BinaryBarrierPayoutAtHitPriceNGreeks
Updated: 31 Oct 2013
Use the table-valued function BinaryBarrierPayoutAtHitPriceNGreeks to calculate the price and other derivatives for binary barrier options having payoffs that are received the moment that the barrier is breached. These calculations 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 1–4:
(1) down-and-in cash-(at-hit)-or-nothing;
(2) up-and-in cash-(at-hit)-or-nothing;
(3) down-and-in asset-(at-hit)-or-nothing; and
(4) up-and-in asset-(at-hit)-or-nothing.
Syntax
SELECT * FROM [wctOptions].[wct].[BinaryBarrierPayoutAtHitPriceNGreeks](
<@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 BinaryBarrierPayoutAtHitPriceNGreeks 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-hit)-or-nothing option.
SELECT *
FROM wct.BinaryBarrierPayoutAtHitPriceNGreeks(
'DI' --BarrierType
,'True' --CashOrNothing
,100 --AssetPrice
,100 --StrikePrice
,92 --BarrierPrice
,2 --Rebate
,.05 --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.BinaryBarrierPayoutAtHitPriceNGreeks(
'DI' --BarrierType
,'True' --CashOrNothing
,100 --AssetPrice
,100 --StrikePrice
,92 --BarrierPrice
,2 --Rebate
,.05 --TimeToMaturity
,.10 --RiskFreeRate
,.05 --DividendRate
,.20 --Volatility
) k
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 0.116492416924189
Delta -0.059603371860284
Gamma 0.0259861715479204
Theta -0.0132918373028393
Vega 0.0257101284791915
Rho -0.00250229140030256
Lambda 0
GammaP 0.0259861715479204
DdeltaDtime 0.00407713935862317
DdeltaDvol -0.00794324465924756
DdeltaDvolDvol 0.00043724976979409
DgammaDvol 0.00090817359812978
DvegaDvol 0.00215365444436455
VegaP 0.051420256958383
PhiRho2 0.00246002848277652
DgammaDspot -0.00862503893306332
DeltaX 0
RiskNeutralDensity 0
DvommaDvol -0.000395468309802832
DgammaDtime 0.0904978465936139
DvegaDtime -0.179924545179075
FuturesOptionsRho -4.22629167655397E-05
CarrySensitivity -0.00246002848277652
ForwardPrice 100.25031276058
ForwardPoints 0.250312760579519
Use CROSS APPLY when there are many input rows.
SELECT CASE n.cash WHEN 'True' THEN 'CASH' ELSE 'ASSET' END as CA,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,'DI','False',100,100,92,92,.50,.10,.05,.20)
,(7,'DI','False',100,100,96,96,.50,.10,.05,.20)
,(8,'DI','False',100,100,100,100,.50,.10,.05,.20)
,(9,'UI','False',100,100,104,104,.50,.10,.05,.20)
,(10,'UI','False',100,100,108,108,.50,.10,.05,.20)
)n(rn,bt,cash,S,X,K,H,T,Rf,Rd,Vol)
CROSS APPLY wct.BinaryBarrierPayoutAtHitPriceNGreeks(bt,cash,S,X,K,H,T,Rf,Rd,Vol)k
Here are the (reformatted) results.
|
CA
|
K
|
Price
|
Delta
|
Gamma
|
Theta
|
Vega
|
Rho
|
|
CASH
|
92
|
1.02275
|
-0.09686
|
0.00645
|
-0.00193
|
0.04833
|
-0.02455
|
|
CASH
|
96
|
1.48240
|
-0.11848
|
0.00573
|
-0.00112
|
0.02987
|
-0.01772
|
|
CASH
|
100
|
2.00000
|
-0.13406
|
0.00435
|
0.00000
|
0.00000
|
0.00000
|
|
CASH
|
104
|
1.59241
|
0.10254
|
0.00031
|
-0.00114
|
0.01505
|
0.01293
|
|
CASH
|
108
|
1.21896
|
0.09434
|
0.00200
|
-0.00206
|
0.02864
|
0.01984
|
|
ASSET
|
92
|
47.04640
|
-4.45551
|
0.29651
|
-0.08889
|
2.22306
|
-1.12919
|
|
ASSET
|
96
|
71.15529
|
-5.68714
|
0.27504
|
-0.05352
|
1.43363
|
-0.85060
|
|
ASSET
|
100
|
100.00000
|
-6.70285
|
0.21757
|
0.00000
|
0.00000
|
0.00000
|
|
ASSET
|
104
|
82.80548
|
5.33224
|
0.01596
|
-0.05935
|
0.78247
|
0.67214
|
|
ASSET
|
108
|
65.82385
|
5.09460
|
0.10818
|
-0.11146
|
1.54646
|
1.07117
|