vanillaOptionEuropean: Pricing Plain-Vanilla (European and American) and Barrier Options (European)

Description

Functions to calculate the theoretical prices and (some) Greeks for plain-vanilla and barrier options.

Usage

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)
barrierOptionEuropean(S, X, H, tau, r, q = 0, v, tauD = 0, D = 0,
                      type = "call",
                      barrier.type = "downin",
                      rebate = 0,
                      greeks = FALSE,
                      model = NULL, ...)

Value

Returns the price (a numeric vector of length one) if greeks is

FALSE, else returns a list.

Arguments

S: spot
X: strike
H: barrier
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 tauD greater than zero and not greater than tau are kept.
D: vector of dividends (in currency units); default is no dividends.
type: call or put; default is call.
barrier.type: string: combination of up/down and in/out, such as downin
rebate: currently not implemented
greeks: compute Greeks? Defaults to TRUE. But see Details for American options.
model: what model to use to value the option. Default is NULL which is equivalent to bsm.
...: parameters passed to pricing model
M: number of time steps in the tree
exercise: european (default) or american
price: numeric; the observed price to be recovered through choice of volatility.
uniroot.control: A list. If there are elements named interval, tol or maxiter, these are passed to uniroot. Any other elements of the list are ignored.
uniroot.info: logical; default is FALSE. If TRUE, the function will return the information returned by uniroot. See paragraph Value below.

Author

Enrico Schumann

Details

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.

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")

Haug, E. (2007) The Complete Guide to Option Pricing Formulas. 2nd edition. McGraw-Hill.

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). https://enricoschumann.net/NMOF.htm#NMOFmanual

Examples

Run this code

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)

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