROA. Such real option is known as the switching option. IV. Switching Options of the Baseball Game: Numerical Examples In the following subsections, we use an example of a single game to analyse the baseball options. Home teamA and guest team B are assumed very competitive teams although the historical statistics slightly favour teamA.1 An example of baseball game To simplify the discussions, we assume two teams are competitive; hence it is difficult to know which team will win the game. Following the standard binomial tree option valuation procedure, we first compute the option value for each node in the last period state, discounting back to the present value for the state before the last one, and continue this process until the option value for the beginning period state is obtained. To employ the baseball options, however, we must first estimate the value of wager volatility following Equation 6. To estimate this value, the AVG, FLP and ERA of home team A and guest team B are shown in Tables 1and 2, while Table 3shows the statistics for all league players’averages. The switching cost from teamA to team B is assumed to be $25. Based upon our hypothetical example, the estimated wager variability for teamA using Equation 6is thus σ2 A=4.369 (see Table 3). For team B, we obtain very similar up- and down-movement factors, i.e. u1.01. Hence the probability for up-and down-movement is also close to 0.5. Forexample,assumingannualrisk-freerateof3%,eachinningis about 10 minutes, we have t=10/(365∗24∗60)=1.90259E-05, hence r=0.03∗10/(365∗24∗60)=5.70776E−07, both are very 1Handicap–point spread is often used in cases where teams are not competitive, we skip this possibility to avoid unnecessary analytical complications. 2Interest rate in the option pricing model is the annual interest rate because time unit in such model is a fraction of a year. In the baseball game, the time duration in each inning is very short (typically 10 minutes or so), hence converting the annual interest rate to per inning equivalent rate results in a near-zero risk-free rate. The near-zero risk-free rate makes the discount rate (1+r)equal to one. As a result, only the expected cost at the kth inning is taken into consideration. Table 2. Team B line-up players’ERA, AVG, FLD Line-up AVG (batting FLD (fielding (player) ERA average) percentage) CF 0.304 1 SS 0.333 1 2B 0.258 0.942 3B 0.176 0.867 DH 0.308 1 1B 0.297 0.969 LF 0.242 0.941 C 0.200 1 RF 0.250 1 DH 0.222 0.917 C01 SP 7.260 RP 0 CP 0 Average 4.300 0.236 0.967 Table 3. All players’average statistics League League League Team A’s Team B’s AVG FLP ERA σ2σ2 0.243 0.805 5.234 4.369 3.224 small numbers. From Table 3, since σ2=4.369 for team A, hence theup-and down-movementfactorsarecalculated asu1.01,'d= 1/u=0.99, respectively. The probabilities of up-movement (and down-movement) are thus p.5. Since the risk-free interest rate and time are very small unit, therefore, for team B u1.01, p.5, which is very similar to team A’s statistics. Based upon the above explanations, the calculated up- and down-movement factors are close to zero irrespective of σand interest rate. This is because the short time horizon renders the impact of σand interest rate negligible.2 This result reflects that both teams are competitive in our hypotheticalexample.Inthereal-worldgames,teamsareoftencompetitiveandtheendingresultsarenoteasytopredict.Themostrecent World Cup soccer game is a good example, in which the ‘favourites’ (e.g.BrazilandArgentina)losttothe‘underdogs’(e.g.Germanyand the Netherlands). Risk is managed by means of purchasing bets for which there are twostrategies:(1)HoldingteamAorBincurringabeginninginvestment of $1000; (2) Switching between teams A and B incurring an additionalcostof(non-recurrent)$25.At the beginningof each new 10 minutes (innings), the casinos obtain an estimate of cash flow in the first four innings. So, for example, at one o’clock estimated cash flows are obtained for the next 30 minutes, i.e. second, third and fourth innings. We assume bettors are allowed to bet only during the first four innings because outcome becomes less uncertain as games progress. To facilitate our discussions and compare with the switchingstrategyinthe numerical example,we furtherassumethat the bettor makes a wager in the first four innings. The wager volatility determines whether investment in team A or B is cheaper. For The real options approach to the baseball game betting 79 Table 4. Deriving cost-minimizing decisions by means of SDP when switching cost =$25 Wager Existing $ Costs of continuing $ Costs of switching paths strategy existing strategy to alternative strategy Cost-minimizing decisions at 1:30 1.03 A 1000 1030 +25 =1055 1.01 A 1000 1010 +25 =1035 0.99 A 1000 990 +25 =1015 0.97 A 1000 970 +25 =995 1.03 B 1030 1000 +25 =1025 1.01 B 1010 1000 +25 =1025 0.99 B 990 1000 +25 =1025 0.97 B 970 1000 +25 =1025 (Expected) Cost-minimizing decisions at 1:20 1.02 A 1000 +1/2(1000 +1000)=2000 1020 +25 +1/2(1025 +1010)=2062.5 1.0 A 1000 +1/2(1000 +1000)=2000 1000 +25 +1/2(1010 +990)=2025 0.98 A 1000 +1/2(1000 +995)=1997.5 980 +25 +1/2(990 +970)=1985 1.02 B 1020 +1/2(1025 +1010)=2037.5 1000 +25 +1/2(1000 +1000)=2025 1000 +1/2(1010 +990)=2000 1.0 B 980 +1/2(990 +970)=1960 1000 +25 +1/2(1000 +1000)=2025 0.98 B 1000 +25 +1/2(1000 +995)=2022.5 (Expected) Cost-minimizing decisions at 1:10 1.01 A 1000 +1/2(2000 +2000)=3000 1010 +25 +1/2(2025 +2000)=3047.5 0.99 A 1000 +1/2(2000 +1985)=2992.5 990 +25 +1/2(2000 +1960)=2995 1.01 B 1010 +1/2(2025 +2000)=3022.5 1000 +25 +1/2(2000 +2000)=3025 0.99 B 990+1/2(2000+1960)=2970 1000+25+1/2(2000+1985)=3017.5 (Expected) Cost-minimizing decisions at 1:00 1.0 A 1000 +1/2(3000 +2992.5)=3996.25 1000 +25 +1/2(3022.5 +2970)=4021.25 1.0 B 1000 +1/2(3022.5 +2970)=3996.25 1000 +25 +1/2(3000 +2992.5)=4021.25 Note: The $ costs of the decision that minimizes (expected) costs is underlined. simplicitytheinitialwagerisassumedtobe$1000atoneo’clockand the sequence follows a basic binomial process with an equal chance of up and down where the up-movement factor is 1.01, while the down-movement factor is 0.99. Hence, expectations in the present case are assumed static (the current rate is expected to prevail in the future).Thepossiblewagerdevelopmentsoverthebettor’s planning horizon are reflected in the binomial tree shown in Fig. 1. Following this procedure, the wager variation of betting on team A over the first four innings can be shown in the following binomial tree as: The statistics shown in Fig. 1can be interpreted as the wagers during different stages of the game given the volatility estimate. For example, the wager increases to 1.03 at 1:30 if one wishes to bet at this stage of the game. This is because the chance for team A to win the game at this point of time is enhanced, given the path of the development. On the other hand, the wager decreases to 0.97 in the event of moving along the down-path consecutively. The wagers shownin differentnodesofthetree thus representthecost of betting if one enters the game betting at different time of the game. These numbers are calculated based upon a hypothetical estimate of σ2, which may vary in the real-world setting. SDP and risk implications In this subsection, the cost-minimizing decisions are determined first and then their risk implications are assessed. Cost-minimizing decisionsare notdirectly obvious,sincethe optimaldecisionsatany time depend on decisions taken in the past due to the existence of switchingcostsanduncertaintyin payoffs.SDP(DixitandPindyck, 1994) is the standard procedure for such decision-making. The first stepis tocalculateoptimal decisionsfor thelastperiod inabinomial treeforallpossiblepayoffsandforthetwoinvestmentstrategies(i.e. switching or status quo). Cost-minimizing decisions for all possible statesare obtainedbycomparingthecostsofcontinuing theexisting strategy to the costs of switching to the other strategy. Then, having obtainedthebestdecisionsfor1:30,thosefor1:20canbedetermined and the procedures go on. This procedure is illustrated in Table 4 which also shows that the optimal decision in each period depends ontheprevailingwagerandtheteamthebettorinvestedbefore.Note that the SDP procedure used in these calculations is the same as the procedure used to determine the value of anAmerican option (Cox et al., 1979). The probability distribution of investment costs, and hence the risk implications of cost minimization can be obtained by tracking the eight possible wager paths (Fig. 1) and determining cost-minimizing decisions in each case (Table 4). If the switching cost is $25, the cost-minimizing decisions are shown in Table 4. The calculations in Table 4show that a costminimizing bettor initially betting on teamA should definitely stick to this strategy until 1:30 and should switch to team B only when the wager path has become 0.97. If the bettor’s initial bet is on team B, it should definitely stick to this strategy until 1:30 and shouldswitchto teamAonlywhen the wagerpath has become1.03. Using the procedure of SDP, we can obtain the minimum expected costs of $3996.25. The total investment is $4000 (assuming $1000 each inning for the ease of comparing with the switching strategy); but because the risk-free interest rate is taken to be near zero, the discount factor 1, then the NPV =4000.As a result, the value of switching option is $4000–$3996.25 =$3.75. If the switching cost is lowered to $10, the cost-minimizing decisions are shown in Table 5. The calculations in Table 5show that a 80 S.