BinomialDiscreteDividendsTree
Updated: 15 Dec 2013
Use the table-valued function BinomialDiscreteDividendsTree to return the option value, intrinsic value, underlying value, and present value of the dividend amounts for each node on a binomial tree for an American or European option paying discrete dividends.
Syntax
SELECT * FROM [wctOptions].[wct].[BinomialDiscreteDividendsTree](
<@CallPut, nvarchar(4000),>
,<@AmEur, nvarchar(4000),>
,<@AssetPrice, float,>
,<@StrikePrice, float,>
,<@TimeToMaturity, float,>
,<@RiskFreeRate, float,>
,<@DividendRate, float,>
,<@Dividend_RangeQuery, nvarchar(max),>
,<@Volatility, float,>
,<@NumberOfSteps, int,>)
Arguments
@CallPut
identifies the option as being a call ('C') or a put ('P'). @CallPut is an expression of type nvarchar or of a type that can be implicitly converted to nvarchar.
@AmEur
identifies the option as being American ('A') or European ('E'). @AmEur is an expression of type nvarchar or of a type that can be implicitly converted to nvarchar.
@AssetPrice
the price of the underlying asset. @AssetPrice is an expression of type float or of a type that can be implicitly converted to float.
@StrikePrice
the exercise price of the option. @StrikePrice is an expression of type float or of a type that can be implicitly converted to float.
@TimeToMaturity
the time to expiration of the option, expressed in years. @TimeToMaturity is an expression of type float or of a type that can be implicitly converted to float.
@RiskFreeRate
the annualized, continuously compounded zero-coupon risk-free rate over the life of the option. @RiskFreeRate is an expression of type float or of a type that can be implicitly converted to float.
@DividendRate
the annualized, continuously compounded zero-coupon dividend rate over the life of the option, used in addition to the discrete dividends. @DividendRate is an expression of type float or of a type that can be implicitly converted to float.
@Dividend_RangeQuery
a string containing an SQL statement which, when executed, provides the function with the times and amounts of the dividends to be used in the calculation. The results of the SQL must contain exactly two columns, the first being the time value, as a float or as a value that implicitly converts to float, and the second being the dividend amount as float, or as a value that implicitly converts to float. @Dividend_RangeQuery is an expression of type nvarchar or of a type that can be implicitly converted to nvarchar.
@Volatility
the volatility of the relative price change of the underlying asset. @Volatility is an expression of type float or of a type that can be implicitly converted to float.
@NumberOfSteps
the number of steps in the binomial tree. @NumberOfSteps is an expression of type int or of a type that can be implicitly converted to int.
Return Types
RETURNS TABLE (
[node] [int] NULL,
[stepno] [int] NULL,
[underlying] [float] NULL,
[intrinsic] [float] NULL,
[price] [float] NULL,
[PVdividends] [float] NULL
)
Remarks
· @Volatility must be greater than zero (@Volatility > 0).
· @TimeToMaturity must be greater than zero (@TimeToMaturity > 0).
· @AssetPrice must be greater than zero (@AssetPrice > 0).
· @StrikePrice must be greater than zero (@StrikePrice > 0).
· @NumberOfSteps must be greater than 1 (@NumberOfSteps > 1)
· Negative time values returned by @Dividend_RangeQuery are ignored.
· Time values returned by @Dividend_RangeQuery that are greater than @TimeToMaturity are ignored.
· If @RiskFreeRate is NULL then @RiskFreeRate is set to zero.
· If @DividendRate is NULL then @DividendRate is set to zero.
Examples
In this example we have an American put where the underlying price is 50, the exercise price is 50, the risk-free rate is 0.10, the volatility is 0.4 and dividends are expected according to the following schedule:
|
T
|
amt
|
|
0.071233
|
0.11
|
|
0.320548
|
0.11
|
|
0.569863
|
0.12
|
|
0.819178
|
0.12
|
The number of steps is 10 and the time-to-maturity is 1 (year).
SELECT *
FROM wct.BinomialDiscreteDividendsTree(
'P', --@CallPut
'A', --@AmEur
50, --@AssetPrice
50, --@StrikePrice
1, --@TimeToMaturity
0.10, --@RiskFreeRate
0, --@DividendRate
'SELECT T, amt
FROM (VALUES
(0.071233,0.11),
(0.320548,0.11),
(0.569863,0.12),
(0.819178,0.12)
)n(T,amt)' --@Dividend_RangeQuery
,0.4 --@Volatility
,10 --@NumberOfSteps
)
This produces the following result which has been reformatted for presentation purposes.
