A real options approach to the baseball game betting

Figures & data

Fig. 1. Numerical example – paths of wager in the first four innings

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

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
C		0	1
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	σ^2	σ^2
0.243	0.805	5.234	4.369	3.224

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
1.01	A	1000
0.99	A	1000	990+25=1015
0.97	A	1000
1.03	B	1030
1.01	B	1010
0.99	B	990
0.97	B	970
(Expected) Cost-minimizing decisions at 1:20
1.02	A
1.0	A
0.98	A
1.02	B
1.0	B
0.98	B
(Expected) Cost-minimizing decisions at 1:10
1.01	A
0.99	A
1.01	B
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
1.0	B

Note: The $ costs of the decision that minimizes (expected) costs is underlined.

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
1.01	A	1000
0.99	A	1000
0.97	A	1000
1.03	B	1030
1.01	B	1010
0.99	B	990
0.97	B	970
(Expected) Cost-minimizing decisions at 1:20
1.02	A
1.0	A
0.98	A
1.02	B
1.0	B
0.98	B
(Expected) Cost-minimizing decisions at 1:10
1.01	A
0.99	A
1.01	B
0.99	B
(Expected) Cost-minimizing decisions at 1:00
1.0	A
1.0	B

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		Switching
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