-Y. Wang Table 5. Deriving cost-minimizing decisions by means of SDP when switching cost =$10 Wager Existing $ Costs of continuing $ Costs of switching to path strategy existing strategy alternative strategy Cost-minimizing decisions at 1:30 1.03 A 1000 1030 +10 =1040 1.01 A 1000 1010 +10 =1020 0.99 A 1000 990 +10 =1000 0.97 A 1000 970 +10 =980 1.03 B 1030 1000 +10 =1010 1.01 B 1010 1000 +10 =1010 0.99 B 990 1000 +10 =1010 0.97 B 970 1000 +10 =1010 (Expected) Cost-minimizing decisions at 1:20 1.02 A 1000 +1/2(1000 +1000)=2000 1020 +10 +1/2(1010 +1010)=2040 1.0 A 1000 +1/2(1000 +1000)=2000 1000 +10 +1/2(1010 +990)=2010 0.98 A 1000 +1/2(1000 +980)=1990 980 +10 +1/2(990 +970)=1970 1.02 B 1020 +1/2(1010 +1010)=2030 1000 +10 +1/2(1000 +1000)=2010 1.0 B 1000 +1/2(1010 +990)=2000 1000 +10 +1/2(1000 +1000)=2010 0.98 B 980 +1/2(990 +970)=1960 1000 +10 +1/2(1000 +980)=2000 (Expected) Cost-minimizing decisions at 1:10 1.01 A 1000 +1/2(2000 +2000)=3000 1010 +10 +1/2(2010 +2000)=3025 0.99 A 1000 +1/2(2000 +1970)=2985 990 +10 +1/2(2000 +1960)=2980 1.01 B 1010 +1/2(2010 +2000)=3015 1000 +10 +1/2(2000 +2000)=3010 0.99 B 990 +1/2(2000 +1960)=2970 1000 +10 +1/2(2000 +1970)=2995 (Expected) Cost-minimizing decisions at 1:00 1.0 A 1000 +1/2(3000 +2980)=3990 1000 +10 +1/2(3010 +2970)=4000 1.0 B 1000 +1/2(3010 +2970)=3990 1000 +10 +1/2(3000 +2980)=4000 Note: The $ costs of the decision that minimizes (expected) costs is underlined. Table 6. The value of flexibility: outcome of cost-minimizing decisions under different wager path scenarios Switching cost =$10 Switching cost =$25 Wager path A (a) B (b) A (c) B (d) Path 1: (1.0)-(1.01)-(1.02)-(1.03) 4000 4010 4000 4035 Path 2: (1.0)-(1.01)-(1.02)-(1.01) 4000 4020 4000 4060 Path 3: (1.0)-(1.01)-(1.0)-(1.01) 4000 4020 4000 4020 Path 4: (1.0)-(1.01)-(1.0)-(0.99) 4000 4020 4000 4000 Path 5: (1.0)-(0.99)-(1.0)-(1.01) 4020 4000 4000 4000 Path 6: (1.0)-(0.99)-(1.0)-(0.99) 4000 3980 4000 3980 Path 7: (1.0)-(0.99)-(0.98)-(0.99) 3980 3960 4020 3960 Path 8: (1.0)-(0.99)-(0.98)-(0.97) 3950 3940 3975 3940 Average costs 3993.75 3993.75 3999.375 3999.375 cost-minimizing bettor initially betting on teamA should definitely stick to this strategy until 1:30 and should switch to team B only when the wager path has become 0.97. If the initial bet is on team A, there is no difference either to switch or to stay status quo at 1:30 whenthewagerpath is0.99.Similarly, iftheinitialbetis on team B, there is no difference whether to switch or to hold onto the original team when the wager path has become 1.01. When the switching costs are lowered from $25 to $10; the value of switching option is changed from $3.75 to $10 ($4000–$3990). Therefore, we find that the lower the switching cost, the higher the value of switching option. The value of flexibility Based upon the statistics shown in Tables 4and 5, Table 6summa- rizes the value of flexibility. The statistics shown in columns (a) and (b) in Table 6can be derived directly from Table 5. Columns (c) and (d) can be derived directly from Table 4. Teams with relatively low switching costs benefit from wager uncertainty: Table 6shows that the lower the switching cost, the lower the investment cost.When switching cost =10, the expected investment cost for a bettor that switches optimally is $3993.75 regardlessofitsinitialstrategy,whereasabettorthatstartswithteam A and sticks to it pays a certain $1000. Moreover, wager volatility should be assessed relative to the degree of uncertainty. When the switching costs are lowered from, say, $25–$10, we find that investing in flexibility decreases expected costs and lowers risk: Comparing (c) and (d) with (a) and (b) shows that investing in lower switching costs yields two benefits: expected investment costs fall (from $3999.375 to $3993.75) and so does risk (the worst scenario costs now $4020 instead of $4060). Hence, the investment in switching should definitely be made if it costs $5.625 or less (i.e. The real options approach to the baseball game betting 81 $3999.375–$3993.75). If it costs more, nevertheless, it may still be worthwhile if enough value is attached to the risk reduction. V. Conclusions Investing in flexibility serves an important purpose, especially if uncertainty is high. The uses of options are based on an assumption of aversion to downside risk. This implies that the objective of managing risk should be to exploit the wager volatility in order to decrease the bettor’s cost, while ensuring that the downside risk remains sufficiently small. In this article, we introduce the concept of baseball options with a switching feature. More specifically, we propose the use of switching options to facilitate the pricing of baseball games. We believe this is the first attempt to apply the real options theory to baseball game scheme. To do so, we first design some rules in our assumptions and define variables that are related to the baseball options. We then use the ROA to demonstrate the decision rules for betting a baseball game. 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