BinaryBarrierOnlyPriceNGreeks
Updated: 31 Oct 2013
Use the table-valued function BinaryBarrierOnlyPriceNGreeks to calculate the price and other derivatives 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.
BinaryBarrierOnlyPriceNGreeks 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 * FROM [wctOptions].[wct].[BinaryBarrierOnlyPriceNGreeks](
<@BarrierType, nvarchar(4000),>
,<@CashOrNothing, bit,>
,<@AssetPrice, float,>
,<@StrikePrice, float,>
,<@BarrierPrice, float,>
,<@Rebate, float,>
,<@TimeToMaturity, float,>
,<@RiskFreeRate, float,>
,<@DividendRate, float,>
,<@Volatility, float,>)
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.
Return Type
RETURNS TABLE (
[Price] [float] NULL,
[Delta] [float] NULL,
[Gamma] [float] NULL,
[Theta] [float] NULL,
[Vega] [float] NULL,
[Rho] [float] NULL,
[Lambda] [float] NULL,
[GammaP] [float] NULL,
[DdeltaDtime] [float] NULL,
[DdeltaDvol] [float] NULL,
[DdeltaDvolDvol] [float] NULL,
[DgammaDvol] [float] NULL,
[DvegaDvol] [float] NULL,
[VegaP] [float] NULL,
[PhiRho2] [float] NULL,
[DgammaDspot] [float] NULL,
[DeltaX] [float] NULL,
[RiskNeutralDensity] [float] NULL,
[DvommaDvol] [float] NULL,
[DgammaDtime] [float] NULL,
[DvegaDtime] [float] NULL,
[FuturesOptionsRho] [float] NULL,
[CarrySensitivity] [float] NULL,
[ForwardPrice] [float] NULL,
[ForwardPoints] [float] NULL
The columns returned by BinaryBarrierOnlyPriceNGreeksPriceNGreeks are measuring the sensitivity of the theoretical value of the option.
|
Column
|
Description
|
|
Price
|
The theoretical value of the option.
|
|
Delta
|
The sensitivity to small changes in the asset price; the first derivative of the option with respect to price.
|
|
Gamma
|
The rate of change in Delta with respect to small changes in the asset price; the second derivative of the option with respect to price.
|
|
Theta
|
The sensitivity to small changes in time; the first derivative of the option with respect to time.
|
|
Vega
|
The sensitivity to small changes in volatility; the first derivative of the option with respect to volatility.
|
|
Rho
|
The sensitivity to small changes in the risk-free rate; the first derivative of the option with respect to the risk-free rate.
|
|
Lambda
|
Delta multiplied by the asset price divided by the theoretical value. If the theoretical value is zero, then lambda is set to zero.
|
|
GammaP
|
Gamma multiplied by asset price divided by strike price.
|
|
DdeltaDtime
|
The instantaneous change in delta over the passage of time; the second derivative, once to asset price and once to time.
|
|
DdeltaDvol
|
The sensitivity of delta with respect to volatility; the second derivative, once to asset price and once to volatility.
|
|
DdeltaDvolDvol
|
The second derivative of delta with respect to volatility; the third derivative, once to asset price and twice to volatility.
|
|
DgammaDvol
|
The rate of change in gamma with respect to changes in volatility; the third derivative, twice to asset price and once to volatility.
|
|
DvegaDvol
|
The rate of change to Vega as the volatility changes; the second derivative with respect to volatility.
|
|
VegaP
|
The percentage change in theoretical value for a 10 per cent change in volatility.
|
|
PhiRho2
|
The sensitivity to a change in the dividend yield (foreign interest rate for a currency option); the first derivative with respect to dividend yield.
|
|
DgammaDspot
|
The rate of change in gamma with respect to change in the asset price; the third derivative with respect to price.
|
|
DeltaX
|
The sensitivity to a change in the strike price; the first derivative with respect to strike price.
|
|
RiskNeutralDensity
|
The sensitivity of DeltaX; the second derivative with respect to strike price.
|
|
DvommaDvol
|
The sensitivity of DvegaDvol to changes in volatility; the third derivative, twice to asset price and once to volatility.
|
|
DgammaDtime
|
The sensitivity of Gamma to the passage of time; the third derivative, twice to asset price and once to time.
|
|
DvegaDtime
|
The sensitivity of Vega to the passage of time; the second derivative, once to volatility and once to time.
|
|
FuturesOptionsRho
|
The sensitivity to change in risk-free rate assuming a cost-of-carry of zero.
|
|
CarrySensitivity
|
-PhiRho2
|
|
ForwardPrice
|
The value of the underlying asset at the expiration date of the option.
|
|
ForwardPoints
|
The difference between the ForwardPrice and the asset price.
|
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.
Example
In this example we calculate the price and Greeks of a down-and-in cash-(at-expiry)-or-nothing option.
SELECT *
FROM wct.BinaryBarrierOnlyPriceNGreeks(
'DI' --BarrierType
,'True' --CashOrNothing
,100 --AssetPrice
,100 --StrikePrice
,92 --BarrierPrice
,2 --Rebate
,.50 --TimeToMaturity
,.10 --RiskFreeRate
,.05 --DividendRate
,.20 --Volatility
)
Here are the first few columns of the resultant table.
In this SELECT we un-pivot the columns returned by the function for ease of viewing the results.
