SQL Server BinaryBarrierPayoutAtHitPriceNGreeks Function

Updated 2023-11-14 21:16:36.643000

Description

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 [westclintech].[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

table

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.

Examples

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 is 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.

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 results.