Today is the second day of UseR! 2008 in Germany, and I will be giving a short talk on a market data interface I developed for R a while back. The confererence is apparently the largest R user conference yet, with over 400 participants, from all areas of industry and academia.
Here are the slides:
For some reason, I couldn’t get the Beamer package to number my contents correctly, so it looks a little strange.
Here is a simple implementation of the Black-Scholes pricing formula in R. This will return a two-element vector containing the calculated call and put price, respectively.
# Black-Scholes Option Value # Call value is returned in values[1], put in values[2] blackscholes <- function(S, X, rf, T, sigma) { values <- c(2) d1 <- (log(S/X)+(rf+sigma^2/2)*T)/sigma*sqrt(T) d2 <- d1 - sigma * sqrt(T) values[1] <- S*pnorm(d1) - X*exp(-rf*T)*pnorm(d2) values[2] <- X*exp(-rf*T) * pnorm(-d2) - S*pnorm(-d1) values }
Example use:
> blackscholes(100,110,.05,1,.2)
[1] 6.040088 10.675325
Following on from the post a few weeks ago on a simple Black-Scholes Excel macro, here is a follow up, with a slightly updated version that also calculates the major greeks (vega, gamma and delta). The formulae are from Hull’s book, or see here for examples of the closed-form greeks.
Usage of the spreadsheet is identical to the previous example, so refer to that post. The only difference is the output, which now looks like this (click to enlarge):
The spreadsheet is available for download here, and the code is below for reference.
Function BlackScholes(SpotPrice As Double, ExercisePrice As Double,
TimeToMaturity As Double, RiskFreeRate As Double, sigma As Double,
Optional DividendYield As Double) As Double()
Dim d1 As Double
Dim d2 As Double
Dim Nd1 As Double
Dim Nd2 As Double
Dim N_dash_d1 As Double
Dim ResultArray() As Double
Dim gamma As Double
Dim vega As Double
Dim theta As Double
ReDim ResultArray(10) As Double
If (IsMissing(DividendYield)) Then
d1 = WorksheetFunction.Ln(SpotPrice / ExercisePrice)
+ ((RiskFreeRate + (0.5 * (sigma ^ 2))) * TimeToMaturity)
Else
d1 = WorksheetFunction.Ln(SpotPrice / ExercisePrice)
+ ((RiskFreeRate - DividendYield + (0.5 * (sigma ^ 2))) * TimeToMaturity)
End If
d1 = d1 / (sigma * (TimeToMaturity ^ (1 / 2)))
d2 = d1 - (sigma * (TimeToMaturity ^ (1 / 2)))
Nd1 = WorksheetFunction.NormSDist(d1)
Nd2 = WorksheetFunction.NormSDist(d2)
'Call Value
If (IsMissing(DividendYield)) Then
ResultArray(0) = (SpotPrice * Nd1) - (ExercisePrice * Exp(-RiskFreeRate * TimeToMaturity) * Nd2)
Else
ResultArray(0) = Exp(-DividendYield * TimeToMaturity)
* (SpotPrice * Nd1) - (ExercisePrice * Exp(-RiskFreeRate * TimeToMaturity) * Nd2)
End If
'Call Delta
ResultArray(1) = Nd1 * Exp(-DividendYield * TimeToMaturity)
'Call Gamma
N_dash_d1 = (1 / ((2 * WorksheetFunction.Pi) ^ (1 / 2)) * (Exp((-d1 ^ 2) / 2)))
gamma = (N_dash_d1 * Exp(-DividendYield * TimeToMaturity)) / (SpotPrice * sigma * (TimeToMaturity ^ (1 / 2)))
ResultArray(2) = gamma
'Call Vega
vega = SpotPrice * (TimeToMaturity ^ (1 / 2)) * N_dash_d1 * Exp(-DividendYield * TimeToMaturity)
ResultArray(3) = vega
'Call Theta
theta = -(SpotPrice * N_dash_d1 * sigma * Exp(-DividendYield * TimeToMaturity)) / (2 * (TimeToMaturity ^ (1 / 2)))
theta = theta + (DividendYield * SpotPrice * Nd1 * Exp(-DividendYield * TimeToMaturity))
theta = theta - (RiskFreeRate * ExercisePrice * Exp(-RiskFreeRate * TimeToMaturity) * Nd2)
ResultArray(4) = theta
'Put Value
If (IsMissing(DividendYield)) Then
ResultArray(5) = Exp(-RiskFreeRate * TimeToMaturity) * ExercisePrice
* (1 - Nd2) - SpotPrice * (1 - Nd1)
Else
ResultArray(5) = Exp(-RiskFreeRate * TimeToMaturity) * ExercisePrice
* WorksheetFunction.NormSDist(-d2) - Exp(-DividendYield * TimeToMaturity)
* SpotPrice * WorksheetFunction.NormSDist(-d1)
End If
'Put delta
ResultArray(6) = (Nd1 - 1) * Exp(-DividendYield * TimeToMaturity)
'Put Gamma
ResultArray(7) = gamma
'Put Vega
ResultArray(8) = vega
'Put Theta
theta = -(SpotPrice * N_dash_d1 * sigma * Exp(-DividendYield * TimeToMaturity)) / (2 * (TimeToMaturity ^ (1 / 2)))
theta = theta - (DividendYield * SpotPrice * WorksheetFunction.NormSDist(-d1) * Exp(-DividendYield * TimeToMaturity))
theta = theta + (RiskFreeRate * ExercisePrice * Exp(-RiskFreeRate * TimeToMaturity) * WorksheetFunction.NormSDist(-d2))
ResultArray(9) = theta
BlackScholes = ResultArray
End Function
The story of the Reuters Globex futures trading system developed for the Chicago Mercantile Exchange is a classic ETS (Electronic Trading System) case study. Originally devised in 1987 and launched in 1992, the product was plagued with technical difficulties and defects, and the general reluctance of the traditionally pit-based futures trading community to accept its merits.
