Saturday, January 15, 2011

Black-Scholes Model

Black-Scholes Option Princing Model for dividend paying underlying assets VBA code (Premium only)


'// Author's blog:
'// Heart of the Finance
'// http://heartofthefinance.blogspot.com/
'//
'// Black-Scholes Option Princing Model for dividend paying underlying assets VBA code (Premium only)
'// version 1.0.0
'// Last update: 1/15/2011

Public Function BlackScholes(CallPutFlag As String, S As Double, K As Double, T As Double, r As Double, q As Double, v As Double) As Double

'S: spot price of an underlying asset
'K: strike price
'T: (T-t); time to maturity from today. i.e. T: time at maturity, t: today
'r: risk-free rate (annualized, continuous compounding)
'q: dividend yield (annualized, continuous compounding)
'v:volatility (standard deviation of return) of an underlying asset (annualized)


Dim d1 As Double, d2 As Double

d1 = (Log(S / K) + (r - q + v ^ 2 / 2) * T) / (v * Sqr(T))
d2 = d1 - v * Sqr(T)

If CallPutFlag = "c" Then

BlackScholes = Exp(-q * T) * S * CND(d1) - K * Exp(-r * T) * CND(d2)

ElseIf CallPutFlag = "p" Then

'BlackScholes = K * Exp(-r * T) * CND(-d2) - S * Exp(-q * T) * CND(-d1)
BlackScholes = K * Exp(-r * T) * (1 - CND(d2)) - S * Exp(-q * T) * (1 - CND(d1))

End If

End Function

'// The cumulative normal distribution function
Public Function CND(X As Double) As Double

Dim L As Double, K As Double
Const a1 = 0.31938153: Const a2 = -0.356563782: Const a3 = 1.781477937:
Const a4 = -1.821255978: Const a5 = 1.330274429

L = Abs(X)
K = 1 / (1 + 0.2316419 * L)
CND = 1 - 1 / Sqr(2 * Application.Pi()) * Exp(-L ^ 2 / 2) * (a1 * K + a2 * K ^ 2 + a3 * K ^ 3 + a4 * K ^ 4 + a5 * K ^ 5)

If X < 0 Then CND = 1 - CND End If End Function


Black-Scholes Option Princing Model for dividend paying underlying assets VBA code (Premium and Greeks)

'// Author's blog:
'// Heart of the Finance
'// http://heartofthefinance.blogspot.com/
'//
'// Black-Scholes Option Princing Model for dividend paying underlying assets VBA code (Premium and Greeks)
'// version 1.0.0
'// Last update: 1/15/2011

'****************************************************************************
'* Cumulative Standard Normal Distribution *
'* (This function provides similar result as NORMSDIST( ) on Excel) *
'****************************************************************************

Function SNorm(z)
c1 = 2.506628
c2 = 0.3193815
c3 = -0.3565638
c4 = 1.7814779
c5 = -1.821256
c6 = 1.3302744
If z > 0 Or z = 0 Then
w = 1
Else: w = -1
End If
y = 1 / (1 + 0.2316419 * w * z)
SNorm = 0.5 + w * (0.5 - (Exp(-z * z / 2) / c1) * _
(y * (c2 + y * (c3 + y * (c4 + y * (c5 + y * c6))))))

End Function

'**********************************************************************
'* Black-Scholes European Call Price Computation *
'**********************************************************************

Function Call_Eur(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

's: spot price of an underlying asset
'k: strike price
't: (T-t); time to maturity. i.e. T: maturity, t: current
'r: risk-free rate (annual rate, continuous compounding)
'q: dividend yield (annual rate, continuous compounding)
'sd:volatility (standard deviation of return) of an underlying asset

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Call_Eur = Exp(-q * t) * s * SNorm(d1) - k * Exp(-r * t) * SNorm(d2)

End Function

'*********************************************************************
'* Black-Scholes European Put Price Computation *
'*********************************************************************

Function Put_Eur(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

'Put-call parity
'Call_Eur_tmp = Exp(-q * t) * s * SNorm(d1) - k * Exp(-r * t) * SNorm(d2)
'Put_Eur = -s * Exp(-q * t) + Call_Eur_tmp + k * Exp(-r * t)

