SQL Server StandardBarrierPriceNGreeks Function

Updated 2023-11-14 21:24:46.300000

Description

Use the table-valued function StandardBarrierPriceNGreeks to calculate the price and Greeks of a European-style Knock-In or Knock-Out option. StandardBarrierPriceNGreeks valuations are based on the formulae published by Mark Rubinstein and Eric Reiner in 1991.

Syntax

SELECT * FROM [westclintech].[wct].[StandardBarrierPriceNGreeks](
  <@CallPut, nvarchar(4000),>
 ,<@BarrierType, nvarchar(4000),>
 ,<@AssetPrice, float,>
 ,<@StrikePrice, float,>
 ,<@BarrierPrice, float,>
 ,<@Rebate, float,>
 ,<@TimeToMaturity, float,>
 ,<@RiskFreeRate, float,>
 ,<@DividendRate, float,>
 ,<@Volatility, float,>)

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.

@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.

@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

table

colName	colDatatype	colDesc
Price	float	The theoretical value of the option.
Delta	float	The sensitivity to small changes in the asset price; the first derivative of the option with respect to price.
Gamma	float	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	float	The sensitivity to small changes in time; the first derivative of the option with respect to time.
Vega	float	The sensitivity to small changes in volatility; the first derivative of the option with respect to volatility.
Rho	float	The sensitivity to small changes in the risk-free rate; the first derivative of the option with respect to the risk-free rate.
Lambda	float	Delta multiplied by the asset price divided by the theoretical value. If the theoretical value is zero, then lambda is set to zero.
GammaP	float	Gamma multiplied by asset price divided by strike price.
DdeltaDtime	float	The instantaneous change in delta over the passage of time; the second derivative, once to asset price and once to time.
DdeltaDvol	float	The sensitivity of delta with respect to volatility; the second derivative, once to asset price and once to volatility.
DdeltaDvolDvol	float	The second derivative of delta with respect to volatility; the third derivative, once to asset price and twice to volatility.
DgammaDvol	float	The rate of change in gamma with respect to changes in volatility; the third derivative, twice to asset price and once to volatility.
DvegaDvol	float	The rate of change to Vega as the volatility changes; the second derivative with respect to volatility.
VegaP	float	The percentage change in theoretical value for a 10 per cent change in volatility.v
PhiRho2	float	The sensitivity to a change in the dividend yield (foreign interest rate for a currency option); the first derivative with respect to dividend yield.
RhoFuturesOption	float	The sensitivity to a change in the cost-of carry; the first derivative with respect to the cost-of-carry.
CarrySensitivity	float	-PhiRho2
DgammaDspot	float	The rate of change in gamma with respect to change in the asset price; the third derivative with respect to price.
DeltaX	float	The sensitivity to a change in the strike price; the first derivative with respect to strike price.
RiskNeutralDensity	float	The sensitivity of DeltaX; the second derivative with respect to strike price.
DvommaDvol	float	The sensitivity of DvegaDvol to changes in volatility; the third derivative, twice to asset price and once to volatility.
DgammaDtime	float	The sensitivity of Gamma to the passage of time; the third derivative, twice to asset price and once to time.
DvegaDtime	float	The sensitivity of Vega to the passage of time; the second derivative, once to volatility and once to time.
ForwardPrice	float	The value of the underlying asset at the expiration date of the option.
ForwardPoints	float	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.

To calculate a single return value use StandardBarrier.

Examples

A down-and-in call.

SELECT *
FROM wct.StandardBarrierPriceNGreeks(   'C',  --PutCall
                                        'DI', --BarrierType
                                        100,  --Asset Price
                                        90,   --Strike Price
                                        97,   --Barrier
                                        2,    --Rebate
                                        0.5,  --Time-to-expiry
                                        .10,  --Risk Free Rate
                                        .05,  --Dividend Rate
                                        .20   --Volatility
                                    );

Here is the resultant table.

