├── pkg ├── .gitignore ├── .Rbuildignore ├── tests │ ├── testthat.R │ └── testthat │ │ ├── test-Bachelier.R │ │ ├── test-BlackScholes.R │ │ ├── test-CEV.R │ │ ├── test-SABR.R │ │ └── test-spread.R ├── NEWS.md ├── README.md ├── NAMESPACE ├── FE-R.Rproj ├── DESCRIPTION ├── man │ ├── CevMassZero.Rd │ ├── CevPrice.Rd │ ├── BachelierPrice.Rd │ ├── BlackScholesImpvol.Rd │ ├── BachelierImpvol.Rd │ ├── SpreadBachelier.Rd │ ├── Nsvh1Choi2019.Rd │ ├── BlackScholesPrice.Rd │ ├── SwitchMargrabe.Rd │ ├── SabrHagan2002.Rd │ ├── SpreadKirk.Rd │ └── SpreadBjerksund2014.Rd └── R │ ├── bachelier_price.R │ ├── blackscholes_impvol.R │ ├── blackscholes_price.R │ ├── bachelier_impvol.R │ ├── cev.R │ ├── sabr.R │ └── spread.R ├── README.md ├── .gitignore └── LICENSE /pkg/.gitignore: -------------------------------------------------------------------------------- 1 | inst/doc 2 | -------------------------------------------------------------------------------- /pkg/.Rbuildignore: -------------------------------------------------------------------------------- 1 | ^.*\.Rproj$ 2 | ^\.Rproj\.user$ 3 | ^LICENSE\.md$ 4 | -------------------------------------------------------------------------------- /pkg/tests/testthat.R: -------------------------------------------------------------------------------- 1 | library(testthat) 2 | library(FER) 3 | 4 | test_check("FER") 5 | -------------------------------------------------------------------------------- /pkg/NEWS.md: -------------------------------------------------------------------------------- 1 | # FER 0.94 2 | 3 | * Added new functions: `CevMassZero`, `Nsvh1Choi2019`, `SpreadBachelier`, 4 | `SpreadBjerksund2014`, `SpreadKirk`, and `SwitchMargrabe` 5 | * Added a `NEWS.md` file to track changes to the package. 6 | -------------------------------------------------------------------------------- /pkg/README.md: -------------------------------------------------------------------------------- 1 | # FER 2 | Financial Engineering in R 3 | 4 | ## Contents 5 | * Black-Scholes model: option price and implied volatility 6 | * Bachelier model: option price and implied volatility 7 | * Constant-Elasticity-of-Variance (CEV) model: option price 8 | * Stochastic-Alpha-Beta-Rho (SABR) model: equivalent BS volatility and price 9 | -------------------------------------------------------------------------------- /pkg/NAMESPACE: -------------------------------------------------------------------------------- 1 | # Generated by roxygen2: do not edit by hand 2 | 3 | export(BachelierImpvol) 4 | export(BachelierPrice) 5 | export(BlackScholesImpvol) 6 | export(BlackScholesPrice) 7 | export(CevMassZero) 8 | export(CevPrice) 9 | export(Nsvh1Choi2019) 10 | export(SabrHagan2002) 11 | export(SpreadBachelier) 12 | export(SpreadBjerksund2014) 13 | export(SpreadKirk) 14 | export(SwitchMargrabe) 15 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # FE-R 2 | Financial Engineering functions in R 3 | 4 | ## R Package 5 | * https://cran.r-project.org/package=FER 6 | 7 | ## Contents 8 | * Black-Scholes option pricing model: price and implied volatility 9 | * Bachelier option pricing model: price and implied volatility 10 | 11 | ## Installation 12 | Install `devtools` package in run 13 | ```R 14 | library(devtools) 15 | devtools::install_github("PyFE/FE-R", subdir="pkg") 16 | ``` 17 | -------------------------------------------------------------------------------- /pkg/FE-R.Rproj: -------------------------------------------------------------------------------- 1 | Version: 1.0 2 | 3 | RestoreWorkspace: No 4 | SaveWorkspace: No 5 | AlwaysSaveHistory: Default 6 | 7 | EnableCodeIndexing: Yes 8 | UseSpacesForTab: Yes 9 | NumSpacesForTab: 2 10 | Encoding: UTF-8 11 | 12 | RnwWeave: Sweave 13 | LaTeX: pdfLaTeX 14 | 15 | AutoAppendNewline: Yes 16 | StripTrailingWhitespace: Yes 17 | 18 | BuildType: Package 19 | PackageUseDevtools: Yes 20 | PackageInstallArgs: --no-multiarch --with-keep.source 21 | PackageRoxygenize: rd,collate,namespace 22 | -------------------------------------------------------------------------------- /pkg/tests/testthat/test-Bachelier.R: -------------------------------------------------------------------------------- 1 | library(FER) 2 | 3 | test_that("Bachelier Price and Impvol", { 4 | strike <- runif(1000, 50, 150) 5 | sigma <- runif(1000, 1, 20) 6 | texp <- runif(1000, 1e-4, 10) 7 | intr <- runif(1000, 0, 0.1) 8 | divr <- runif(1000, 0, 0.1) 