In grfiv/ustreasuries: Treasury Data Download, Fixed-Income Models and Derivative Analyses
knitr::opts_chunk$set(collapse=TRUE, # hadley
comment = "#>", # hadley
error=TRUE, purl=FALSE, # to be able to see errors
fig.width=7.25, fig.height=6) # nice-sized pictures
library(ustreasuries)
-
EuroCall Calculate the price of a European call option with or without dividends
-
EuroPut Calculate the price of a European put option with or without dividends
# Hull 7th edition Ch 13 P 294
Stock <- 42 # S_0
Exercise <- 40 # K
Time <- 0.50 # T
Interest <- 0.10 # r
Yield <- 0 # q
sigma <- 0.20
ans_c <- EuroCall(Stock, Exercise, Time, Interest, Yield, sigma)
ans_p <- EuroPut(Stock, Exercise, Time, Interest, Yield, sigma)
writeLines(paste0("c = ", round(ans_c,2), ", p = ", round(ans_p,2)))
-
EuroCallVol / EuroPutVol Implied Volatility for European Options
# Hull, 7th edition ch 13 p296
Stock <- 21 # S_0
Exercise <- 20 # K
Time <- 0.25 # T
Interest <- 0.10 # r
Yield <- 0 # q
Call_price <- 1.875
Put_price <- 1.875
callvol <- EuroCallVol(Stock, Exercise, Time, Interest, Yield, Call_price)
putvol <- EuroPutVol(Stock, Exercise, Time, Interest, Yield, Put_price)
writeLines(paste0("Implied Call Volatility: ", round(callvol*100, 1), "% per annum\n",
"Implied Put Volatility: ", round(putvol*100, 1), "% per annum"))
Greeks
Relationship between Delta, Theta and Gamma
$\Theta + rS\Delta + \frac{1}{2} \sigma^2 S^2 \Gamma = r \Pi$
-
DeltaCall Change in call-option price given change in asset price
# Hull, 7th edition Ch 17 p 363
library(ustreasuries)
Stock <- 49
Exercise <- 50
Time <- 20/52
Interest <- 0.05
Yield <- 0
sigma <- 0.20
ans <- DeltaCall(Stock, Exercise, Time, Interest, Yield, sigma)
writeLines(paste0("Delta call: when the asset price changes by Delta_S,\n",
" the option price changes by Delta_S*",round(ans, 3)))
-
DeltaPut Change in put-option price given change in asset price
# Hull, 7th edition Ch 17 p 362,3
Stock <- 49 # S_0
Exercise <- 50 # K
Time <- 20/52 # T
Interest <- 0.05 # r
Yield <- 0 # q
sigma <- 0.20
dcall <- DeltaCall(Stock, Exercise, Time, Interest, Yield, sigma)
dput <- DeltaPut(Stock, Exercise, Time, Interest, Yield, sigma)
writeLines(paste0("Delta put: when the asset price changes by Delta_S,\n",
" the option price changes by Delta_S*",round(dput, 3),
"\nDelta put = Delta call - 1? ", dput == dcall-1))
x_axis <- c()
y_axis_call <- c()
y_axis_put <- c()
for (Stock in seq(0, 100)) {
x_axis <- c(x_axis, Stock)
y_axis_call <- c(y_axis_call, DeltaCall(Stock, Exercise, Time, Interest, Yield, sigma))
y_axis_put <- c(y_axis_put, DeltaPut(Stock, Exercise, Time, Interest, Yield, sigma))
}
plot(x_axis, y_axis_call, ylim=c(-1.5,1.5), type="l",col="red",
xlab="Stock Price", ylab="Delta",
main="Variation of Delta with Asset Price",
sub="K = 50, r = 5%, q = 0, sigma = 20%")
lines(x_axis, y_axis_put,col="blue")
abline(h=0)
abline(h=-1, lty=2)
abline(h=1, lty=2)
legend("topright",c("DeltaCall","DeltaPut"),lty=c(1,1),col=c("red","blue"))
abline(v=Exercise)
text(x=Exercise, y=-1.25, labels="K", cex=1.25)
#dev.copy(png,filename="../Wiki_images/deltaplot.png");
#dev.off ();
-
ThetaCall the decay in the value of a call or a portfolio of calls as time passes
-
ThetaPut the decay in the value of a put or a portfolio of puts as time passes
# Hull, 7th edition Ch 17 p 367
Stock <- 49 # S_0
Exercise <- 50 # K
Time <- 20/52 # T
Interest <- 0.05 # r
Yield <- 0 # q
