Black Implied Volatility for European Options

	σ=blkimpv(F0,K,r,T,Price,FlagIsCall=true,xtol=1e-14,ytol=1e-15)

Where:

	F0         = Value of the Forward.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	Price      = Price of the Option.
	FlagIsCall = true for Call Options, false for Put Options.

	σ          = implied volatility of the European Option.

Example

julia> blkimpv(10.0,10.0,0.01,2.0,2.0)
0.36568658096623635

Black Price for European Options

	Price=blkprice(F0,K,r,T,σ,FlagIsCall=true)

Where:

	F0         = Value of the Forward.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility.
	FlagIsCall = true for Call Options, false for Put Options.

	Price      = price of the European Option.

Example

julia> blkprice(10.0,10.0,0.01,2.0,0.2)
1.1023600107733191

Black Price for Binary European Options

	Price=blsbin(S0,K,r,T,σ,FlagIsCall=true)

Where:

	F0         = Value of the Forward.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility.
	FlagIsCall = true for Call Options, false for Put Options.

	Price      = price of the Binary European Option.

Example

julia> blsbin(10.0,10.0,0.01,2.0,0.2)
0.4624714677292208

Black & Scholes Delta for European Options

	Δ=blsdelta(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Δ          = delta of the European Option.

Example

julia> blsdelta(10.0,10.0,0.01,2.0,0.2,0.01)
0.5452173371920436

Black & Scholes Gamma for European Options

	Γ=blsgamma(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Γ          = gamma of the European Option.

Example

julia> blsgamma(10.0,10.0,0.01,2.0,0.2,0.01)
0.13687881535712826

Black & Scholes Implied Volatility for European Options

	σ=blsimpv(S0,K,r,T,Price,d=0.0,FlagIsCall=true,xtol=1e-14,ytol=1e-15)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	Price      = Price of the Option.
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	σ          = implied volatility of the European Option.

Example

julia> blsimpv(10.0,10.0,0.01,2.0,2.0)
0.3433730534290586

Black & Scholes Lambda for European Options

	Λ=blslambda(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Λ          = lambda of the European Option.

Example

julia> blslambda(10.0,10.0,0.01,2.0,0.2,0.01)
4.945909973725978

Black & Scholes Price for European Options

	Price=blsprice(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Price      = price of the European Option.

Example

julia> blsprice(10.0,10.0,0.01,2.0,0.2,0.01)
1.1023600107733191

Black & Scholes Psi for European Options

	Ψ=blspsi(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Ψ          = psi of the European Option.

Example

julia> blspsi(10.0,10.0,0.01,2.0,0.2,0.01)
-10.904346743840872

Black & Scholes Rho for European Options

	ρ=blsrho(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	ρ          = rho of the European Option.

Example

julia> blsrho(10.0,10.0,0.01,2.0,0.2,0.01)
8.699626722294234

Black & Scholes Theta for European Options

	Θ=blstheta(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Θ          = theta of the European Option.

Example

julia> blstheta(10.0,10.0,0.01,2.0,0.2,0.01)
-0.26273403060652334

Black & Scholes Vanna for European Options

	Vanna=blsvanna(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	Vanna        = vanna of the European Option.

Example

julia> blsvanna(10.0,10.0,0.01,2.0,0.2,0.01)
0.2737576307142566

Black & Scholes Vega for European Options

	ν=blsvega(S0,K,r,T,σ,d=0.0,FlagIsCall=true)

Where:

	S0         = Value of the Underlying.
	K          = Strike Price of the Option.
	r          = Zero Rate.
	T          = Time to Maturity of the Option.
	σ          = Implied Volatility
	d          = Implied Dividend of the Underlying.
	FlagIsCall = true for Call Options, false for Put Options.

	ν          = vega of the European Option.

Example

julia> blsvega(10.0,10.0,0.01,2.0,0.2,0.01)
5.475152614285131

Actual Number of days between two dates

	ndays=daysact(Date1,Date2)

Where:

	Date1 = Start date.
	Date2 = End date.

	ndays      = Actual Number of days between Start Date and End Date.

Example

julia> daysact(Date(1996,10,12),Date(1998,1,10))
455

From Excel Number Format to Date

	Date=fromExcelNumberToDate(ExcelNumber)

Where:

	ExcelNumber = Integer representing a date in the excel format.

	Date      = Date representing the input in the Julia object format.

Example

julia> fromExcelNumberToDate(45000)
2023-03-15

Cumulative Distribution Function of a Standard Gaussian Random Variable

	y=normcdf(x)

Where:

	x         = point of evaluation.

	y      = probability that a standard gaussian random variable is below x.

Example

julia> normcdf(0.0)
0.5

Probability Distribution Function of a Standard Gaussian Random Variable

	y=normpdf(x)

Where:

	x         = point of evaluation.

	y      = value.

Example

julia> normpdf(0.0)
0.3989422804014327

Fraction of year between two Dates according the following convention

	yfr=yearfrac(Date1,Date2,basis)

Where:

	Date1 = Start date.
	Date2 = End date.
	basis = Integer representing the following conventions:
			- 0 = (ACT/ACT)
			- 1 = (30/360 SIA)
			- 2 = (ACT/360)
			- 3 = (ACT/365)
			- 4 = (30/360 PSA)
			- 5 = (30/360 ISDA)
			- 6 = (30E/360)
			- 7 = (ACT/365 JPN)

	yfr      = fraction of year between start and end date according to basis.

Example

julia> yearfrac(Date(1996,10,12),Date(1998,1,10),1)
1.2444444444444445