9 | cp <- ifelse(runif(1000)>0.5, 1L, -1L) 10 | 11 | p <- FER::BachelierPrice(strike, 100, texp, sigma, cp=cp, intr=intr, divr=divr) 12 | iv <- FER::BachelierImpvol(p, strike, 100, texp, cp=cp, intr=intr, divr=divr) 13 | p2 <- FER::BachelierPrice(strike, 100, texp, iv, cp=cp, intr=intr, divr=divr) 14 | expect_equal(p, p2, tolerance = 1e-8) 15 | }) 16 | -------------------------------------------------------------------------------- /pkg/tests/testthat/test-BlackScholes.R: -------------------------------------------------------------------------------- 1 | library(FER) 2 | 3 | test_that("Black-Scholes Price and Impvol", { 4 | strike <- runif(1000, 50, 150) 5 | sigma <- runif(1000, 0.001, 2) 6 | texp <- runif(1000, 1e-4, 10) 7 | intr <- runif(1000, 0, 0.1) 8 | divr <- runif(1000, 0, 0.1) 9 | cp <- ifelse(runif(100)>0.5, 1L, -1L) 10 | 11 | p <- FER::BlackScholesPrice(strike, 100, texp, sigma, cp=cp, intr=intr, divr=divr) 12 | iv <- FER::BlackScholesImpvol(p, strike, 100, texp, cp=cp, intr=intr, divr=divr) 13 | p2 <- FER::BlackScholesPrice(strike, 100, texp, iv, cp=cp, intr=intr, divr=divr) 14 | expect_equal(p, p2, tolerance = 1e-8) 15 | }) 16 | -------------------------------------------------------------------------------- /pkg/tests/testthat/test-CEV.R: -------------------------------------------------------------------------------- 1 | library(FER) 2 | 3 | test_that("CEV Reference Price", { 4 | value <- round( FER::CevPrice( 5 | strike=seq(0.4, 2, 0.4)*0.05, spot=0.05, texp=1, sigma=0.4, beta=0.3), 6 | digits=5 ) 7 | value2 <- c(0.04608, 0.04229, 0.03868, 0.03525, 0.03203) 8 | expect_equal(value, value2, tolerance = 1e-6) 9 | }) 10 | 11 | test_that("CEV Put Price vs MassZero", { 12 | value1 <- FER::CevPrice( 13 | strike=0.001, spot=1, texp=1:10, sigma=0.5, beta=0.2, cp=-1 14 | )*1000 15 | value2 <- FER::CevMassZero(spot=1, texp=1:10, sigma=0.5, beta=0.2) 16 | expect_equal(value1, value2, tolerance = 1e-5) 17 | }) 18 | 19 | 20 | -------------------------------------------------------------------------------- /pkg/DESCRIPTION: -------------------------------------------------------------------------------- 1 | Package: FER 2 | Title: Financial Engineering in R 3 | Version: 0.94 4 | Authors@R: person("Jaehyuk", "Choi", email = "pyfe@eml.cc", role = c("aut", "cre")) 5 | Description: R implementations of standard financial engineering codes; 6 | vanilla option pricing models such as Black-Scholes, Bachelier, CEV, and 7 | SABR. 8 | URL: https://github.com/PyFE/FE-R 9 | BugReports: https://github.com/PyFE/FE-R/issues 10 | Depends: 11 | R (>= 3.3.1) 12 | NeedsCompilation: no 13 | License: GPL (>= 2) 14 | Encoding: UTF-8 15 | LazyData: true 16 | RoxygenNote: 7.1.1 17 | Imports: 18 | stats, 19 | statmod 20 | Suggests: 21 | testthat (>= 3.0.0) 22 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ### User 2 | .DS_Store 3 | 4 | # History files 5 | .Rhistory 6 | .Rapp.history 7 | 8 | # Session Data files 9 | .RData 10 | 11 | # User-specific files 12 | .Ruserdata 13 | 14 | # Example code in package build process 15 | *-Ex.R 16 | 17 | # Output files from R CMD build 18 | /*.tar.gz 19 | 20 | # Output files from R CMD check 21 | /*.Rcheck/ 22 | 23 | # RStudio files 24 | .Rproj.user/ 25 | 26 | # produced vignettes 27 | vignettes/*.html 28 | vignettes/*.pdf 29 | 30 | # OAuth2 token, see https://github.com/hadley/httr/releases/tag/v0.3 31 | .httr-oauth 32 | 33 | # knitr and R markdown default cache directories 34 | *_cache/ 35 | /cache/ 36 | 37 | # Temporary files created by R markdown 38 | *.utf8.md 39 | *.knit.md 40 | 41 | # R Environment Variables 42 | .Renviron 43 | .Rproj.user 44 | -------------------------------------------------------------------------------- /pkg/man/CevMassZero.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/cev.R 3 | \name{CevMassZero} 4 | \alias{CevMassZero} 5 | \title{Calculate the mass at zero under the CEV model} 6 | \usage{ 7 | CevMassZero( 8 | spot, 9 | texp = 1, 10 | sigma, 11 | beta = 0.5, 12 | intr = 0, 13 | divr = 0, 14 | forward = spot * exp(-divr * texp)/df, 15 | df = exp(-intr * texp) 16 | ) 17 | } 18 | \arguments{ 19 | \item{spot}{(vector of) spot price} 20 | 21 | \item{texp}{(vector of) time to expiry} 22 | 23 | \item{sigma}{(vector of) volatility} 24 | 25 | \item{beta}{beta} 26 | 27 | \item{intr}{interest rate} 28 | 29 | \item{divr}{dividend rate} 30 | 31 | \item{forward}{forward price. If given, \code{forward} overrides \code{spot}} 32 | 33 | \item{df}{discount factor. If given, \code{df} overrides \code{intr}} 34 | } 35 | \value{ 36 | mass at zero 37 | } 38 | \description{ 39 | Calculate the mass at zero under the CEV model 40 | } 41 | \examples{ 42 | spot <- 100 43 | texp <- 1.2 44 | beta <- 0.5 45 | sigma <- 2 46 | FER::CevMassZero(spot, texp, sigma, beta) 47 | 48 | } 49 | -------------------------------------------------------------------------------- /pkg/R/bachelier_price.R: -------------------------------------------------------------------------------- 1 | #' Calculate Bachelier model option price 2 | #' 3 | #' @param strike (vector of) strike price 4 | #' @param spot (vector of) spot price 5 | #' @param texp (vector of) time to expiry 6 | #' @param sigma (vector of) volatility 7 | #' @param intr interest rate (domestic interest rate) 8 | #' @param divr dividend/convenience yield (foreign interest rate) 9 | #' @param cp call/put sign. \code{1} for call, \code{-1} for put. 10 | #' @param forward forward price. If given, \code{forward} overrides \code{spot} 11 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 12 | #' @return option price 13 | #' @export 14 | #' 15 | #' @references Choi, J., Kim, K., & Kwak, M. (2009). 16 | #' Numerical Approximation of the Implied Volatility Under Arithmetic Brownian 17 | #' Motion. Applied Mathematical Finance, 16(3), 261-268. 18 | #' \doi{10.1080/13504860802583436} 19 | #' 20 | #' @examples 21 | #' spot <- 100 22 | #' strike <- seq(80,125,5) 23 | #' texp <- 1.2 24 | #' sigma <- 20 25 | #' intr <- 0.05 26 | #' FER::BachelierPrice(strike, spot, texp, sigma, intr=intr) 27 | #' 28 | #' @seealso \code{\link{BachelierImpvol}} 29 | #' 30 | BachelierPrice <- function( 31 | strike=forward, spot, texp=1, sigma, 32 | intr=0, divr=0, cp=1L, 33 | forward=spot*exp(-divr*texp)/df, df=exp(-intr*texp) 34 | ){ 35 | stdev <- sigma*sqrt(texp) 36 | stdev[stdev < 1e-32] <- 1e-32 37 | dn <- cp*(forward-strike)/stdev 38 | 39 | #Option Price 40 | price <- df * (cp*(forward-strike)*stats::pnorm(dn) + stdev*stats::dnorm(dn)) 41 | return( price ) 42 | } 43 | -------------------------------------------------------------------------------- /pkg/man/CevPrice.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/cev.R 3 | \name{CevPrice} 4 | \alias{CevPrice} 5 | \title{Calculate the constant elasticity of variance (CEV) model option price} 6 | \usage{ 7 | CevPrice( 8 | strike = forward, 9 | spot, 10 | texp = 1, 11 | sigma, 12 | beta = 0.5, 13 | intr = 0, 14 | divr = 0, 15 | cp = 1L, 16 | forward = spot * exp(-divr * texp)/df, 17 | df = exp(-intr * texp) 18 | ) 19 | } 20 | \arguments{ 21 | \item{strike}{(vector of) strike price} 22 | 23 | \item{spot}{(vector of) spot price} 24 | 25 | \item{texp}{(vector of) time to expiry} 26 | 27 | \item{sigma}{(vector of) volatility} 28 | 29 | \item{beta}{elasticity parameter} 30 | 31 | \item{intr}{interest rate (domestic interest rate)} 32 | 33 | \item{divr}{dividend/convenience yield (foreign interest rate)} 34 | 35 | \item{cp}{call/put sign. \code{1} for call, \code{-1} for put.} 36 | 37 | \item{forward}{forward price. If given, \code{forward} overrides \code{spot}} 38 | 39 | \item{df}{discount factor. If given, \code{df} overrides \code{intr}} 40 | } 41 | \value{ 42 | option price 43 | } 44 | \description{ 45 | Calculate the constant elasticity of variance (CEV) model option price 46 | } 47 | \examples{ 48 | spot <- 100 49 | strike <- seq(80,125,5) 50 | texp <- 1.2 51 | beta <- 0.5 52 | sigma <- 2 53 | FER::CevPrice(strike, spot, texp, sigma, beta) 54 | 55 | } 56 | \references{ 57 | Schroder, M. (1989). Computing the constant elasticity 58 | of variance option pricing formula. Journal of Finance, 59 | 44(1), 211-219. \doi{10.1111/j.1540-6261.1989.tb02414.x} 60 | } 61 | -------------------------------------------------------------------------------- /pkg/man/BachelierPrice.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/bachelier_price.R 3 | \name{BachelierPrice} 4 | \alias{BachelierPrice} 5 | \title{Calculate Bachelier model option price} 6 | \usage{ 7 | BachelierPrice( 8 | strike = forward, 9 | spot, 10 | texp = 1, 11 | sigma, 12 | intr = 0, 13 | divr = 0, 14 | cp = 1L, 15 | forward = spot * exp(-divr * texp)/df, 16 | df = exp(-intr * texp) 17 | ) 18 | } 19 | \arguments{ 20 | \item{strike}{(vector of) strike price} 21 | 22 | \item{spot}{(vector of) spot price} 23 | 24 | \item{texp}{(vector of) time to expiry} 25 | 26 | \item{sigma}{(vector of) volatility} 27 | 28 | \item{intr}{interest rate (domestic interest rate)} 29 | 30 | \item{divr}{dividend/convenience yield (foreign interest rate)} 31 | 32 | \item{cp}{call/put sign. \code{1} for call, \code{-1} for put.