sigma <- 0.20
thcall <- ThetaCall(Stock, Exercise, Time, Interest, Yield, sigma)
thput <- ThetaPut(Stock, Exercise, Time, Interest, Yield, sigma)
rKe <- Interest * Exercise * exp(-Interest*Time)
#--------------------------------------------------------------------
writeLines(paste0("ThetaCall: ", round(thcall, 2), "\n",
"ThetaPut: ", round(thput, 2), "\n",
"per calendar day: ", round(thcall/365, 4), " (call)", "\n",
"per trading day: ", round(thcall/252, 4), " (call)", "\n\n",
"ThetaPut is always greater than ThetaCall by an amount rKe:", "\n",
"Diff: ",thput-thcall,"\n",
"rKe: ",rKe))
x_axis <- c()
y_axis_call <- c()
y_axis_put <- c()
for (Stock in seq(0, 100)) {
x_axis <- c(x_axis, Stock)
y_axis_call <- c(y_axis_call, ThetaCall(Stock, Exercise, Time, Interest, Yield, sigma))
y_axis_put <- c(y_axis_put, ThetaPut(Stock, Exercise, Time, Interest, Yield, sigma))
}
plot(x_axis, y_axis_call, type="l", col="black",
xlab="Stock Price", ylab="Theta",
ylim=c(floor(min(y_axis_call)), ceiling(max(y_axis_put))),
main="Variation in Theta as Stock Price Changes",
sub=expression(paste0("K = 50, r = 5%, q = 0 ", sigma, " = 20%")))
abline(v=Exercise)
text(x=Exercise, y=0, labels="K", cex=1.5)
abline(h=0, lty=2)
abline(h=-Interest*Exercise*exp(-Interest*Time), lty=2)
text(x=5, y=-Interest*Exercise*exp(-Interest*Time), pos=3, labels=expression(-rKe^{-rT}))
lines(x=x_axis, y=y_axis_put, lty=5, col="red")
legend("topright",c("ThetaPut","ThetaCall"),lty=c(5,1),col=c("red","black"))
#dev.copy(png,filename="../Wiki_images/thetacall.png");
#dev.off ();
-
Gamma the change in Delta with respect to asset price
# Hull, 7th edition Ch 17 p 371,2
Stock <- 49 # S_0
Exercise <- 50 # K
Time <- 20/52 # T
Interest <- 0.05 # r
Yield <- 0 # q
sigma <- 0.20
gamma <- Gamma(Stock, Exercise, Time, Interest, Yield, sigma)
round(gamma, 3) # 0.066
x_axis <- c()
y_axis <- c()
for (Stock in seq(0, 100)) {
x_axis <- c(x_axis, Stock)
y_axis <- c(y_axis, Gamma(Stock, Exercise, Time, Interest, Yield, sigma))
}
plot(x_axis, y_axis, type="l", col="black",
xlab="Stock Price", ylab="Gamma",
main="Variation in Gamma as Stock Price Changes",
sub="K = 50, r = 5%, q = 0, sigma = 20%")
abline(v=Exercise)
text(x=Exercise, y=0.01, labels="K", cex=1.5)
#dev.copy(png,filename="../Wiki_images/gamma.png");
#dev.off ();
-
Vega the sensitivity to changes in the volatility of the underlying
# Hull, 7th edition Ch 17 p 375
Stock <- 49 # S_0
Exercise <- 50 # K
Time <- 20/52 # T
Interest <- 0.05 # r
Yield <- 0 # q
sigma <- 0.20
vega <- Vega(Stock, Exercise, Time, Interest, Yield, sigma)
writeLines(paste0("The value of Vega is ", round(vega,1), "\n",
"Therefore, a 1% change in the volatility from 20% to 21%", "\n",
"will raise the price of the option by this amount:", "\n",
"1% x ", round(vega,1), " = ", round((0.01 * vega), 3),
", from ", Stock, " to ", Stock+round((0.01 * vega), 3)))
x_axis <- c()
y_axis <- c()
for (Stock in seq(0, 100,0.01)) {
x_axis <- c(x_axis, Stock)
y_axis <- c(y_axis, Vega(Stock, Exercise, Time, Interest, Yield, sigma))
}
plot(x_axis, y_axis, type="l", col="black",
xlab="Stock Price", ylab="Vega",
main="Variation in Vega as Stock Price Changes",
sub="K = 50, r = 5%, q = 0, sigma = 20%")
abline(v=Exercise)
text(x=Exercise, y=0.01, labels="K", cex=1.5)
#dev.copy(png,filename="../Wiki_images/vegaprice.png");
#dev.off ();
Stock <- 49 # S_0
x_axis <- c()
y_axis <- c()