On its first day of business in June 1992, 2,063 futures contracts traded on CME Globex. Today, an average of more than one million contracts a day are traded on CME Globex, accounting for over 50% of total CME volume.
See http://www.cme.com/trading/get/abt/welglobex951.html
The orginal product was beset with issues, and there were still serious doubts as to whether an ETS could ever replace the open outcry system:
See http://sigchi.org/chi95/proceedings/papers/jll_bdy.htm
However, the success of Globex (although still not without its issues - see http://www.finextra.com/fullstory.asp?id=13768) is a strong validation of the potential efficiency gains that can be realised on an ETS platform.
[UPDATE: After reading this post, see here for an updated version of the spreadsheet].
The Black-Scholes option valuation formula for an option paying a continuous dividend yield is the following:
Where
and
Attached is a simple Excel function that calculates the Black-Scholes option value for a specific set of input parameters. Currently, it just calculates the call value - if you use it as an array function, it will return a 4-element array with call value, call delta, put value, put delta, respectively. You could extend it pretty easily to calculate the rest of the Greeks.
Here is the code for the function (To create an Excel function, press ALT-F11 in your workbook and select Insert>Module. Note that the dividend yield parameter is optional.
Function BlackScholes(SpotPrice As Double, ExercisePrice As Double,
TimeToMaturity As Double, RiskFreeRate As Double, sigma As Double,
Optional DividendYield As Double) As Double()
Dim d1 As Double
Dim d2 As Double
Dim Nd1 As Double
Dim Nd2 As Double
Dim ResultArray() As Double
ReDim ResultArray(4) As Double
If (IsMissing(DividendYield)) Then
d1 = WorksheetFunction.Ln(SpotPrice / ExercisePrice) +
((RiskFreeRate + (0.5 * (sigma ^ 2))) * TimeToMaturity)
Else
d1 = WorksheetFunction.Ln(SpotPrice / ExercisePrice) +
((RiskFreeRate - DividendYield + (0.5 * (sigma ^ 2))) * TimeToMaturity)
End If
d1 = d1 / (sigma * (TimeToMaturity ^ (1 / 2)))
d2 = d1 - (sigma * (TimeToMaturity ^ (1 / 2)))
Nd1 = WorksheetFunction.NormSDist(d1)
Nd2 = WorksheetFunction.NormSDist(d2)
'Call Value
If (IsMissing(DividendYield)) Then
ResultArray(0) = (SpotPrice * Nd1)
- (ExercisePrice * Exp(-RiskFreeRate * TimeToMaturity) * Nd2)
Else
ResultArray(0) = Exp(-DividendYield * TimeToMaturity) * (SpotPrice * Nd1)
- (ExercisePrice * Exp(-RiskFreeRate * TimeToMaturity) * Nd2)
End If
'Call Delta
ResultArray(1) = Nd1
'Put Value
If (IsMissing(DividendYield)) Then
ResultArray(2) = Exp(-RiskFreeRate * TimeToMaturity)
* ExercisePrice * (1 - Nd2) - SpotPrice * (1 - Nd1)
Else
ResultArray(2) = Exp(-RiskFreeRate * TimeToMaturity) * ExercisePrice
* WorksheetFunction.NormSDist(-d2) - Exp(-DividendYield * TimeToMaturity)
* SpotPrice * WorksheetFunction.NormSDist(-d1)
End If
'Put delta
ResultArray(3) = -WorksheetFunction.NormSDist(-d1)
BlackScholes = ResultArray
End Function
Save this, and set up the input parameters in Excel. Select a range of 4 cells, and then click the f(x) function selection button. Choose from the list of User Defined functions, and then select BlackScholes. Excel will prompt you for the input parameters:
When you are finished inputing the parameters, press CTRL+SHIFT+ENTER to execute the function. Excel will populate the four cells with the calculated option values:
A sample workbook is attached:BlackScholes.xls