Issue short: I have values for Asian Options which I'm trying to replicate using a self-build vba calculator. The values I have to hit is from FinCAD and I'm using a discrete arithmetic average rate option based on Haug, Haug and Margrabe and also trying the Curran version (implemented in VBA). All code below:
More information: I'm sure we calculate using the informations differently since my value is quite different from he's. Underneath is the exact input used:
- Asian Call Option (average price not strike)
- Valuation date: 06-04-2017
- Averaging period: 01-05-2017 until 31-05-2017 (May)
- Underlying price (per 05-04-2017): 515.25
- Strike: 515
- Volatility: 20,669%
- Risk free rate: 0,980%
I use working days (in my country) so:
- Time to next average point (t1): 15 days (from 06-04-2017 to 01-05-2017)
- Time to maturity (T): 35 days (from 06-04-2017 to 31-05-2017)
My values (after using 252):
- Asian Value (Haug, Haug and Margrabe): 12.7763
- Asian Value (Curran): 12.7753
- BS european value: 16.2952
He's value:
- 18.64
I simply can't understand the difference. It's not just about the days used. Maybe it's the volatility. Also because my BS European value is lower than he's value and that should not be the case. I have also validated my values according to free available calculators on www. I just got the info that FinCAD is using their function, aaGeo_Asian, so I tried a geometric version but still don't get they numbers. I do not have information on the specifics in their function. However it could look like they also use projected future values and not "just" the current price of the underlying.
Public Function DiscreteAsianHHM(runFlag As String, CallPutFlag As String,
TimeFlag As String, ContinuousFlag As String, S As Double, SA As Double, X
As Double, _
input_t1 As Double, input_t As Double, n As Double, m As Double,
input_r As Double, input_b As Double, v As Double) As Double
Dim d1 As Double, d2 As Double, h As Double, EA As Double, EA2 As Double
Dim vA As Double, OptionValue As Double
Dim t1 As Double, T As Double
Dim r As Double, b As Double
'Making sure of the chosen input time (years/days)
t1 = TimeConvert(TimeFlag, input_t1)
T = TimeConvert(TimeFlag, input_t)
'Converting to continious compounding rate
r = CCRConvert(ContinuousFlag, input_r)
b = CCRConvert(ContinuousFlag, input_b)
' Calculate either Asian (A) or plain European (E) by BS formula
If runFlag = "A" Then
h = (T - t1) / (n - 1)
If b = 0 Then
EA = S
Else
EA = S / n * Exp(b * t1) * (1 - Exp(b * h * n)) / (1 - Exp(b * h))
End If
'If we are in the averaging period and way ITM
If m > 0 Then
If SA > n / m * X Then ' Exercise is certain for call, put must be
out-of-the-money
If CallPutFlag = "Put" Then
DiscreteAsianHHM = 0
ElseIf CallPutFlag = "Call" Then
SA = SA * m / n + EA * (n - m) / n
DiscreteAsianHHM = (SA - X) * Exp(-r * T)
End If
Exit Function
End If
End If
If m = n - 1 Then ' Only one fix left use Black-Scholes weighted with
time
X = n * X - (n - 1) * SA
DiscreteAsianHHM = GBlackScholes(CallPutFlag, S, X, T, r, b, v) * 1
/ n
Exit Function
End If
If b = 0 Then
EA2 = S * S * Exp(v * v * t1) / (n * n) _
* ((1 - Exp(v * v * h * n)) / (1 - Exp(v * v * h)) _
+ 2 / (1 - Exp(v * v * h)) * (n - (1 - Exp(v * v * h * n)) / (1 -
Exp(v * v * h))))
Else
EA2 = S * S * Exp((2 * b + v * v) * t1) / (n * n) _
* ((1 - Exp((2 * b + v * v) * h * n)) / (1 - Exp((2 * b + v * v)
* h)) _
+ 2 / (1 - Exp((b + v * v) * h)) * ((1 - Exp(b * h * n)) / (1 -
Exp(b * h)) _
- (1 - Exp((2 * b + v * v) * h * n)) / _
(1 - Exp((2 * b + v * v) * h))))
End If
vA = Sqr((Log(EA2) - 2 * Log(EA)) / T)
OptionValue = 0
'If we are in the averaging period we need to adjust the strike price
If m > 0 Then
X = n / (n - m) * X - m / (n - m) * SA
End If
d1 = (Log(EA / X) + vA ^ 2 / 2 * T) / (vA * Sqr(T))
d2 = d1 - vA * Sqr(T)
If CallPutFlag = "Call" Then
OptionValue = Exp(-r * T) * (EA * CND(d1) - X * CND(d2))
ElseIf (CallPutFlag = "Put") Then
OptionValue = Exp(-r * T) * (X * CND(-d2) - EA * CND(-d1))
End If
DiscreteAsianHHM = OptionValue * (n - m) / n
ElseIf runFlag = "E" Then
'Generalized BS model
DiscreteAsianHHM = GBlackScholes(CallPutFlag, S, X, T, r, b, v)
End If
End Function
Public Function TimeConvert(TimeFlag As String, input_t As Double) As Double
If TimeFlag = "Days" Then
TimeConvert = input_t / 252
Else
TimeConvert = input_t
End If
End Function
'The generalized Black and Scholes formula
Public Function GBlackScholes(CallPutFlag As String, S As Double, X _
As Double, T As Double, r As Double, b As Double, v As Double)
As Double
Dim d1 As Double, d2 As Double
'I thought this was a mistake but note that q=r-b. r is risk free rate, b is
cost of carry and q is dividend
d1 = (Log(S / X) + (b + v ^ 2 / 2) * T) / (v * Sqr(T))
d2 = d1 - v * Sqr(T)
If CallPutFlag = "Call" Then
GBlackScholes = S * Exp((b - r) * T) * CND(d1) - X * Exp(-r * T) * CND(d2)
ElseIf CallPutFlag = "Put" Then
GBlackScholes = X * Exp(-r * T) * CND(-d2) - S * Exp((b - r) * T) * CND(-d1)
End If
End Function
Public Function CCRConvert(ContinuousFlag As String, rate As Double) As
Double
If ContinuousFlag = "Continuous" Then
CCRConvert = rate
ElseIf ContinuousFlag = "Annual" Then
CCRConvert = Log(1 + rate)
ElseIf ContinuousFlag = "Semi-annual" Then
CCRConvert = 2 * Log(1 + rate / 2)
ElseIf ContinuousFlag = "Quarterly" Then
CCRConvert = 4 * Log(1 + rate / 4)
ElseIf ContinuousFlag = "Monthly" Then
CCRConvert = 12 * Log(1 + rate / 12)
ElseIf ContinuousFlag = "Daily" Then
CCRConvert = 252 * Log(1 + rate / 252)
End If
End Function
'Using the build in cummulative normal distribution
Function CND(X As Double) As Double
CND = WorksheetFunction.Norm_Dist(X, 0, 1, True)
End Function