for (sigma in seq(0.05, 0.75, 0.01)) {
x_axis <- c(x_axis, sigma)
y_axis <- c(y_axis, Vega(Stock, Exercise, Time, Interest, Yield, sigma))
}
plot(x_axis, y_axis, type="l", col="black",
xlab="sigma", ylab="Vega",
main="Variation in Vega as Volatility Changes",
sub="K = 50, r = 5%, q = 0, S = 49")
abline(h=12.10548, lty=2)
abline(v=0.20, lty=2)
#dev.copy(png,filename="../Wiki_images/vegasigma.png")
#dev.off ()
-
RhoCall the sensitivity to changes in risk-free rate of interest
-
RhoPut the sensitivity to changes in the risk-free rate of return
# Hull, 7th edition Ch 17 p 376
Stock <- 49 # S_0
Exercise <- 50 # K
Time <- 20/52 # T
Interest <- 0.05 # r
Yield <- 0 # q
sigma <- 0.20
rhocall <- RhoCall(Stock, Exercise, Time, Interest, Yield, sigma)
rhoput <- RhoPut(Stock, Exercise, Time, Interest, Yield, sigma)
writeLines(paste0("RhoCall: ", round(rhocall, 2), "\n",
"RhoPut: ", round(rhoput, 2)))
x_axis <- c()
y_axis_call <- c()
y_axis_put <- c()
for (Stock in seq(0, 100, 0.01)) {
x_axis <- c(x_axis, Stock)
y_axis_call <- c(y_axis_call, RhoCall(Stock, Exercise, Time, Interest, Yield, sigma))
y_axis_put <- c(y_axis_put, RhoPut(Stock, Exercise, Time, Interest, Yield, sigma))
}
plot(x_axis, y_axis_call, type="l", col="black",
xlab="Stock Price", ylab="Rho",
ylim=c(floor(min(y_axis_put)),ceiling(max(y_axis_call))),
main="Variation in Rho as Stock Price Changes",
sub="K = 50, r = 5%, q = 0, sigma = 20%")
lines(x=x_axis, y=y_axis_put, lty=5, col="red")
abline(v=Exercise)
text(x=Exercise, y=0.01, labels="K", cex=1.5)
abline(h=0, lty=2)
legend("topleft", c("RhoCall","RhoPut"), lty=c(1,5), col=c("black","red"))
#dev.copy(png,filename="../Wiki_images/rhoprice.png");
#dev.off ()
grfiv/ustreasuries documentation built on May 17, 2019, 8:36 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set(collapse=TRUE, # hadley comment = "#>", # hadley error=TRUE, purl=FALSE, # to be able to see errors fig.width=7.25, fig.height=6) # nice-sized pictures
library(ustreasuries)
-
EuroCall Calculate the price of a European call option with or without dividends
-
EuroPut Calculate the price of a European put option with or without dividends
# Hull 7th edition Ch 13 P 294 Stock <- 42 # S_0 Exercise <- 40 # K Time <- 0.50 # T Interest <- 0.10 # r Yield <- 0 # q sigma <- 0.20 ans_c <- EuroCall(Stock, Exercise, Time, Interest, Yield, sigma) ans_p <- EuroPut(Stock, Exercise, Time, Interest, Yield, sigma) writeLines(paste0("c = ", round(ans_c,2), ", p = ", round(ans_p,2)))
-
EuroCallVol / EuroPutVol Implied Volatility for European Options
# Hull, 7th edition ch 13 p296 Stock <- 21 # S_0 Exercise <- 20 # K Time <- 0.25 # T Interest <- 0.10 # r Yield <- 0 # q Call_price <- 1.875 Put_price <- 1.875 callvol <- EuroCallVol(Stock, Exercise, Time, Interest, Yield, Call_price) putvol <- EuroPutVol(Stock, Exercise, Time, Interest, Yield, Put_price) writeLines(paste0("Implied Call Volatility: ", round(callvol*100, 1), "% per annum\n", "Implied Put Volatility: ", round(putvol*100, 1), "% per annum"))
Greeks
Relationship between Delta, Theta and Gamma
$\Theta + rS\Delta + \frac{1}{2} \sigma^2 S^2 \Gamma = r \Pi$
-
DeltaCall Change in call-option price given change in asset price
# Hull, 7th edition Ch 17 p 363 library(ustreasuries) Stock <- 49 Exercise <- 50 Time <- 20/52 Interest <- 0.05 Yield <- 0 sigma <- 0.20 ans <- DeltaCall(Stock, Exercise, Time, Interest, Yield, sigma) writeLines(paste0("Delta call: when the asset price changes by Delta_S,\n", " the option price changes by Delta_S*",round(ans, 3)))
-
