Pricing Plain-Vanilla Options (European and American)
Functions to calculate the theoretical prices and (some) Greeks for plain vanilla options.
vanillaOptionEuropean(S, X, tau, r, q, v, tauD = 0, D = 0,
type = "call", greeks = TRUE,
model = NULL, ...)
vanillaOptionAmerican(S, X, tau, r, q, v, tauD = 0, D = 0,
type = "call", greeks = TRUE, M = 101)
vanillaOptionImpliedVol(exercise = "european", price, S, X, tau, r,
q = 0, tauD = 0, D = 0, type = "call",
M = 101,
uniroot.control = list(), uniroot.info = FALSE)S |
spot |
X |
strike |
tau |
time-to-maturity in years |
r |
risk-free rate |
q |
continuous dividend yield, see Details. |
v |
variance (volatility squared) |
tauD |
vector of times-to-dividends in years. Only dividends with
|
D |
vector of dividends (in currency units); default is no dividends. |
type |
call or put; default is call. |
greeks |
compute Greeks? Defaults to |
model |
what model to use to value the option. Default is |
... |
parameters passed to pricing model |
M |
number of time steps in the tree |
exercise |
|
price |
numeric; the observed price to be recovered through choice of volatility. |
uniroot.control |
A list. If there are elements named
|
uniroot.info |
logical; default is |
For European options the formula of Messrs Black, Scholes and Merton is
used. It can be used for equities (set q equal to the dividend
yield), futures (Black, 1976; set q equal to r),
currencies (Garman and Kohlhagen, 1983; set q equal to the
foreign risk-free rate). For future-style options (e.g. options on the German Bund
future), set q and r equal to zero.
The Greeks are provided in their raw (‘textbook’) form with only one exception: Theta is made negative. For practical use, the other Greeks are also typically adjusted: Theta is often divided by 365 (or some other yearly day count); Vega and Rho are divided by 100 to give the sensitivity for one percentage-point move in volatility/the interest rate. Raw Gamma is not much use if not adjusted for the actual move in the underlier.
For European options the Greeks are computed through the respective analytic expressions. For American options only Delta, Gamma and Theta are computed because they can be directly obtained from the binomial tree; other Greeks need to be computed through a finite difference (see Examples).
For the European-type options, the function understands vectors of inputs, except for dividends. American options are priced via a Cox-Ross-Rubinstein tree; no vectorisation is implemented here.
The implied volatility is computed with uniroot from the
stats package (the default search interval is c(0.00001,
2); it can be changed through uniroot.control).
Dividends (D) are modelled via the escrowed-dividend model.
Returns the price (a numeric vector of length one) if greeks is
FALSE, else returns a list.
If greeks is TRUE, the function will return a list with
named elements (value, delta and so on). Prior to version
0.26-3, the first element of this list was named price.
Enrico Schumann
Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. https://www.elsevier.com/books/numerical-methods-and-optimization-in-finance/gilli/978-0-12-815065-8
Schumann, E. (2019) Financial Optimisation with R (NMOF Manual). http://enricoschumann.net/NMOF.htm#NMOFmanual
S <- 100; X <- 100; tau <- 1; r <- 0.02; q <- 0.06; vol <- 0.3
unlist(vanillaOptionEuropean(S, X, tau, r, q, vol^2, type = "put"))
S <- 100; X <- 110; tau <- 1; r <- 0.1; q <- 0.06; vol <- 0.3; type <- "put"
unlist(vanillaOptionAmerican(S, X, tau, r, q, vol^2, type = type,
greeks = TRUE))
## compute rho for 1% move
h <- 0.01
(vanillaOptionAmerican(S, X, tau, r + h, q, vol^2,
type = type, greeks = FALSE) -
vanillaOptionAmerican(S, X, tau, r, q, vol^2,
type = type, greeks = FALSE)) / (h*100)
## compute vega for 1% move
h <- 0.01
(vanillaOptionAmerican(S, X, tau, r, q,(vol + h)^2,
type = type, greeks = FALSE) -
vanillaOptionAmerican(S, X, tau, r, q, vol^2,
type = type, greeks = FALSE)) / (h*100)
S <- 100; X <- 100
tau <- 1; r <- 0.05; q <- 0.00
D <- c(1,2); tauD <- c(0.3,.6)
type <- "put"
v <- 0.245^2 ## variance, not volatility
p <- vanillaOptionEuropean(S = S, X = X, tau, r, q, v = v,
tauD = tauD, D = D, type = type, greeks = FALSE)
vanillaOptionImpliedVol(exercise = "european", price = p,
S = S, X = X, tau = tau, r = r, q = q, tauD = tauD, D = D, type = type)
p <- vanillaOptionAmerican(S = S, X = X, tau, r, q, v = v,
tauD = tauD, D = D, type = type, greeks = FALSE)
vanillaOptionImpliedVol(exercise = "american", price = p,
S = S, X = X, tau = tau, r = r, q = q, tauD = tauD, D = D, type =
type, uniroot.control = list(interval = c(0.01, 0.5)))
## compute implied q
S <- 100; X <- 100
tau <- 1; r <- 0.05; q <- 0.072
v <- 0.22^2 ## variance, not volatility
call <- vanillaOptionEuropean(S=S, X = X, tau=tau, r=r, q=q, v=v,
type = "call", greeks = FALSE)
put <- vanillaOptionEuropean(S=S, X = X, tau=tau, r=r, q=q, v=v,
type = "put", greeks = FALSE)
# ... the simple way
-(log(call + X * exp(-tau*r) - put) - log(S)) / tau
# ... the complicated way :-)
volDiffCreate <- function(exercise, call, put, S, X, tau, r) {
f <- function(q) {
cc <- vanillaOptionImpliedVol(exercise = exercise, price = call,
S = S, X = X, tau = tau, r = r, q = q, type = "call")
pp <- vanillaOptionImpliedVol(exercise = exercise, price = put,
S = S, X = X, tau = tau, r = r, q = q, type = "put")
abs(cc - pp)
}
f
}
f <- volDiffCreate(exercise = "european",
call = call, put = put, S = S, X = X, tau = tau, r)
optimise(f,interval = c(0, 0.2))$minimum
##
S <- 100; X <- 100
tau <- 1; r <- 0.05; q <- 0.072
v <- 0.22^2 ## variance, not volatility
vol <- 0.22
vanillaOptionEuropean(S=S, X = X, tau=tau, r=r, q=q, v=v, ## with variance
type = "call", greeks = FALSE)
vanillaOptionEuropean(S=S, X = X, tau=tau, r=r, q=q, vol=vol, ## with vol
type = "call", greeks = FALSE)
vanillaOptionEuropean(S=S, X = X, tau=tau, r=r, q=q, vol=vol, ## with vol
type = "call", greeks = FALSE, v = 0.2^2)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.