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XLeratorDB/financial-options Documentation

SQL server function for pricing binary barrier options


BinaryBarrierOnly

Updated: 31 Oct 2013


Use BinaryBarrierOnly to calculate the price or Greeks for binary barrier options having payoffs that are received only at expiration. Down-and-in and up-and-in options have positive payoffs if the barrier is breached at some time before expiration. Down-and-out and up-and-out options have positive payoffs if the barrier is not breached before expiration.
BinaryBarrierOnly valuations are based on the formulae published by Mark Rubinstein and Eric Reiner in 1991. In their July 31, 1991 paper Binary Options, they enumerated 28 different types of binary barrier options. This function deals with options numbers 5–12:
(5)    down-and-in cash-(at-expiry)-or-nothing;
(6)    up-and-in cash-(at-expiry)-or-nothing;
(7)    down-and-in asset-(at-expiry)-or-nothing;
(8)    up-and-in asset-(at-expiry)-or-nothing;
(9)    down-and-out cash-or-nothing;
(10)   up-and-out cash-or-nothing;
(11)   down-and-out asset-or-nothing; and
(12)   up-and-out asset-or-nothing.
Syntax
SELECT [wctOptions].[wct].[BinaryBarrierOnly](
  <@BarrierType, nvarchar(4000),>
 ,<@CashOrNothing, bit,>
 ,<@AssetPrice, float,>
 ,<@StrikePrice, float,>
 ,<@BarrierPrice, float,>
 ,<@Rebate, float,>
 ,<@TimeToMaturity, float,>
 ,<@RiskFreeRate, float,>
 ,<@DividendRate, float,>
 ,<@Volatility, float,>
 ,<@ReturnValue, nvarchar(4000),>)
Arguments
@BarrierType
identifies the type of barrier as 'UI' (Up-and-In), 'UO' (Up-and-Out), 'DI' (Down-and-In), or 'DO' (Down-and-out). @BarrierType must be of a type nvarchar or of a type that implicitly converts to nvarchar.
@CashOrNothing
identifies the option as either a cash-or-nothing ('True') or asset-or-nothing ('False') binary barrier. @CashOrNothing is an expression of type bit or of a type that can be implicitly converted to bit.
@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.
@BarrierPrice
For a knock-in option, @BarrierPrice is the value at which the option comes into existence if the @AssetPrice crosses the barrier. For a knock-out option, @BarrierPrice is the value at which the option is extinguished if the @AssetPrice crosses the barrier. @BarrierPrice must be of a type float or of a type that implicitly converts to float.
@Rebate
An amount paid to the buyer of the option in the event that the barrier is never breached. @Rebate must be of a type float or of a type that implicitly converts 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 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 continuously compounded zero coupon dividend rate over the life of the option. For currency options, @DividendRate should be the foreign risk-free zero coupon rate. @DividendRate is an expression of type float or of a type that can be implicitly converted to float.
@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.
@ReturnValue
identifies the calculation to be performed. @ReturnValue is an expression of type nvarchar or of a type that can be implicitly converted to nvarchar. For a full description of the return values, see BinaryBarrierOnlyPriceNGreeks. @ReturnValue is not case-sensitive. The following values are acceptable for @ReturnValue

@ReturnValue
Returns
'P','PRICE'
Price
'D','DELTA'
Delta
'G','GAMMA'
Gamma
'T','THETA'
Theta
'V','VEGA'
Vega
'R','RHO'
Rho
'L','LAMBDA'
Lambda
'DDDV','VANNA','DVEGADSPOT','DDELTADVOL'
DdeltaDvol
'DVV','DDELTADVOLDVOL'
DdeltaDvolDvol
'DT','CHARM','DDELTADTIME'
DdeltaDtime
'GV','ZOMMA','DGAMMADVOL'
DgammaDvol
'GP','GAMMAP'
GammaP
'DVDV','VOMMA','VOLGA','DVEGADVOL'
DvegaDvol
'VP','VEGAP'
VegaP
'PR2','PHIRHO2'
PhiRho2
'S','SPEED','DGAMMADSPOT'
DgammaDspot
'DX','DELTAX'
Delta X
'GX','GAMMAX','DX','RND','RISKNEUTRALDENSITY'
Risk Neutral Density
'VVV','ULTIMA','DVOMMADVOL'
DvommaDvol
'VT','VETA','DVEGADTIME'
DvegaDtime
'GT','COLOR','DGAMMADTIME'
DgammaDtime
'FR','RHOFUTURESOPTIONS','FUTURESOPTIONSRHO'
Futures Options Rho
'B','CARRYSENSITIVITY'
Carry Sensitivity

