BlackScholesMertonIV

Updated 2023-11-16 16:43:20.820000

Syntax

SELECT [westclintech].[wct].[BlackScholesMertonIV] (
  <@CallPut, nvarchar(4000),>
 ,<@AssetPrice, float,>
 ,<@StrikePrice, float,>
 ,<@TimeToMaturity, float,>
 ,<@RiskFreeRate, float,>
 ,<@DividendRate, float,>
 ,<@Price, float,>)

Description

Use the scalar function BlackScholesMertonIV to calculate an implied volatility for a European-style option using the Black-Scholes-Merton option pricing formula.

Arguments

@Price

the observed price of the option. @Price 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 interest rate. @DividendRate 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 of return over the life of the option. @RiskFreeRate is an expression of type float or of a type that can be implicitly converted 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.

@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.

@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.

@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.

Return Type

float

Remarks

@TimeToMaturity must be greater than zero (@TimeToMaturity > 0).

@AssetPrice must be greater than zero (@AssetPrice > 0).

@StrikePrice must be greater than zero (@StrikePrice > 0).

@Price must be greater than zero (@Price > 0).

If @DividendRate is NULL then @DividendRate = 0.

If @RiskFreeRate is NULL then @RiskFreeRate = 0.

To calculate the price or Greeks of a European option use BLACKSCHOLESMERTON or BINOMIALEURO.

To calculate the implied volatility of a European option using the binomial method, use BINOMIALEUROIV.

To calculate the implied volatility on American options use BJERKSUNDSTENSLANDIV or BINOMIALAMERICANIV.

Examples

Calculate the implied volatility for a call option on 2012-09-04, expiring on 2012-12-15, with a current asset price of 99.5, a strike price of 100 and an observed price of 4.15. The risk free rate is 2% and the dividend rate is 0.5%.

SELECT ROUND(wct.BlackScholesMertonIV(   'C',                                                --PutCall
                                         99.5,                                               --Asset Price
                                         100,                                                --Strike Price
                                         datediff(d, '2012-09-04', '2012-12-15') / 365.0000, --Time-to-expiry
                                         .02,                                                --Risk Free Rate
                                         .005,                                               --Dividend Rate
                                         4.15                                                --Price
                                     ),
             6
            ) as Volatility;

This produces the following result.

{"columns":[{"field":"Volatility","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"Volatility":"0.199999"}]}

In the following SELECT, we collect the implied volatility for a series of strike prices on EUR/USD option which expire in 7 days time. The opion price is the midpoint of the bid and ask and the 7 day EURIBOR rate is 0.0335700% and the 7 day LIBOR rate is 0.1870000%.

SELECT wct.BlackScholesMertonIV(   'C',
                                   127.91,                                --Asset price    
                                   n.s,                                   --Strike   
                                   Cast(7 as float) / cast(365 as float), --Time
                                   0.000335699,                           --Continuously Compouned EURIBOR
                                   0.001869966,                           --Continuously Compouned LIBOR
                                   (call_bid + call_ask) / 2              --Price
                               ) as [Call IV],
       cast(n.call_bid as money) as Bid,
       cast(n.call_ask as money) as Ask,
       n.s as Strike,
       cast(n.put_bid as money) as Bid,
       cast(n.put_ask as money) as Ask,
       wct.BlackScholesMertonIV(   'P',
                                   127.91,                                --Asset price
                                   n.s,                                   --Strike
                                   Cast(7 as float) / cast(365 as float), --Time
                                   0.000335699,                           --Continuously Compouned EURIBOR
                                   0.001869966,                           --Continuously Compouned LIBIBOR
                                   (put_bid + put_ask) / 2                --Price
                               ) as [Put IV]
FROM
(
    SELECT 6.85,
           7.00,
           121,
           NULL,
           .04
    UNION ALL
    SELECT 5.85,
           6.00,
           122,
           NULL,
           .04
    UNION ALL
    SELECT 4.85,
           5.00,
           123,
           NULL,
           .04
    UNION ALL
    SELECT 3.85,
           4.00,
           124,
           NULL,
           .04
    UNION ALL
    SELECT 2.91,
           2.99,
           125,
           .02,
           .05
    UNION ALL
    SELECT 2.00,
           2.06,
           126,
           .10,
           .13
    UNION ALL
    SELECT 1.20,
           1.25,
           127,
           .29,
           .32
    UNION ALL
    SELECT 0.59,
           0.62,
           128,
           .68,
           .71
    UNION ALL
    SELECT 0.23,
           0.25,
           129,
           1.31,
           1.35
    UNION ALL
    SELECT 0.06,
           0.09,
           130,
           2.13,
           2.19
    UNION ALL
    SELECT 0.01,
           .04,
           131,
           3.05,
           3.15
) n(call_bid, call_ask, s, put_bid, put_ask);