DeltaPut Change in put-option price given change in asset price
# Hull, 7th edition Ch 17 p 362,3 Stock <- 49 # S_0 Exercise <- 50 # K Time <- 20/52 # T Interest <- 0.05 # r Yield <- 0 # q sigma <- 0.20 dcall <- DeltaCall(Stock, Exercise, Time, Interest, Yield, sigma) dput <- DeltaPut(Stock, Exercise, Time, Interest, Yield, sigma) writeLines(paste0("Delta put: when the asset price changes by Delta_S,\n", " the option price changes by Delta_S*",round(dput, 3), "\nDelta put = Delta call - 1? ", dput == dcall-1))
x_axis <- c() y_axis_call <- c() y_axis_put <- c() for (Stock in seq(0, 100)) { x_axis <- c(x_axis, Stock) y_axis_call <- c(y_axis_call, DeltaCall(Stock, Exercise, Time, Interest, Yield, sigma)) y_axis_put <- c(y_axis_put, DeltaPut(Stock, Exercise, Time, Interest, Yield, sigma)) } plot(x_axis, y_axis_call, ylim=c(-1.5,1.5), type="l",col="red", xlab="Stock Price", ylab="Delta", main="Variation of Delta with Asset Price", sub="K = 50, r = 5%, q = 0, sigma = 20%") lines(x_axis, y_axis_put,col="blue") abline(h=0) abline(h=-1, lty=2) abline(h=1, lty=2) legend("topright",c("DeltaCall","DeltaPut"),lty=c(1,1),col=c("red","blue")) abline(v=Exercise) text(x=Exercise, y=-1.25, labels="K", cex=1.25) #dev.copy(png,filename="../Wiki_images/deltaplot.png"); #dev.off ();
-
ThetaCall the decay in the value of a call or a portfolio of calls as time passes
-
ThetaPut the decay in the value of a put or a portfolio of puts as time passes
# Hull, 7th edition Ch 17 p 367 Stock <- 49 # S_0 Exercise <- 50 # K Time <- 20/52 # T Interest <- 0.05 # r Yield <- 0 # q sigma <- 0.20 thcall <- ThetaCall(Stock, Exercise, Time, Interest, Yield, sigma) thput <- ThetaPut(Stock, Exercise, Time, Interest, Yield, sigma) rKe <- Interest * Exercise * exp(-Interest*Time) #-------------------------------------------------------------------- writeLines(paste0("ThetaCall: ", round(thcall, 2), "\n", "ThetaPut: ", round(thput, 2), "\n", "per calendar day: ", round(thcall/365, 4), " (call)", "\n", "per trading day: ", round(thcall/252, 4), " (call)", "\n\n", "ThetaPut is always greater than ThetaCall by an amount rKe:", "\n", "Diff: ",thput-thcall,"\n", "rKe: ",rKe))
x_axis <- c() y_axis_call <- c() y_axis_put <- c() for (Stock in seq(0, 100)) { x_axis <- c(x_axis, Stock) y_axis_call <- c(y_axis_call, ThetaCall(Stock, Exercise, Time, Interest, Yield, sigma)) y_axis_put <- c(y_axis_put, ThetaPut(Stock, Exercise, Time, Interest, Yield, sigma)) } plot(x_axis, y_axis_call, type="l", col="black", xlab="Stock Price", ylab="Theta", ylim=c(floor(min(y_axis_call)), ceiling(max(y_axis_put))), main="Variation in Theta as Stock Price Changes", sub=expression(paste0("K = 50, r = 5%, q = 0 ", sigma, " = 20%"))) abline(v=Exercise) text(x=Exercise, y=0, labels="K", cex=1.5) abline(h=0, lty=2) abline(h=-Interest*Exercise*exp(-Interest*Time), lty=2) text(x=5, y=-Interest*Exercise*exp(-Interest*Time), pos=3, labels=expression(-rKe^{-rT})) lines(x=x_axis, y=y_axis_put, lty=5, col="red") legend("topright",c("ThetaPut","ThetaCall"),lty=c(5,1),col=c("red","black")) #dev.copy(png,filename="../Wiki_images/thetacall.png"); #dev.off ();
-
Gamma the change in Delta with respect to asset price