Return Type
float
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).
·         If @ReturnValue is NULL, then @ReturnValue is set to 'P'.
·         If @DividendRate is NULL then @DividendRate = 0.
·         If @RiskFreeRate is NULL @RiskFreeRate = 0.
·         @BarrierPrice must be greater than zero (@BarrierPrice > 0).
·         @Rebate must be greater than or equal to zero (@Rebate >= 0).
·         If @Rebate is NULL, then @Rebate = 0.
·         @BarrierPrice assumes continuous monitoring.
·         To convert a non-continuous @BarrierPrice use the AdjustedBarrier function.
·         Use BinaryBarrierOnlyPriceNGreeks to calculate the price and all the Greeks.
Example
In this example we calculate the theoretical value (price) of a down-and-in cash-(at-expiry)-or-nothing option.
SELECT wct.BinaryBarrierOnly(
 'DI'          --BarrierType
,'True'        --CashOrNothing
,100           --AssetPrice
,100           --StrikePrice
,92            --BarrierPrice
,2             --Rebate
,.50           --TimeToMaturity
,.10           --RiskFreeRate
,.05           --DividendRate
,.20           --Volatility
,'P'           --ReturnValue
) as Price
This produces the following result.
                 Price
----------------------
     0.990650335835426
This SQL returns delta for the same option.
SELECT wct.BinaryBarrierOnly(
 'DI'          --BarrierType
,'True'        --CashOrNothing
,100           --AssetPrice
,100           --StrikePrice
,92            --BarrierPrice
,2             --Rebate
,.50           --TimeToMaturity
,.10           --RiskFreeRate
,.05           --DividendRate
,.20           --Volatility
,'D'           --ReturnValue
) as 'Delta'
This produces the following result.
                 Delta
----------------------
    -0.059603371860284
This SELECT statement reproduces the table for cash-or-nothing where Call/Put = Both from the Binary Options paper by Mark Rubenstein and Eric Reiner, July 31, 1991.
SELECT *
FROM (
   SELECT 'OUT' as [OUT/IN]
   ,n.T
   ,n2.H
   ,wct.BinaryBarrierOnly(CASE WHEN n2.H <= 100 THEN 'DO' ELSE 'UO' END,'True',100,100,n2.H,2,n.T,.10,.05,.20,'P') as Price
   FROM (VALUES (.05),(.50),(.95))n(T)
   CROSS APPLY(VALUES (92),(96),(100),(104),(108))n2(H)
   UNION ALL
   SELECT 'IN'
   ,n.T
   ,n2.H
   ,wct.BinaryBarrierOnly(CASE WHEN n2.H <= 100 THEN 'DI' ELSE 'UI' END,'True',100,100,n2.H,2,n.T,.10,.05,.20,'P') as Price
   FROM (VALUES (.05),(.50),(.95))n(T)
   CROSS APPLY(VALUES (92),(96),(100),(104),(108))n2(H)
   ) D
PIVOT(SUM(Price) for T in([.05],[.50],[.95])) as P
ORDER BY 1
Here are the results of this query, reformatted for ease of viewing.

OUT/IN
H
0.05
0.5
0.95
IN
92
0.116333
0.99065
1.139267
IN
96
0.697207
1.42435
1.468934
IN
100
1.990025
1.902459
1.818746
IN
104
0.779568
1.529522
1.571909
IN
108
0.179699
1.179427
1.331495
OUT
92
1.873692
0.911809
0.679479
OUT
96
1.292818
0.478108
0.349812
OUT
100
0
0
0
OUT
104
1.210457
0.372937
0.246837
OUT
108
1.810326
0.723031
0.487251

This SELECT statement reproduces the table for asset-or-nothing where Call/Put = Both from the Binary Options paper by Mark Rubenstein and Eric Reiner, July 31, 1991.
SELECT *
FROM (
   SELECT 'OUT' as [OUT/IN]
   ,n.T
   ,n2.H
   ,wct.BinaryBarrierOnly(CASE WHEN n2.H <= 100 THEN 'DO' ELSE 'UO' END,'False',100,100,n2.H,n2.H,n.T,.10,.05,.20,'P') as Price
   FROM (VALUES (.05),(.50),(.95))n(T)
   CROSS APPLY(VALUES (92),(96),(100),(104),(108))n2(H)
   UNION ALL
   SELECT 'IN'
   ,n.T
   ,n2.H
   ,wct.BinaryBarrierOnly(CASE WHEN n2.H <= 100 THEN 'DI' ELSE 'UI'    END,'False',100,100,n2.H,n2.H,n.T,.10,.05,.20,'P') as Price
   FROM (VALUES (.05),(.50),(.95))n(T)
   CROSS APPLY(VALUES (92),(96),(100),(104),(108))n2(H)
   ) D
PIVOT(SUM(Price) for T in([.05],[.50],[.95])) as P
ORDER BY 1
Here are the results of this query, reformatted for ease of viewing.

OUT/IN
H
0.05
0.5
0.95
IN
92
5.354976833
46.30137593
54.22773205
IN
96
33.50814925
69.74709336
73.41124416
IN
100
99.75031224
97.5309912
95.36104731
IN
104
40.58998192
81.15264996
85.12583223
IN
108
9.710977743
64.74642884
74.46348397
OUT
92
94.39533541
51.22961527
41.13331526
OUT
96
66.24216299
27.78389785
21.94980315
OUT
100
0
0
0
OUT
104
59.16033032
16.37834124
10.23521508
OUT
108
90.0393345
32.78456236
20.89756334

 


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