} 33 | 34 | \item{forward}{forward price. If given, \code{forward} overrides \code{spot}} 35 | 36 | \item{df}{discount factor. If given, \code{df} overrides \code{intr}} 37 | } 38 | \value{ 39 | option price 40 | } 41 | \description{ 42 | Calculate Bachelier model option price 43 | } 44 | \examples{ 45 | spot <- 100 46 | strike <- seq(80,125,5) 47 | texp <- 1.2 48 | sigma <- 20 49 | intr <- 0.05 50 | FER::BachelierPrice(strike, spot, texp, sigma, intr=intr) 51 | 52 | } 53 | \references{ 54 | Choi, J., Kim, K., & Kwak, M. (2009). 55 | Numerical Approximation of the Implied Volatility Under Arithmetic Brownian 56 | Motion. Applied Mathematical Finance, 16(3), 261-268. 57 | \doi{10.1080/13504860802583436} 58 | } 59 | \seealso{ 60 | \code{\link{BachelierImpvol}} 61 | } 62 | -------------------------------------------------------------------------------- /pkg/man/BlackScholesImpvol.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/blackscholes_impvol.R 3 | \name{BlackScholesImpvol} 4 | \alias{BlackScholesImpvol} 5 | \title{Calculate Black-Scholes implied volatility} 6 | \usage{ 7 | BlackScholesImpvol( 8 | price, 9 | strike = forward, 10 | spot, 11 | texp = 1, 12 | intr = 0, 13 | divr = 0, 14 | cp = 1L, 15 | forward = spot * exp(-divr * texp)/df, 16 | df = exp(-intr * texp) 17 | ) 18 | } 19 | \arguments{ 20 | \item{price}{(vector of) option price} 21 | 22 | \item{strike}{(vector of) strike price} 23 | 24 | \item{spot}{(vector of) spot price} 25 | 26 | \item{texp}{(vector of) time to expiry} 27 | 28 | \item{intr}{interest rate (domestic interest rate)} 29 | 30 | \item{divr}{dividend/convenience yield (foreign interest rate)} 31 | 32 | \item{cp}{call/put sign. \code{1} for call, \code{-1} for put.} 33 | 34 | \item{forward}{forward price. If given, \code{forward} overrides \code{spot}} 35 | 36 | \item{df}{discount factor. If given, \code{df} overrides \code{intr}} 37 | } 38 | \value{ 39 | Black-Scholes implied volatility 40 | } 41 | \description{ 42 | Calculate Black-Scholes implied volatility 43 | } 44 | \examples{ 45 | spot <- 100 46 | strike <- 100 47 | texp <- 1.2 48 | sigma <- 0.2 49 | intr <- 0.05 50 | price <- 20 51 | FER::BlackScholesImpvol(price, strike, spot, texp, intr=intr) 52 | 53 | } 54 | \references{ 55 | Giner, G., & Smyth, G. K. (2016). statmod: Probability Calculations 56 | for the Inverse Gaussian Distribution. The R Journal, 8(1), 339-351. 57 | \doi{10.32614/RJ-2016-024} 58 | } 59 | \seealso{ 60 | \code{\link{BlackScholesPrice}} 61 | } 62 | -------------------------------------------------------------------------------- /pkg/man/BachelierImpvol.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/bachelier_impvol.R 3 | \name{BachelierImpvol} 4 | \alias{BachelierImpvol} 5 | \title{Calculate Bachelier model implied volatility} 6 | \usage{ 7 | BachelierImpvol( 8 | price, 9 | strike = forward, 10 | spot, 11 | texp = 1, 12 | intr = 0, 13 | divr = 0, 14 | cp = 1L, 15 | forward = spot * exp(-divr * texp)/df, 16 | df = exp(-intr * texp) 17 | ) 18 | } 19 | \arguments{ 20 | \item{price}{(vector of) option price} 21 | 22 | \item{strike}{(vector of) strike price} 23 | 24 | \item{spot}{(vector of) spot price} 25 | 26 | \item{texp}{(vector of) time to expiry} 27 | 28 | \item{intr}{interest rate (domestic interest rate)} 29 | 30 | \item{divr}{dividend/convenience yield (foreign interest rate)} 31 | 32 | \item{cp}{call/put sign. \code{1} for call, \code{-1} for put.} 33 | 34 | \item{forward}{forward price. If given, \code{forward} overrides \code{spot}} 35 | 36 | \item{df}{discount factor. If given, \code{df} overrides \code{intr}} 37 | } 38 | \value{ 39 | Bachelier implied volatility 40 | } 41 | \description{ 42 | Calculate Bachelier model implied volatility 43 | } 44 | \examples{ 45 | spot <- 100 46 | strike <- 100 47 | texp <- 1.2 48 | sigma <- 20 49 | intr <- 0.05 50 | price <- 20 51 | FER::BachelierImpvol(price, strike, spot, texp, intr=intr) 52 | 53 | } 54 | \references{ 55 | Choi, J., Kim, K., & Kwak, M. (2009). 56 | Numerical Approximation of the Implied Volatility Under Arithmetic Brownian 57 | Motion. Applied Mathematical Finance, 16(3), 261-268. 