callHestoncf {NMOF}

Price of a European Call under the Heston Model

Description

Computes the price of a European Call under the Heston model (and the equivalent Black–Scholes–Merton volatility)

Usage

callHestoncf(S, X, tau, r, q, v0, vT, rho, k, sigma, implVol = FALSE,
             ...,
             uniroot.control = list(), uniroot.info = FALSE)

Arguments

S

current stock price

X

strike price

tau

time to maturity

r

risk-free rate

q

dividend rate

v0

current variance

vT

long-run variance (theta in Heston's paper)

rho

correlation between spot and variance

k

speed of mean-reversion (kappa in Heston's paper)

sigma

volatility of variance. A value smaller than 0.01 is replaced with 0.01.

implVol

compute equivalent Black–Scholes–Merton volatility? Default is FALSE.

...

named arguments, passed to integrate

uniroot.control

A list. If there are elements named interval, tol or maxiter, these are passed to uniroot. 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 section Value below.

Details

The function computes the value of a plain vanilla European call under the Heston model. Put values can be computed through put–call-parity.

If implVol is TRUE, the function will compute the implied volatility necessary to obtain the same price 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).

Note that the function takes variances as inputs (not volatilities).

Value

Returns the value of the call (numeric) under the Heston model or, if implVol is TRUE, a list of the value and the implied volatility. If uniroot.info is TRUE, then instead of only the computed volatility, the complete output of uniroot is included in the result.

Note

If implVol is TRUE, the function will return a list with elements named value and impliedVol. Prior to version 0.26-3, the first element was named callPrice.

Author(s)

Enrico Schumann

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. 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). http://enricoschumann.net/NMOF.htm#NMOFmanual

See Also

callCF, EuropeanCall

Examples

S <- 100; X <- 100; tau <- 1; r <- 0.02; q <- 0.01
v0  <- 0.2^2  ## variance, not volatility
vT  <- 0.2^2  ## variance, not volatility
rho <- -0.7; k <- 0.2; sigma <- 0.5

## get Heston price and BSM implied volatility
result <- callHestoncf(S = S, X = X, tau = tau, r = r, q = q,
                       v0 = v0, vT = vT, rho = rho, k = k,
                       sigma = sigma, implVol = TRUE)

## Heston price
result[[1L]]

## price BSM with implied volatility
vol <- result[[2L]]
d1 <- (log(S/X) + (r - q + vol^2 / 2)*tau) / (vol*sqrt(tau))
d2 <- d1 - vol*sqrt(tau)
callBSM <- S * exp(-q * tau) * pnorm(d1) -
           X * exp(-r * tau) * pnorm(d2)
callBSM  ## should be (about) the same as result[[1L]]