Option Pricing and Greeks Estimation for Asian and European Options

Description

The Price, Delta and Gamma of European and Asian Options under Geometric Brownian Motion are calculated using the Black-Scholes formula and Efficient Monte Carlo and Randomized Quasi Monte Carlo Algorithms.

Details

The OptionPricing package calculates the Price, Delta and Gamma for European options using the Black-Scholes formula (see BS_EC). The price, Delta and Gamma for Asian call options under geometric Brownian motion are calculated using a very efficient Monte Carlo and randomized quasi-Monte Carlo algorithm (see AsianCall). The function AsianCall_AppLord implements a high-quality approximation for the price of an Asian option.

Author(s)

Kemal Dingec, Wolfgang Hormann

Examples

# 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=c("best"),
  sampling=c("QMC"),metpar=list(maxiter=100,tol=1.e-14,cvmethod="splitting"),
  sampar=list(nout=50,n=2039,a=1487,baker=TRUE,genmethod="pca"))

# Calculation of the Price of an Asian option using a good approximation  
  AsianCall_AppLord(T = 1, d = 12, K = 100, r = 0.05, sigma = 0.2, S0 = 100)


# standard settings for an efficient simulation using MC and variance reduction
AsianCall(T=1,d=12,K=170,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))
# Calculation of the approximate price, a bit different to the above result 
  AsianCall_AppLord(T = 1, d = 12, K = 170, r = 0.05, sigma = 0.2, S0 = 100)
  
# Calculation of the Price of an Asian option using a good approximation  
  AsianCall_AppLord(T = 1, d = 12, K = 100, r = 0.05, sigma = 0.2, S0 = 100)

#Price, Delta and Gamma of European options using Black-Scholes  
BS_EC(K=100, r = 0.05, sigma = 0.2, T = 0.25, S0 = 100)
BS_EP(K=100, r = 0.05, sigma = 0.2, T = 0.25, S0 = 100)

Calculates the Price, Delta and Gamma of an Asian Option

Description

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

Usage

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

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

# 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))

					  

Asian Options - Approximation

Description

The price of an arithmetic average Asian option is computed using the approximation method of Lord.

Usage

AsianCall_AppLord(T=1, d=12, K=100, r=0.05, sigma=0.1, S0=100, all=TRUE)

Arguments

Details

AsianCall_AppLord() uses a sophisticated approximation of Lord (2006).

Value

returns the approximate price.

Author(s)

Kemal Dingec, Wolfgang Hormann

References

Lord, R., Partially Exact and Bounded Approximations for Arithmetic Asian Options, Journal of Computational Finance, Vol. 10, No. 2, pp. 1-52, 2006

See Also

OptionPricing-package

Examples

AsianCall_AppLord(T = 1, d = 12, K = 100, r = 0.05, sigma = 0.25, S0 = 100, all = TRUE)

Black-Scholes Formula for European Call and Put

Description

Calculates the Price, Delta and Gamma of an European Call or Put option using the Black-Scholes formula.

Usage

BS_EC( T = 0.25, K = 100, r = 0.05, sigma = 0.2, S0 = 100 )
BS_EP( T = 0.25, K = 100, r = 0.05, sigma = 0.2, S0 = 100 )

Arguments

Value

Returns a vector containing the option price, Delta and Gamma

Author(s)

Wolfgang Hormann

See Also

OptionPricing-package

Examples

BS_EC(K=100, r = 0.05, sigma = 0.2, T = 0.25, S0 = 100)
BS_EP(K=100, r = 0.05, sigma = 0.2, T = 0.25, S0 = 100)