Price	Delta	Gamma	Theta	Vega	Rho	Lambda	GammaP	DdeltaDtime	DdeltaDvol	DdeltaDvolDvol	DgammaDvol	DvegaDvol	VegaP	PhiRho2	RhoFuturesOption	CarrySensitivity	DgammaDspot	DeltaX	RiskNeutralDensity	DvommaDvol	DgammaDtime	DvegaDtime	ForwardPrice	ForwardPoints
8.59507043337689	-0.64902584297144	0.0520920195867802	-0.0173153378580277	0.258665904442879	0.156984606560862	-7.55114048223624	0.0520920195867802	0.000306085750352931	0.0135552475910572	-0.650285444001763	-0.00416152947657622	-0.00199687534063742	0.517331808885757	-0.199959958364104	-0.0379626818832079	0.199959958364104	12980.5178819353	-0.582783834168765	0.0207556638542883	0.000564868690818621	-5.0437133921443E-05	-0.0699760441980818	102.531512052443	2.53151205244289

In this SELECT we un-pivot the columns returned by the function for ease of viewing the results.

SELECT *
FROM wct.StandardBarrierPriceNGreeks(   'C',  --PutCall
                                        'DI', --BarrierType
                                        100,  --Asset Price
                                        90,   --Strike Price
                                        97,   --Barrier
                                        2,    --Rebate
                                        0.5,  --Time-to-expiry
                                        .10,  --Risk Free Rate
                                        .05,  --Dividend Rate
                                        .20   --Volatility
                                    );

This produces the following result.

Return Value	Value
Price	8.59507043337689
Delta	-0.649025843078022
Gamma	0.052092730129516
Theta	-0.0173153378580135
Vega	0.258665904443234
Rho	0.156984606562283
Lambda	-7.55114048347627
GammaP	0.052092730129516
DdeltaDtime	0.000306086723699144
DdeltaDvol	0.0135552546964846
DdeltaDvolDvol	-0.650285444002208
DgammaDvol	-0.00416152983184759
DvegaDvol	-0.00199687541169169
VegaP	0.517331808886468
PhiRho2	-0.199959958365525
RhoFuturesOption	-0.0379626818828527
CarrySenstivity	0.199959958365525
DgammaDspot	12980.5178819264
DeltaX	-0.582783834275347
RiskNeutralDensity	0.0207577954824956
DvommaDvol	0.000564868690799969
DgammaDtime	-5.04370755206702E-05
DvegaDtime	-0.0699760427380625
ForwardPrice	102.531512052443
ForwardPoints	2.5315120524428

Use CROSS APPLY to with StandardBarrierPriceNGreeks when there are multiple rows of input data.

SELECT n.rn,
       k.*
FROM
(
    VALUES
        (1, 'C', 'DI', 100, 97, 95, 2, 0.75, 0.005, 0.009, 0.16),
        (2, 'P', 'DI', 100, 97, 95, 2, 0.75, 0.005, 0.009, 0.16),
        (3, 'C', 'DO', 100, 97, 95, 2, 0.75, 0.005, 0.009, 0.16),
        (4, 'P', 'DO', 100, 97, 95, 2, 0.75, 0.005, 0.009, 0.16),
        (5, 'C', 'UI', 100, 103, 105, 2, 0.75, 0.005, 0.009, 0.16),
        (6, 'P', 'UI', 100, 103, 105, 2, 0.75, 0.005, 0.009, 0.16),
        (7, 'C', 'UO', 100, 103, 105, 2, 0.75, 0.005, 0.009, 0.16),
        (8, 'P', 'UO', 100, 103, 105, 2, 0.75, 0.005, 0.009, 0.16)
) n (rn, z, bt, S, X, H, K, T, Rf, Rd, Vol)
    CROSS APPLY wct.StandardBarrierPriceNGreeks(z, bt, S, X, H, K, T, Rf, Rd, Vol)
              k;