58 | \doi{10.1080/13504860802583436} 59 | } 60 | \seealso{ 61 | \code{\link{BachelierPrice}} 62 | } 63 | -------------------------------------------------------------------------------- /pkg/tests/testthat/test-SABR.R: -------------------------------------------------------------------------------- 1 | library(FER) 2 | 3 | # Table 1 in Antonov & Spector (2012). https://ssrn.com/abstract=2026350 4 | test_that("SabrHagan2002 BS Volatility Table 1", { 5 | value <- round( FER::SabrHagan2002( 6 | strike=seq(0.1, 2, 0.1), spot=1, texp=10, sigma=0.25, 7 | vov=0.3, rho=-0.8, beta=0.3), 8 | digits = 4 ) 9 | value2 <- c( 10 | 0.7176, 0.5725, 0.4886, 0.4293, 0.3835, 11 | 0.3462, 0.3148, 0.2876, 0.2638, 0.2427, 12 | 0.2238, 0.2068, 0.1916, 0.1781, 0.1663, 13 | 0.1562, 0.1478, 0.1412, 0.136, 0.1322) 14 | expect_equal(value, value2, tolerance = 1e-5) 15 | }) 16 | 17 | # Table 13 in Antonov & Spector (2012). https://ssrn.com/abstract=2026350 18 | test_that("SabrHagan2002 BS Volatility Table 13", { 19 | value <- round( FER::SabrHagan2002( 20 | strike=seq(0.1, 2, 0.1), spot=1, texp=20, sigma=0.25, 21 | vov=0.3, rho=-0.5, beta=0.3), 22 | digits=4 ) 23 | value2 <- c( 24 | 0.7603, 0.5973, 0.5078, 0.4463, 0.3997, 25 | 0.3626, 0.332, 0.3063, 0.2844, 0.2658, 26 | 0.2498, 0.2364, 0.2251, 0.2158, 0.2083, 27 | 0.2023, 0.1976, 0.1941, 0.1914, 0.1895) 28 | expect_equal(value, value2, tolerance = 1e-5) 29 | }) 30 | 31 | 32 | test_that("NSVh Put-Call Parity", { 33 | spot <- 120 34 | strike <- 120 + (-3:3)*10 35 | texp <- 2.3 36 | sigma <- 20 37 | vov <- 0.2 38 | rho <- -0.5 39 | intr <- 0.1 40 | divr <- 0.05 41 | 42 | val = exp(-divr*texp)*spot - exp(-intr*texp)*strike 43 | c <- FER::Nsvh1Choi2019(strike, spot, texp, sigma, vov, rho, 44 | intr=intr, divr=divr, cp=1L) 45 | p <- FER::Nsvh1Choi2019(strike, spot, texp, sigma, vov, rho, 46 | intr=intr, divr=divr, cp=-1L) 47 | expect_equal(c - p, val, tolerance = 1e-12) 48 | }) 49 | 50 | -------------------------------------------------------------------------------- /pkg/R/blackscholes_impvol.R: -------------------------------------------------------------------------------- 1 | #' Calculate Black-Scholes implied volatility 2 | #' 3 | #' @param price (vector of) option price 4 | #' @param strike (vector of) strike price 5 | #' @param spot (vector of) spot price 6 | #' @param texp (vector of) time to expiry 7 | #' @param intr interest rate (domestic interest rate) 8 | #' @param divr dividend/convenience yield (foreign interest rate) 9 | #' @param cp call/put sign. \code{1} for call, \code{-1} for put. 10 | #' @param forward forward price. If given, \code{forward} overrides \code{spot} 11 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 12 | #' @return Black-Scholes implied volatility 13 | #' 14 | #' @references Giner, G., & Smyth, G. K. (2016). statmod: Probability Calculations 15 | #' for the Inverse Gaussian Distribution. The R Journal, 8(1), 339-351. 16 | #' \doi{10.32614/RJ-2016-024} 17 | #' 18 | #' @export 19 | #' 20 | #' @examples 21 | #' spot <- 100 22 | #' strike <- 100 23 | #' texp <- 1.2 24 | #' sigma <- 0.2 25 | #' intr <- 0.05 26 | #' price <- 20 27 | #' FER::BlackScholesImpvol(price, strike, spot, texp, intr=intr) 28 | #' 29 | #' @seealso \code{\link{BlackScholesPrice}} 30 | #' 31 | BlackScholesImpvol <- function( 32 | price, strike=forward, spot, texp=1, 33 | intr=0, divr=0, cp=1L, 34 | forward=spot*exp(-divr*texp)/df, df=exp(-intr*texp) 35 | ){ 36 | timeval <- (price/df - pmax(cp*(forward-strike), 0))/pmin(forward, strike) 37 | # when the time value is very slightly negative, we correct to give zero vol. 38 | timeval[price>0 & -8*.Machine$double.eps0 & 10)) 42 | 43 | price <- cp*df*(forward*term1 - strike*term2) 44 | 45 | return(price) 46 | } 47 | 48 | 49 | #' Calculate the mass at zero under the CEV model 50 | #' 51 | #' @param spot (vector of) spot price 52 | #' @param texp (vector of) time to expiry 53 | #' @param sigma (vector of) volatility 54 | #' @param beta beta 55 | #' @param intr interest rate 56 | #' @param divr dividend rate 57 | #' @param forward forward price. If given, \code{forward} overrides \code{spot} 58 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 59 | #' @return mass at zero 60 | #' 61 | #' @export 62 | #' 63 | #' @examples 64 | #' spot <- 100 65 | #' texp <- 1.2 66 | #' beta <- 0.5 67 | #' sigma <- 2 68 | #' FER::CevMassZero(spot, texp, sigma, beta) 69 | #' 70 | CevMassZero <- function( 71 | spot, texp=1, sigma, beta=0.5, intr=0, divr=0, 72 | forward=spot*exp(-divr*texp)/df, df=exp(-intr*texp) 73 | ){ 74 | betac <- 1.0 - beta 75 | scale <- (betac*sigma)^2*texp 76 | x <- 0.5*forward^(2*betac)/scale 77 | deg <- 0.5/betac # degree of freedom 78 | 79 | mass <- stats::pgamma(x, deg, lower.tail=F) 80 | 81 | return(mass) 82 | } 83 | -------------------------------------------------------------------------------- /pkg/R/sabr.R: -------------------------------------------------------------------------------- 1 | #' Calculate the equivalent BS volatility (Hagan et al. 2002) for 2 | #' the Stochatic-Alpha-Beta-Rho (SABR) model 3 | #' 4 | #' @param strike (vector of) strike price 5 | #' @param spot (vector of) spot price 6 | #' @param texp (vector of) time to expiry 7 | #' @param sigma (vector of) volatility 8 | #' @param vov (vector of) vol-of-vol 9 | #' @param rho (vector of) correlation 10 | #' @param beta (vector of) beta 11 | #' @param intr interest rate (domestic interest rate) 12 | #' @param divr convenience rate (foreign interest rate) 13 | #' @param cp call/put sign. \code{NULL} for BS vol (default), \code{1} for call price, \code{-1} for put price. 14 | #' @param forward forward price. If given, \code{forward} overrides \code{spot} 15 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 16 | #' @return BS volatility or option price based on \code{cp} 17 | #' 18 | #' @references Hagan, P. S., Kumar, D., Lesniewski, A. S., & Woodward, D. E. (2002). 19 | #' Managing Smile Risk. Wilmott, September, 84-108. 20 | #' 21 | #' @export 22 | #' 23 | #' @examples 24 | #' sigma <- 0.25 25 | #' vov <- 0.3 26 | #' rho <- -0.8 27 | #' beta <- 0.3 28 | #' texp <- 10 29 | #' strike <- seq(0.1, 2, 0.1) 30 | #' FER::SabrHagan2002(strike, 1, texp, sigma, vov, rho, beta) 31 | #' FER::SabrHagan2002(strike, 1, texp, sigma, vov, rho, beta, cp=1) 32 | #' 33 | SabrHagan2002 <- function( 34 | strike=forward, spot, texp=1, sigma, vov=0, rho=0, beta=1, 35 | intr=0, divr=0, cp=NULL, 36 | forward=spot*exp(-divr*texp)/df, df=exp(-intr*texp) 37 | ){ 38 | betac <- 1 - beta 39 | betac2 <- betac*betac 40 | rho2 <- rho*rho 41 | 42 | powFwdStrk <- (forward*strike)^(betac/2) 43 | logFwdStrk <- log(forward/strike) 44 | logFwdStrk2 <- logFwdStrk^2 45 | 46 | pre1 <- powFwdStrk*( 1 + betac2/24 * logFwdStrk2*(1 + betac2/80 * logFwdStrk2) ) 47 | 48 | pre2alp0 <- (2-3*rho2)*vov^2/24 49 | pre2alp1 <- vov*rho*beta/4/powFwdStrk 50 | pre2alp2 <- betac2/24/powFwdStrk^2 51 | 52 | pre2 <- 1 + texp*( pre2alp0 + sigma*(pre2alp1 + pre2alp2*sigma) ) 53 | 54 | zz <- powFwdStrk*logFwdStrk*vov/sigma # need to make sure sig > 0 55 | yy <- sqrt(1 + zz*(zz-2*rho)) 56 | 57 | xx_zz <- 1 + (zz/2)*(rho + zz*((rho2-1/3) + (5*rho2-3)/4*rho*zz)) 58 | 59 | I <- (zz >= 1e-5) 60 | xx_zz[I] <- log( (yy[I] + (zz[I]-rho))/(1-rho) ) / zz[I] 61 | I <- (zz <= -1e-5) 62 | xx_zz[I] <- log( (1+rho)/(yy[I] - (zz[I]-rho)) ) / zz[I] 63 | 64 | vol.bs <- sigma*pre2/(pre1*xx_zz) # BS volatility 65 | 66 | if(is.null(cp)){ 67 | return(vol.bs) 68 | } else { 69 | # if cp is specified, calculate option price 70 | p <- df * BlackScholesPrice(strike, forward, texp, sigma=vol.bs, cp=cp) 71 | return(p) 72 | } 73 | } 74 | 75 | 76 | #' Calculate the option price under the NSVh model with lambda=1 (Choi et al. 2019) 77 | #' 78 | #' @param strike (vector of) strike price 79 | #' @param spot (vector of) spot price 80 | #' @param texp (vector of) time to expiry 81 | #' @param sigma (vector of) volatility 82 | #' @param vov (vector of) vol-of-vol 83 | #' @param rho (vector of) correlation 84 | #' @param intr interest rate 85 | #' @param divr dividend rate 86 | #' @param cp \code{1} (default) for call price, \code{-1} for put price 87 | #' @param forward forward price. If given, \code{forward} overrides \code{spot} 88 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 89 | #' @return option price 90 | #' 91 | #' @references Choi, J., Liu, C., & Seo, B. K. (2019). Hyperbolic normal 92 | #' stochastic volatility model. Journal of Futures Markets, 39(2), 186–204. 