American options pricing with Least Squares Monte Carlo method

Description

The package compiles functions that calculate prices of American put options with Least Squares Monte Carlo method. The option types are plain vanilla American put, Asian American put, and Quanto American put. The pricing algorithms include variance reduction techniques such as Antithetic Variates and Control Variates. Additional functions are given to derive "price surfaces" at different volatilities and strikes, create 3-D plots, quickly generate Geometric Brownian motion, and calculate prices of European options with Black & Scholes analytical solution.

Details

The Least Squares Monte Carlo is an approach developed to approximate the value of American options. It combines regression modeling and Monte Carlo simulation. The key feature of this method is estimation of the conditional expectation of the future pay-offs by a regression model (for details see Longstaff & Schwartz, 2000). The main pricing functions employing this method in the package are: AmerPutLSM, AsianAmerPutLSM, and QuantoAmerPutLSM. Pricing functions that include variance reduction methods are: AmerPutLSM_AV, QuantoAmerPutLSM_AV (Antithetic Variates) and AmerPutLSM_CV (Control Variates, with Black & Scholes solution for European put used as the control). All these functions are based on Geometric Brownian motion as a price process. They can be used with tailored summary, print, and price functions. The "price surfaces" at different volatilities and strikes can be derived using the functions AmerPutLSMPriceSurf, AsianAmerPutLSMPriceSurf, and QuantoAmerPutLSMPriceSurf, and plotted with tailored plot function. For general reading on option pricing with Monte Carlo methods see Glasserman (2004).

Author(s)

Mikhail A. Beketov

Maintainer: Mikhail A. Beketov <mikhail.beketov@gmx.de>

References

Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. Springer.

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

See Also

Functions: AmerPutLSM, AmerPutLSM_AV, AmerPutLSM_CV, AsianAmerPutLSM, QuantoAmerPutLSM, and QuantoAmerPutLSM_AV.

Examples

Put<-AmerPutLSM(Spot=14.2, Strike=16.5, n=200, m=50)
summary(Put)
price(Put)
plot(AmerPutLSMPriceSurf(vols = (seq(0.1, 1.5, 0.2)), n=200, m=10, 
strikes = (seq(0.5, 1.9, 0.2))), color = divPalette(150, "RdBu"))

Calculating the price of plain vanilla American put

Description

The function calculates the price of plain vanilla American put with Least Squares Monte Carlo method. The regression model included in the algorithm is quadratic polynomial (Longstaff & Schwartz, 2000).

Usage

AmerPutLSM(Spot = 1, sigma = 0.2, n = 1000, m = 365, Strike = 1.1, r = 0.06, 
dr = 0, mT = 1)

## S3 method for class 'AmerPut'
print(x, ...)
## S3 method for class 'AmerPut'
summary(object, ...)

Arguments

Value

The function returns an object of the class AmerPut that is a list comprising the price calculated, option type, and the entry parameters. Class-specific print function gives the option type information and the price. The price as a single number can be derived using the price function. An overview of the entire object can be seen using the summary function.

Author(s)

Mikhail A. Beketov

References

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

See Also

Functions: price, AmerPutLSM_AV, AmerPutLSM_CV, AsianAmerPutLSM, and QuantoAmerPutLSM.

Examples

AmerPutLSM()
put<-AmerPutLSM(Spot=14.2, Strike=16.5, n=500, m=100)
put
summary(put)
price(put)
put$price

Deriving a table of American put prices at different volatilities and strikes

Description

The function calculates the prices at different volatilities and strikes using the AmerPutLSM function.

Usage

AmerPutLSMPriceSurf(Spot = 1, vols = (seq(0.1, 2, 0.1)), n = 1000, m = 365, 
strikes = (seq(0.5, 2.5, 0.1)), r = 0.06, dr = 0, mT = 1)

## S3 method for class 'PriceSurface'
summary(object, ...)
## S3 method for class 'PriceSurface'
plot(x, color = divPalette(800, "RdBu"), ...)

Arguments

Value

The function returns an object of the class PriceSurface that is a matrix of prices at different volatilities and strikes. Class-specific summary function gives the sequences of volatilities and strikes used, as well as maximum, minimum, and average prices. Class-specific plot function constructs a 3-D plot of the price surface.

Author(s)

Mikhail A. Beketov

See Also

Functions: AmerPutLSM, AsianAmerPutLSMPriceSurf, and QuantoAmerPutLSMPriceSurf.

Examples

surface<-AmerPutLSMPriceSurf(vols = (seq(0.1, 1.5, 0.2)), n=200, m=10, 
strikes = (seq(0.5, 1.9, 0.2)))
summary(surface)
plot(surface, color = divPalette(150, "RdBu"))

Pricing plain vanilla American put with Antithetic Variates

Description

The function calculates the price of a plain vanilla American put with Least Squares Monte Carlo method with Antithetic Variates (Glasserman, 2004). The regression model included in the algorithm is quadratic polynomial (Longstaff & Schwartz, 2000).