SELECT n.*
FROM wct.BinaryBarrierOnlyPriceNGreeks(
'DI' --BarrierType
,'True' --CashOrNothing
,100 --AssetPrice
,100 --StrikePrice
,92 --BarrierPrice
,2 --Rebate
,.50 --TimeToMaturity
,.10 --RiskFreeRate
,.05 --DividendRate
,.20 --Volatility
)
CROSS APPLY(VALUES
('Price',Price)
,('Delta',Delta)
,('Gamma',Gamma)
,('Theta',Theta)
,('Vega',Vega)
,('Rho',Rho)
,('Lambda',Lambda)
,('GammaP',GammaP)
,('DdeltaDtime',DdeltaDtime)
,('DdeltaDvol',DdeltaDvol)
,('DdeltaDvolDvol',DdeltaDvolDvol)
,('DgammaDvol',DgammaDvol)
,('DvegaDvol',DvegaDvol)
,('VegaP',VegaP)
,('PhiRho2',PhiRho2)
,('DgammaDspot',DgammaDspot)
,('DeltaX',DeltaX)
,('RiskNeutralDensity',RiskNeutralDensity)
,('DvommaDvol',DvommaDvol)
,('DgammaDtime',DgammaDtime)
,('DvegaDtime',DvegaDtime)
,('FuturesOptionsRho',FuturesOptionsRho)
,('CarrySensitivity',CarrySensitivity)
,('ForwardPrice',ForwardPrice)
,('ForwardPoints',ForwardPoints)
)n([Return Value], Value)
This produces the following result.
Return Value Value
------------------ ----------------------
Price 0.990650335835426
Delta -0.0921821992078131
Gamma 0.00581755754680557
Theta -0.00166113004141066
Vega 0.0461322337422132
Rho -0.0269553299150849
Lambda 0
GammaP 0.00581755754680557
DdeltaDtime -0.000161429127295386
DdeltaDvol 0.00276123440778875
DdeltaDvolDvol 6.69511645234877E-05
DgammaDvol -0.000727088304008916
DvegaDvol -0.00369859602500355
VegaP 0.0922644674844264
PhiRho2 0.0220020781965924
DgammaDspot 9.76890790482798E-06
DeltaX 0
RiskNeutralDensity 0
DvommaDvol 0.000347967943149729
DgammaDtime -0.000231926402740369
DvegaDtime 0.00528397407259708
FuturesOptionsRho -0.0049532516812445
CarrySensitivity -0.0220020781965924
ForwardPrice 102.531512052443
ForwardPoints 2.53151205244289
Use CROSS APPLY when there are many input rows.
SELECT n.bt,n.K,k.Price,K.Delta,K.Gamma,K.Theta,K.Vega,K.Rho
FROM (VALUES
(1,'DI','True',100,100,92,2,.50,.10,.05,.20)
,(2,'DI','True',100,100,96,2,.50,.10,.05,.20)
,(3,'DI','True',100,100,100,2,.50,.10,.05,.20)
,(4,'UI','True',100,100,104,2,.50,.10,.05,.20)
,(5,'UI','True',100,100,108,2,.50,.10,.05,.20)
,(6,'DO','True',100,100,92,2,.50,.10,.05,.20)
,(7,'DO','True',100,100,96,2,.50,.10,.05,.20)
,(8,'DO','True',100,100,100,2,.50,.10,.05,.20)
,(9,'UO','True',100,100,104,2,.50,.10,.05,.20)
,(10,'UO','True',100,100,108,2,.50,.10,.05,.20)
)n(rn,bt,cash,S,X,K,H,T,Rf,Rd,Vol)
CROSS APPLY wct.BinaryBarrierOnlyPriceNGreeks(bt,cash,S,X,K,H,T,Rf,Rd,Vol)k
Here are the (reformatted) results.
|
bt
|
K
|
Price
|
Delta
|
Gamma
|
Theta
|
Vega
|
Rho
|
|
DI
|
92
|
0.99065
|
-0.09218
|
0.00582
|
-0.00166
|
0.04613
|
-0.02696
|
|
DI
|
96
|
1.42435
|
-0.11077
|
0.00480
|
-0.00072
|
0.02807
|
-0.02274
|
|
DI
|
100
|
1.90246
|
-0.12221
|
0.00306
|
0.00052
|
0.00000
|
-0.00951
|
|
UI
|
104
|
1.52952
|
0.09516
|
-0.00030
|
-0.00072
|
0.01380
|
0.00623
|
|
UI
|
108
|
1.17943
|
0.08926
|
0.00152
|
-0.00174
|
0.02693
|
0.01528
|
|
DO
|
92
|
0.91181
|
0.09218
|
-0.00582
|
0.00218
|
-0.04613
|
0.01744
|
|
DO
|
96
|
0.47811
|
0.11077
|
-0.00480
|
0.00125
|
-0.02807
|
0.01323
|
|
DO
|
100
|
0.00000
|
0.12221
|
-0.00306
|
0.00000
|
0.00000
|
0.00000
|
|
UO
|
104
|
0.37294
|
-0.09516
|
0.00030
|
0.00124
|
-0.01380
|
-0.01574
|
|
UO
|
108
|
0.72303
|
-0.08926
|
-0.00152
|
0.00226
|
-0.02693
|
-0.02479
|