93 | #' \doi{10.1002/fut.21967} 94 | #' 95 | #' @export 96 | #' 97 | #' @examples 98 | #' 99 | #' spot <- 100 100 | #' strike <- seq(80,125,5) 101 | #' texp <- 1.2 102 | #' sigma <- 20 103 | #' vov <- 0.2 104 | #' rho <- -0.5 105 | #' strike <- seq(0.1, 2, 0.1) 106 | #' 107 | #' FER::Nsvh1Choi2019(strike, spot, texp, sigma, vov, rho) 108 | #' 109 | Nsvh1Choi2019 <- function( 110 | strike=forward, spot, texp=1, sigma, vov=0, rho=0, 111 | intr=0, divr=0, cp=1L, 112 | forward=spot*exp(-divr*texp)/df, df=exp(-intr*texp) 113 | ){ 114 | rhoc <- sqrt(1-rho*rho) 115 | vov.sqt <- vov * sqrt(texp) 116 | vov.var <- exp(0.5*texp*vov^2) 117 | 118 | d <- asinh(((forward-strike)*vov/sigma - vov.var*rho)/rhoc) + atanh(rho) 119 | d <- d / vov.sqt 120 | 121 | p <- (forward - strike - sigma/vov*rho*vov.var) * stats::pnorm(cp*d) 122 | p <- p + 0.5*sigma/vov*vov.var* 123 | ((1+rho)*stats::pnorm(cp*(d+vov.sqt)) - (1-rho)*stats::pnorm(cp*(d-vov.sqt))) 124 | p <- df * cp * p 125 | return(p) 126 | } 127 | 128 | -------------------------------------------------------------------------------- /pkg/R/spread.R: -------------------------------------------------------------------------------- 1 | #' Margrabe's formula for exhange option price 2 | #' 3 | #' @description The payout of the exchange option is 4 | #' \code{max(S1_T - S2_T, 0)} where \code{S1_T} and \code{S2_T} are the 5 | #' prices at expiry \code{T} of assets 1 and 2 respectively. 6 | #' 7 | #' @param spot1 (vector of) spot price of asset 1 8 | #' @param spot2 (vector of) spot price of asset 2 9 | #' @param texp (vector of) time to expiry 10 | #' @param sigma1 (vector of) volatility of asset 1 11 | #' @param sigma2 (vector of) volatility of asset 2 12 | #' @param corr correlation 13 | #' @param intr interest rate 14 | #' @param divr1 dividend rate of asset 1 15 | #' @param divr2 dividend rate of asset 2 16 | #' @param cp call/put sign. \code{1} for call, \code{-1} for put. 17 | #' @param forward1 forward price of asset 1. If given, overrides \code{spot1} 18 | #' @param forward2 forward price of asset 2. If given, overrides \code{spot2} 19 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 20 | #' @return option price 21 | #' 22 | #' @export 23 | #' 24 | #' @references Margrabe, W. (1978). The value of an option to exchange one 25 | #' asset for another. The Journal of Finance, 33(1), 177–186. 26 | #' 27 | #' @seealso \code{\link{SpreadKirk}} 28 | #' 29 | #' @examples 30 | #' 31 | #' FER::SwitchMargrabe(100, 120, 1.3, 0.2, 0.3, -0.5) 32 | #' 33 | SwitchMargrabe <- function( 34 | spot1, spot2, texp=1, sigma1, sigma2, corr, 35 | intr=0, divr1=0, divr2=0, cp=1L, 36 | forward1=spot1*exp(-divr1*texp)/df, 37 | forward2=spot2*exp(-divr2*texp)/df, 38 | df=exp(-intr*texp) 39 | ){ 40 | vol.spread <- sqrt(sigma1^2 - 2*corr*sigma1*sigma2 + sigma2^2) 41 | val <- df * BlackScholesPrice( 42 | strike=forward2, forward=forward1, texp=texp, sigma=vol.spread, cp=cp) 43 | return(val) 44 | } 45 | 46 | #' Kirk's approximation for spread option 47 | #' 48 | #' @description The payout of the spread option is 49 | #' \code{max(S1_T - S2_T - K, 0)} where \code{S1_T} and \code{S2_T} are the 50 | #' prices at expiry \code{T} of assets 1 and 2 respectively and \code{K} is 51 | #' the strike price. 52 | #' 53 | #' @param strike (vector of) strike price 54 | #' @param spot1 (vector of) spot price of asset 1 55 | #' @param spot2 (vector of) spot price of asset 2 56 | #' @param texp (vector of) time to expiry 57 | #' @param sigma1 (vector of) volatility of asset 1 58 | #' @param sigma2 (vector of) volatility of asset 2 59 | #' @param corr correlation 60 | #' @param intr interest rate 61 | #' @param divr1 dividend rate of asset 1 62 | #' @param divr2 dividend rate of asset 2 63 | #' @param cp call/put sign. \code{1} for call, \code{-1} for put. 64 | #' @param forward1 forward price of asset 1. If given, overrides \code{spot1} 65 | #' @param forward2 forward price of asset 2. If given, overrides \code{spot2} 66 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 67 | #' @return option price 68 | #' 69 | #' @export 70 | #' 71 | #' @references Kirk, E. (1995). Correlation in the energy markets. In Managing 72 | #' Energy Price Risk (First, pp. 71–78). Risk Publications. 