callMerton {NMOF}

Price of a European Call under Merton's Jump–Diffusion Model

Description

Computes the price of a European Call under Merton's jump–diffusion model (and the equivalent Black–Scholes–Merton volatility)

Usage

callMerton(S, X, tau, r, q, v, lambda, muJ, vJ, N, implVol = FALSE)

Arguments

S

current stock price

X

strike price

tau

time to maturity

r

risk-free rate

q

dividend rate

v

variance

lambda

jump intensity

muJ

mean jump-size

vJ

variance of log jump-size

N

The number of jumps. See Details.

implVol

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

Details

The function computes the value of a plain-vanilla European call under Merton's jump–diffusion model. Put values can be computed through put–call-parity (see putCallParity). If implVol is TRUE, the function also computes the implied volatility necessary to obtain the same price under Black–Scholes–Merton. The implied volatility is computed with uniroot from the stats package.

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

The number of jumps N typically can be set 10 or 20. (Just try to increase N and see how the results change.)

Value

Returns the value of the call (numeric) or, if implVol is TRUE, a list of the value and the implied volatility.

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

Merton, R.C. (1976) Option Pricing when Underlying Stock Returns are Discontinuous. Journal of Financial Economics 3(1–2), 125–144.

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 <- 0.0075; q <- 0.00
v <- 0.2^2
lambda <- 1; muJ <- -0.2; vJ <- 0.6^2
N <- 20

## jumps can make a difference
callMerton(S, X, tau, r, q, v, lambda, muJ, vJ, N, implVol = TRUE)
callCF(cf = cfMerton, S = S, X = X, tau = tau, r = r, q = q,
       v = v, lambda = lambda, muJ = muJ, vJ = vJ, implVol = TRUE)
vanillaOptionEuropean(S,X,tau,r,q,v, greeks = FALSE)

lambda <- 0 ## no jumps
callMerton(S, X, tau, r, q, v, lambda, muJ, vJ, N, implVol = FALSE)
vanillaOptionEuropean(S,X,tau,r,q,v, greeks = FALSE)

lambda <- 1; muJ <- 0; vJ <- 0.0^2  ## no jumps, either
callMerton(S, X, tau, r, q, v, lambda, muJ, vJ, N, implVol = FALSE)
vanillaOptionEuropean(S,X,tau,r,q,v, greeks = FALSE)