Calculates the Price, Delta and Gamma of an Asian Option

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Description

Prices arithmetic average Asian Call options under geometric Brownian motion. It also estimates the sensitivities Delta and Gamma.

Usage

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AsianCall(T=1,d=12,K=100,r=0.05,sigma=0.2,S0=100,method=c("best","naive"),
          sampling=c("QMC","MC"),
          metpar=list(maxiter=100,tol=1.e-14,cvmethod="splitting"),
          sampar=list(nout=50,seq.type="korobov",n=2039,a=1487,
                      baker=TRUE,genmethod="pca"))

Arguments

T

time to maturity (in years)

d

number of control points

K

strike price

r

risk free interest rate

sigma

volatility

S0

starting stockprice

method

selects the simulation method;
method "best" uses a variance reduction method based on effective control variates and conditional Monte Carlo and is very effective.
method "naive" is mainly provided for comparison purposes.

sampling

sampling QMC uses the Quasi Monte Carlo method Korobov lattice for the simulation.
sampling MC uses standard Monte Carlo for the simulation.

metpar

list holding extra parameters related to the simulation method
For method="best":
maxiter= maximal no of iterations for Newton method
tol= error tolerance for Newton method
for sampling="QMC":
cvmethod=c("splitting","direct") NOT necessary for method = "naive"
"splitting" ... estimates CV coefficients using lm with bootstrap
"direct" ... estimates CV coefficients using lm and the full sample
for sampling="MC":
np ... sample size for pilot run for CV; NOT necessary for method = "naive"

sampar

list holding several parameters related to the sampling method;
for sampling="MC" the list sampar only contains the total samplesize n;

for sampling="QMC" the list sampar contains the elements:

nout

number of independent "randomized" copies of the Korobov lattice

n

number of points of the Korobov lattice

a

important constant for the construction of the Korobov lattice

baker

TRUE/FALSE, indicates if Baker transform should be used for making the integrand periodic

genmethod

= c("pca", "std","pcamain","lt","ltpca"),
note that for method=="naive" only genmethod=c("pca","std") can be used.

genmethod="pca"

principal component analysis

genmethod="std"

standard

genmethod="pcamain"

use only first dirnum main directions of the PCA

genmethod="lt"

uses a transform for the first dirnum

genmethod="ltpca"

combination of lt with pca

dirnum

number of main directions, only used for genmethod="pcamain" or "lt"

Details

Method best (see the reference Dingec and Hormann below) is a very efficient simulation algorithm using multiple Control Variates and conditional MonteCarlo to calculate the the price, delta and gamma of Asian call options under geometric Brownian motion. It is especially effective when QMC is selected as sampling method. As QMC method Korobov Lattice rules are used. For good parameter values see Table 1 of (L'Ecuyer, Lemieux).

Value

returns a matrix holding the price and greeks. The estimated Asian Call price and its estimated delta and gamma form the first column vector, the respective 95 percent error bounds are given in the second column.

Author(s)

Kemal Dingec, Wolfgang Hormann

References

K. D. Dingec and W. Hormann. Improved Monte Carlo and Quasi-Monte Carlo Methods for the Price and the Greeks of Asian Options, Proceedings of the 2014 Winter Simulation Conference A. Tolk, S. D. Diallo, I. O. Ryzhov, L. Yilmaz, S. Buckley, and J. A. Miller, eds.

L'Ecuyer, P., and C. Lemieux. 2000. Variance Reduction via Lattice Rules. Management Science 46 (9): 1214-1235.

See Also

OptionPricing-package

Examples

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# standard settings for an efficient simulation using QMC and variance reduction
AsianCall(T=1,d=12,K=100,r=0.05,sigma=0.2,S0=100,method="best",
  sampling="QMC",metpar=list(maxiter=100,tol=1.e-14,cvmethod="splitting"),
  sampar=list(nout=50,n=2039,a=1487,baker=TRUE,genmethod="pca"))
					   
# efficient Monte Carlo version of the above simulation
AsianCall(T=1,d=12,K=100,r=0.05,sigma=0.2,S0=100,method="best",
  sampling="MC",metpar=list(maxiter=100,tol=1.e-14,np=1000),
  sampar=list(n=10^5))

# simple QMC version without variance reduction

AsianCall(T=1,d=12,K=100,r=0.05,sigma=0.2,S0=100,method="naive",
  sampling="QMC",
  sampar=list(nout=50,n=2039,a=1487,baker=TRUE,genmethod="pca"))
					  
# naive Monte Carlo version
AsianCall(T=1,d=12,K=100,r=0.05,sigma=0.2,S0=100,method="naive",
  sampling="MC",sampar=list(n=10^5))