callCF: Price a Plain-Vanilla Call with the Characteristic Function

Description

Price a European plain-vanilla call with the characteric function.

Usage

callCF(cf, S, X, tau, r, q = 0, ...,
       implVol = FALSE, uniroot.control = list(), uniroot.info = FALSE)
cfBSM(om, S, tau, r, q, v)
cfMerton(om, S, tau, r, q, v, lambda, muJ, vJ)
cfBates(om, S, tau, r, q, v0, vT, rho, k, sigma, lambda, muJ, vJ)
cfHeston(om, S, tau, r, q, v0, vT, rho, k, sigma)
cfVG(om, S, tau, r, q, nu, theta, sigma)

Value

Returns the value of the call (numeric) under the respective model or, if implVol is TRUE, a list of the value and the implied volatility. (If, in addition, uniroot.info is TRUE, the information provided by uniroot is also returned.)

Arguments

cf: characteristic function
S: spot
X: strike
tau: time to maturity
r: the interest rate
q: the dividend rate
...: arguments passed to the characteristic function
implVol: logical: compute implied vol?
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.
om: a (usually complex) argument
v0: a numeric vector of length one
vT: a numeric vector of length one
v: a numeric vector of length one
rho: a numeric vector of length one
k: a numeric vector of length one
sigma: a numeric vector of length one
lambda: a numeric vector of length one
muJ: a numeric vector of length one
vJ: a numeric vector of length one
nu: a numeric vector of length one
theta: a numeric vector of length one

Author

Enrico Schumann

Details

The function computes the value of a plain vanilla European call under different models, using the representation of Bakshi/Madan. Put values can be computed through put--call parity (see putCallParity).

If implVol is TRUE, the function will compute the implied volatility necessary to obtain the same value under Black--Scholes--Merton. 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.

The function uses variances as inputs (not volatilities).

The function is not vectorised (but see the NMOF Manual for examples of how to efficiently price more than one option at once).

References

Bates, David S. (1996) Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. Review of Financial Studies 9 (1), 69--107.

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")

Heston, S.L. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bonds and Currency options. Review of Financial Studies 6 (2), 327--343.

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 <- 0.08
v0 <- 0.2^2  ## variance, not volatility
vT <- 0.2^2  ## variance, not volatility
v <- vT
rho <- -0.3; k <- .2
sigma <- 0.3

## jump parameters (Merton and Bates)
lambda <- 0.1
muJ <- -0.2
vJ <- 0.1^2

## get Heston price and BSM implied volatility
callHestoncf(S, X, tau, r, q, v0, vT, rho, k, sigma, implVol = FALSE)
callCF(cf = cfHeston, S=S, X=X, tau=tau, r=r, q = q,
       v0 = v0, vT = vT, rho = rho, k = k, sigma = sigma, implVol = FALSE)

## Black-Scholes-Merton
callCF(cf = cfBSM, S=S, X=X, tau = tau, r = r, q = q,
       v = v, implVol = TRUE)

## Bates
callCF(cf = cfBates, S = S, X = X, tau = tau, r = r, q = q,
       v0 = v0, vT = vT, rho = rho, k = k, sigma = sigma,
       lambda = lambda, muJ = muJ, vJ = vJ, implVol = FALSE)

## Merton
callCF(cf = cfMerton, S = S, X = X, tau = tau, r = r, q = q,
       v = v, lambda = lambda, muJ = muJ, vJ = vJ, implVol = FALSE)

## variance gamma
nu <- 0.1; theta <- -0.1; sigma <- 0.15
callCF(cf = cfVG, S = S, X = X, tau = tau, r = r, q = q,
       nu = nu, theta = theta, sigma = sigma, implVol = FALSE)

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