# Hull, 7th edition Ch 17 p 371,2 Stock <- 49 # S_0 Exercise <- 50 # K Time <- 20/52 # T Interest <- 0.05 # r Yield <- 0 # q sigma <- 0.20 gamma <- Gamma(Stock, Exercise, Time, Interest, Yield, sigma) round(gamma, 3) # 0.066
x_axis <- c() y_axis <- c() for (Stock in seq(0, 100)) { x_axis <- c(x_axis, Stock) y_axis <- c(y_axis, Gamma(Stock, Exercise, Time, Interest, Yield, sigma)) } plot(x_axis, y_axis, type="l", col="black", xlab="Stock Price", ylab="Gamma", main="Variation in Gamma as Stock Price Changes", sub="K = 50, r = 5%, q = 0, sigma = 20%") abline(v=Exercise) text(x=Exercise, y=0.01, labels="K", cex=1.5) #dev.copy(png,filename="../Wiki_images/gamma.png"); #dev.off ();
-
Vega the sensitivity to changes in the volatility of the underlying
# Hull, 7th edition Ch 17 p 375 Stock <- 49 # S_0 Exercise <- 50 # K Time <- 20/52 # T Interest <- 0.05 # r Yield <- 0 # q sigma <- 0.20 vega <- Vega(Stock, Exercise, Time, Interest, Yield, sigma) writeLines(paste0("The value of Vega is ", round(vega,1), "\n", "Therefore, a 1% change in the volatility from 20% to 21%", "\n", "will raise the price of the option by this amount:", "\n", "1% x ", round(vega,1), " = ", round((0.01 * vega), 3), ", from ", Stock, " to ", Stock+round((0.01 * vega), 3)))
x_axis <- c() y_axis <- c() for (Stock in seq(0, 100,0.01)) { x_axis <- c(x_axis, Stock) y_axis <- c(y_axis, Vega(Stock, Exercise, Time, Interest, Yield, sigma)) } plot(x_axis, y_axis, type="l", col="black", xlab="Stock Price", ylab="Vega", main="Variation in Vega as Stock Price Changes", sub="K = 50, r = 5%, q = 0, sigma = 20%") abline(v=Exercise) text(x=Exercise, y=0.01, labels="K", cex=1.5) #dev.copy(png,filename="../Wiki_images/vegaprice.png"); #dev.off ();
Stock <- 49 # S_0 x_axis <- c() y_axis <- c() for (sigma in seq(0.05, 0.75, 0.01)) { x_axis <- c(x_axis, sigma) y_axis <- c(y_axis, Vega(Stock, Exercise, Time, Interest, Yield, sigma)) } plot(x_axis, y_axis, type="l", col="black", xlab="sigma", ylab="Vega", main="Variation in Vega as Volatility Changes", sub="K = 50, r = 5%, q = 0, S = 49") abline(h=12.10548, lty=2) abline(v=0.20, lty=2) #dev.copy(png,filename="../Wiki_images/vegasigma.png") #dev.off ()
-
RhoCall the sensitivity to changes in risk-free rate of interest
-
RhoPut the sensitivity to changes in the risk-free rate of return
# Hull, 7th edition Ch 17 p 376 Stock <- 49 # S_0 Exercise <- 50 # K Time <- 20/52 # T Interest <- 0.05 # r Yield <- 0 # q sigma <- 0.20 rhocall <- RhoCall(Stock, Exercise, Time, Interest, Yield, sigma) rhoput <- RhoPut(Stock, Exercise, Time, Interest, Yield, sigma) writeLines(paste0("RhoCall: ", round(rhocall, 2), "\n", "RhoPut: ", round(rhoput, 2)))
x_axis <- c() y_axis_call <- c() y_axis_put <- c() for (Stock in seq(0, 100, 0.01)) { x_axis <- c(x_axis, Stock) y_axis_call <- c(y_axis_call, RhoCall(Stock, Exercise, Time, Interest, Yield, sigma)) y_axis_put <- c(y_axis_put, RhoPut(Stock, Exercise, Time, Interest, Yield, sigma)) } plot(x_axis, y_axis_call, type="l", col="black", xlab="Stock Price", ylab="Rho", ylim=c(floor(min(y_axis_put)),ceiling(max(y_axis_call))), main="Variation in Rho as Stock Price Changes", sub="K = 50, r = 5%, q = 0, sigma = 20%") lines(x=x_axis, y=y_axis_put, lty=5, col="red") abline(v=Exercise) text(x=Exercise, y=0.01, labels="K", cex=1.5) abline(h=0, lty=2) legend("topleft", c("RhoCall","RhoPut"), lty=c(1,5), col=c("black","red")) #dev.copy(png,filename="../Wiki_images/rhoprice.png"); #dev.off ()
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.