Put_Eur = k * Exp(-r * t) * (1 - SNorm(d2)) - s * Exp(-q * t) * (1 - SNorm(d1))

End Function


'**********************************************************************
'* Black-Scholes European Call Delta Computation *
'**********************************************************************

Function Call_Eur_Delta(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Call_Eur_Delta = Exp(-q * t) * SNorm(d1)

End Function

'*********************************************************************
'* Black-Scholes European Put Delta Computation *
'*********************************************************************

Function Put_Eur_Delta(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Put_Eur_Delta = -Exp(-q * t) * (1 - SNorm(d1))

End Function

'**********************************************************************
'* Black-Scholes European Call Gamma Computation *
'**********************************************************************

Function Call_Eur_Gamma(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Call_Eur_Gamma = Exp(-q * t) / (s * sd * t ^ 0.5) * Exp(-d1 ^ 2 / 2) / (2 * Application.WorksheetFunction.Pi()) ^ 0.5

End Function

'**********************************************************************
'* Black-Scholes European Put Gamma Computation *
'**********************************************************************

Function Put_Eur_Gamma(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Put_Eur_Gamma = Exp(-q * t) / (s * sd * t ^ 0.5) * Exp(-d1 ^ 2 / 2) / (2 * Application.WorksheetFunction.Pi()) ^ 0.5

End Function

'**********************************************************************
'* Black-Scholes European Call Vega Computation *
'**********************************************************************

Function Call_Eur_Vega(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Call_Eur_Vega = Exp(-q * t) * s * (t ^ 0.5) * Exp(-d1 ^ 2 / 2) / (2 * Application.WorksheetFunction.Pi()) ^ 0.5

End Function

'**********************************************************************
'* Black-Scholes European Put Vega Computation *
'**********************************************************************

Function Put_Eur_Vega(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Put_Eur_Vega = Exp(-q * t) * s * (t ^ 0.5) * Exp(-d1 ^ 2 / 2) / (2 * Application.WorksheetFunction.Pi()) ^ 0.5

End Function

'**********************************************************************
'* Black-Scholes European Call Theta Computation *
'**********************************************************************

Function Call_Eur_Theta(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Call_Eur_Theta = q * Exp(-q * t) * s * SNorm(d1) - Exp(-q * t) * s * sd / (2 * (t) ^ 0.5) * Exp(-d1 ^ 2 / 2) / (2 * Application.WorksheetFunction.Pi()) ^ 0.5 - k * r * Exp(-r * t) * SNorm(d2)

End Function

'**********************************************************************
'* Black-Scholes European Put Theta Computation *
'**********************************************************************

Function Put_Eur_Theta(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Put_Eur_Theta = q * Exp(-q * t) * s * (-1) * SNorm(-d1) - Exp(-q * t) * s * sd / (2 * (t) ^ 0.5) * Exp(-d1 ^ 2 / 2) / (2 * Application.WorksheetFunction.Pi()) ^ 0.5 + k * r * Exp(-r * t) * SNorm(-d2)

End Function


'**********************************************************************
'* Black-Scholes European Call Rho Computation *
'**********************************************************************

Function Call_Eur_Rho(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Call_Eur_Rho = k * t * Exp(-r * t) * SNorm(d2)

End Function


'**********************************************************************
'* Black-Scholes European Put Rho Computation *
'**********************************************************************

Function Put_Eur_Rho(s, k, t, r, q, sd)
Dim a As Single
Dim b As Single
Dim c As Single
Dim d1 As Single
Dim d2 As Single

a = Log(s / k)
b = (r - q + 0.5 * sd ^ 2) * t
c = sd * (t ^ 0.5)
d1 = (a + b) / c
d2 = d1 - sd * (t ^ 0.5)

Put_Eur_Rho = -k * t * Exp(-r * t) * SNorm(-d2)

End Function




Ref.
http://en.wikipedia.org/wiki/Black-Scholes
http://www.worldscibooks.com/etextbook/p556/p556_chap04.pdf
http://www.espenhaug.com/black_scholes.html
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