73 | #' 74 | #' @seealso \code{\link{SwitchMargrabe}} 75 | #' 76 | #' @examples 77 | #' 78 | #' FER::SpreadKirk((-2:2)*10, 100, 120, 1.3, 0.2, 0.3, -0.5) 79 | #' 80 | SpreadKirk <- function( 81 | strike=0, spot1, spot2, texp=1, sigma1, sigma2, corr, 82 | intr=0, divr1=0, divr2=0, cp=1L, 83 | forward1=spot1*exp(-divr1*texp)/df, 84 | forward2=spot2*exp(-divr2*texp)/df, 85 | df=exp(-intr*texp) 86 | ){ 87 | k.plus = pmax(strike, 0) 88 | k.minus = pmin(strike, 0) 89 | 90 | sigma1.shift <- sigma1*forward1/(forward1 - k.minus) 91 | sigma2.shift <- sigma2*forward2/(forward2 + k.plus) 92 | 93 | val <- df * SwitchMargrabe( 94 | forward1=forward1-k.minus, forward2=forward2+k.plus, texp=texp, 95 | sigma1=sigma1.shift, sigma2=sigma2.shift, corr=corr, cp=cp) 96 | return(val) 97 | } 98 | 99 | #' Spread option pricing method by Bjerksund & Stensland (2014) 100 | #' 101 | #' @description The payout of the spread option is 102 | #' \code{max(S1_T - S2_T - K, 0)} where \code{S1_T} and \code{S2_T} are the 103 | #' prices at expiry \code{T} of assets 1 and 2 respectively and \code{K} is 104 | #' the strike price. 105 | #' 106 | #' @param strike (vector of) strike price 107 | #' @param spot1 (vector of) spot price of asset 1 108 | #' @param spot2 (vector of) spot price of asset 2 109 | #' @param texp (vector of) time to expiry 110 | #' @param sigma1 (vector of) volatility of asset 1 111 | #' @param sigma2 (vector of) volatility of asset 2 112 | #' @param corr correlation 113 | #' @param intr interest rate 114 | #' @param divr1 dividend rate of asset 1 115 | #' @param divr2 dividend rate of asset 2 116 | #' @param cp call/put sign. \code{1} for call, \code{-1} for put. 117 | #' @param forward1 forward price of asset 1. If given, overrides \code{spot1} 118 | #' @param forward2 forward price of asset 2. If given, overrides \code{spot2} 119 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 120 | #' @return option price 121 | #' 122 | #' @export 123 | #' 124 | #' @references Bjerksund, P., & Stensland, G. (2014). Closed form spread option 125 | #' valuation. Quantitative Finance, 14(10), 1785–1794. 126 | #' \doi{10.1080/14697688.2011.617775} 127 | #' 128 | #' @examples 129 | #' 130 | #' FER::SpreadBjerksund2014((-2:2)*10, 100, 120, 1.3, 0.2, 0.3, -0.5) 131 | #' 132 | SpreadBjerksund2014 <- function( 133 | strike=0, spot1, spot2, texp=1, sigma1, sigma2, corr, 134 | intr=0, divr1=0, divr2=0, cp=1L, 135 | forward1=spot1*exp(-divr1*texp)/df, 136 | forward2=spot2*exp(-divr2*texp)/df, 137 | df=exp(-intr*texp) 138 | ){ 139 | std11 <- sigma1^2 * texp 140 | std12 <- sigma1*sigma2 * texp 141 | std22 <- sigma2^2 * texp 142 | 143 | a <- forward2 + strike 144 | b <- forward2/a 145 | std <- sqrt(std11 - 2*b*corr*std12 + b^2*std22) 146 | 147 | d3 <- log(forward1/a) 148 | d1 <- ( d3 + 0.5*std11 - b*(corr*std12 - 0.5*b*std22) ) / std 149 | d2 <- ( d3 - 0.5*std11 + corr*std12 + b*(0.5*b - 1)*std22 ) / std 150 | d3 <- ( d3 - 0.5*std11 + 0.5*b^2*std22 ) / std 151 | val <- cp*(forward1*stats::pnorm(cp*d1) - forward2*stats::pnorm(cp*d2) 152 | - strike*stats::pnorm(cp*d3)) 153 | return(df*val) 154 | } 155 | 156 | #' Spread option under the Bachelier model 157 | #' 158 | #' @description The payout of the spread option is 159 | #' \code{max(S1_T - S2_T - K, 0)} where \code{S1_T} and \code{S2_T} are the 160 | #' prices at expiry \code{T} of assets 1 and 2 respectively and \code{K} is 161 | #' the strike price. 162 | #' 163 | #' @param strike (vector of) strike price 164 | #' @param spot1 (vector of) spot price of asset 1 165 | #' @param spot2 (vector of) spot price of asset 2 166 | #' @param texp (vector of) time to expiry 167 | #' @param sigma1 (vector of) Bachelier volatility of asset 1 168 | #' @param sigma2 (vector of) Bachelier volatility of asset 2 169 | #' @param corr correlation 170 | #' @param intr interest rate 171 | #' @param divr1 dividend rate of asset 1 172 | #' @param divr2 dividend rate of asset 2 173 | #' @param cp call/put sign. \code{1} for call, \code{-1} for put. 174 | #' @param forward1 forward price of asset 1. If given, overrides \code{spot1} 175 | #' @param forward2 forward price of asset 2. If given, overrides \code{spot2} 176 | #' @param df discount factor. If given, \code{df} overrides \code{intr} 177 | #' @return option price 178 | #' 179 | #' @export 180 | #' 181 | #' @examples 182 | #' 183 | #' FER::SpreadBachelier((-2:2)*10, 100, 120, 1.3, 20, 36, -0.5) 184 | #' 185 | SpreadBachelier <- function( 186 | strike=0, spot1, spot2, texp=1, sigma1, sigma2, corr, 187 | intr=0, divr1=0, divr2=0, cp=1L, 188 | forward1=spot1*exp(-divr1*texp)/df, 189 | forward2=spot2*exp(-divr2*texp)/df, 190 | df=exp(-intr*texp) 191 | ){ 192 | forward = forward1 - forward2 193 | sigma = sqrt(sigma1^2 - 2*corr*sigma1*sigma2 + sigma2^2) 194 | 195 | val <- BachelierPrice(strike, , texp, sigma, cp=cp, forward=forward, df=df) 196 | return(val) 197 | } 198 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------