|
node
|
stepno
|
underlying
|
intrinsic
|
price
|
PVdividends
|
|
0
|
0
|
50
|
0
|
6.012745681
|
0.439663671
|
|
0
|
1
|
44.00544108
|
5.994558916
|
8.538523152
|
0.333765472
|
|
1
|
1
|
56.57678575
|
0
|
3.685860644
|
0.333765472
|
|
0
|
2
|
38.81981374
|
11.18018626
|
11.78110746
|
0.337119871
|
|
1
|
2
|
49.8974562
|
0.1025438
|
5.56748544
|
0.337119871
|
|
2
|
2
|
64.16391291
|
0
|
1.936708336
|
0.337119871
|
|
0
|
3
|
34.2507646
|
15.7492354
|
15.7492354
|
0.340507982
|
|
1
|
3
|
44.01218359
|
5.987816406
|
8.171636845
|
0.340507982
|
|
2
|
3
|
56.58352826
|
0
|
3.155898956
|
0.340507982
|
|
3
|
3
|
72.77366484
|
0
|
0.794420441
|
0.340507982
|
|
0
|
4
|
30.11415998
|
19.88584002
|
19.88584002
|
0.233052691
|
|
1
|
4
|
38.71574656
|
11.28425344
|
11.601084
|
0.233052691
|
|
2
|
4
|
49.79338902
|
0.20661098
|
5.012393116
|
0.233052691
|
|
3
|
4
|
64.05984573
|
0
|
1.420598017
|
0.233052691
|
|
4
|
4
|
82.43304995
|
0
|
0.2037783
|
0.233052691
|
|
0
|
5
|
26.56608838
|
23.43391162
|
23.43391162
|
0.235394909
|
|
1
|
5
|
34.14565153
|
15.85434847
|
15.85434847
|
0.235394909
|
|
2
|
5
|
43.90707052
|
6.092929479
|
7.711940855
|
0.235394909
|
|
3
|
5
|
56.47841519
|
0
|
2.497448863
|
0.235394909
|
|
4
|
5
|
72.66855177
|
0
|
0.405934689
|
0.235394909
|
|
5
|
5
|
93.51918705
|
0
|
0.012051837
|
0.235394909
|
|
0
|
6
|
23.31953126
|
26.68046874
|
26.68046874
|
0.117398478
|
|
1
|
6
|
29.99850577
|
20.00149423
|
20.00149423
|
0.117398478
|
|
2
|
6
|
38.60009234
|
11.39990766
|
11.39990766
|
0.117398478
|
|
3
|
6
|
49.67773481
|
0.322265193
|
4.293266445
|
0.117398478
|
|
4
|
6
|
63.94419152
|
0
|
0.807878337
|
0.117398478
|
|
5
|
6
|
82.31739573
|
0
|
0.024743819
|
0.117398478
|
|
6
|
6
|
105.9795173
|
0
|
0
|
0.117398478
|
|
0
|
7
|
20.5638798
|
29.4361202
|
29.4361202
|
0.118578352
|
|
1
|
7
|
26.44927183
|
23.55072817
|
23.55072817
|
0.118578352
|
|
2
|
7
|
34.02883497
|
15.97116503
|
15.97116503
|
0.118578352
|
|
3
|
7
|
43.79025396
|
6.209746036
|
7.155866083
|
0.118578352
|
|
4
|
7
|
56.36159863
|
0
|
1.606205356
|
0.118578352
|
|
5
|
7
|
72.55173521
|
0
|
0.050801928
|
0.118578352
|
|
6
|
7
|
93.40237049
|
0
|
0
|
0.118578352
|
|
7
|
7
|
120.2550773
|
0
|
0
|
0.118578352
|
|
0
|
8
|
18.13580143
|
31.86419857
|
31.86419857
|
0.119770085
|
|
1
|
8
|
23.32190286
|
26.67809714
|
26.67809714
|
0.119770085
|
|
2
|
8
|
30.00087737
|
19.99912263
|
19.99912263
|
0.119770085
|
|
3
|
8
|
38.60246395
|
11.39753605
|
11.39753605
|
0.119770085
|
|
4
|
8
|
49.68010641
|
0.319893587
|
3.190014155
|
0.119770085
|
|
5
|
8
|
63.94656313
|
0
|
0.104302245
|
0.119770085
|
|
6
|
8
|
82.31976734
|
0
|
0
|
0.119770085
|
|
7
|
8
|
105.9818889
|
0
|
0
|
0.119770085
|
|
8
|
8
|
136.4553975
|
0
|
0
|
0.119770085
|
|
0
|
9
|
15.87540229
|
34.12459771
|
34.12459771
|
0
|
|
1
|
9
|
20.44530145
|
29.55469855
|
29.55469855
|
0
|
|
2
|
9
|
26.33069348
|
23.66930652
|
23.66930652
|
0
|
|
3
|
9
|
33.91025662
|
16.08974338
|
16.08974338
|
0
|
|
4
|
9
|
43.67167561
|
6.328324388
|
6.328324388
|
0
|
|
5
|
9
|
56.24302028
|
0
|
0.214144596
|
0
|
|
6
|
9
|
72.43315686
|
0
|
0
|
0
|
|
7
|
9
|
93.28379214
|
0
|
0
|
0
|
|
8
|
9
|
120.136499
|
0
|
0
|
0
|
|
9
|
9
|
154.7190359
|
0
|
0
|
0
|
|
0
|
10
|
13.98911852
|
36.01088148
|
36.01088148
|
0
|
|
1
|
10
|
18.01603134
|
31.98396866
|
31.98396866
|
0
|
|
2
|
10
|
23.20213278
|
26.79786722
|
26.79786722
|
0
|
|
3
|
10
|
29.88110729
|
20.11889271
|
20.11889271
|
0
|
|
4
|
10
|
38.48269386
|
11.51730614
|
11.51730614
|
0
|
|
5
|
10
|
49.56033633
|
0.439663671
|
0.439663671
|
0
|
|
6
|
10
|
63.82679304
|
0
|
0
|
0
|
|
7
|
10
|
82.19999725
|
0
|
0
|
0
|
|
8
|
10
|
105.8621188
|
0
|
0
|
0
|
|
9
|
10
|
136.3356274
|
0
|
0
|
0
|
|
10
|
10
|
175.5812515
|
0
|
0
|
0
|