Usage

AmerPutLSM_AV(Spot = 1, sigma = 0.2, n = 1000, m = 365, Strike = 1.1, r = 0.06, 
dr = 0, mT = 1)

## S3 method for class 'AmerPutAV'
print(x, ...)
## S3 method for class 'AmerPutAV'
summary(object, ...)

Arguments

Value

The function returns an object of the class AmerPutAV that is a list comprising the price calculated and the entry parameters. Class-specific print function gives the option type information and the price. The price as a single number can be derived using the price function. An overview of the entire object can be seen using the summary function.

Author(s)

Mikhail A. Beketov

References

Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. Springer.

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

See Also

Functions: price, AmerPutLSM, AmerPutLSM_CV, AsianAmerPutLSM, and QuantoAmerPutLSM.

Examples

AmerPutLSM_AV(n=500, m=50)
put<-AmerPutLSM_AV(Spot=14.2, Strike=16.5, n=200, m=50)
put
summary(put)
price(put)
put$price

Pricing plain vanilla American put with Control Variates

Description

The function calculates the price of a plain vanilla American put with Least Squares Monte Carlo method with Control Variates (Glasserman, 2004). Black & Scholes solution for European put is used as the control. The regression model included in the algorithm is quadratic polynomial (Longstaff & Schwartz, 2000).

Usage

AmerPutLSM_CV(Spot = 1, sigma = 0.2, n = 1000, m = 365, Strike = 1.1, r = 0.06, 
dr = 0, mT = 1)

## S3 method for class 'AmerPutCV'
print(x, ...)
## S3 method for class 'AmerPutCV'
summary(object, ...)

Arguments

Value

The function returns an object of the class AmerPutCV that is a list comprising the price calculated and the entry parameters. Class-specific print function gives the option type information and the price. The price as a single number can be derived using the price function. An overview of the entire object can be seen using the summary function.

Author(s)

Mikhail A. Beketov

References

Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. Springer.

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

See Also

Functions: price, AmerPutLSM, AmerPutLSM_AV, AsianAmerPutLSM, and QuantoAmerPutLSM.

Examples

AmerPutLSM_CV()
put<-AmerPutLSM_CV(Spot=14.2, Strike=16.5, n=200, m=50)
put
summary(put)
price(put)
put$price

Calculating the price of Asian American put

Description

The function calculates the price of Asian American put with Least Squares Monte Carlo method (pay-off based on arithmetic mean). The regression model included in the algorithm is quadratic polynomial (Longstaff & Schwartz, 2000).

Usage

AsianAmerPutLSM(Spot = 1, sigma = 0.2, n = 1000, m = 365, Strike = 1.1, r = 0.06, 
dr = 0, mT = 1)

## S3 method for class 'AsianAmerPut'
print(x, ...)
## S3 method for class 'AsianAmerPut'
summary(object, ...)

Arguments

Value

The function returns an object of the class AsianAmerPut that is a list comprising the price calculated, option type, and the entry parameters. Class-specific print function gives the option type information and the price. The price as a single number can be derived using the price function. An overview of the entire object can be seen using the summary function.

Author(s)

Mikhail A. Beketov

References

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

See Also

Functions: price, AmerPutLSM, AmerPutLSM_CV, AmerPutLSM_AV, and QuantoAmerPutLSM.

Examples

AsianAmerPutLSM(n=500, m=100)
put<-AsianAmerPutLSM(Spot=14.2, Strike=16.5, n=500, m=50)
put
summary(put)
price(put)
put$price

Deriving a table of Asian American put prices at different volatilities and strikes

Description

The function calculates the prices at different volatilities and strikes using the AsianAmerPutLSM function.

Usage

AsianAmerPutLSMPriceSurf(Spot = 1, vols = (seq(0.1, 2, 0.1)), n = 1000, m = 365, 
strikes = (seq(0.5, 2.5, 0.1)), r = 0.06, dr = 0, mT = 1)

Arguments

Value

The function returns an object of the class PriceSurface that is a matrix of prices at different volatilities and strikes. Class-specific summary function gives the sequences of volatilities and strikes used, as well as maximum, minimum, and average prices. Class-specific plot function constructs a 3-D plot of the price surface.

Author(s)

Mikhail A. Beketov

See Also

Functions: AsianAmerPutLSM, summary.PriceSurface, plot.PriceSurface, AmerPutLSMPriceSurf, and QuantoAmerPutLSMPriceSurf.

Examples

surface<-AsianAmerPutLSMPriceSurf(vols = (seq(0.1, 1.5, 0.2)), n=200, m=10, 
strikes = (seq(0.5, 1.9, 0.2)))
summary(surface)
plot(surface, color = divPalette(150, "RdBu"))

Black & Scholes solution for European put and call

Description

Pricing plain vanilla American put and call options using Black & Scholes solution.

Usage

EuPutBS(Spot, sigma, Strike, r, dr, mT)

EuCallBS(Spot, sigma, Strike, r, dr, mT)

Arguments

Value

The function returns the price as a single number (class "numeric").

See Also

AmerPutLSM_CV

Examples

EuPutBS(1, 0.2, 1, 0.06, 0, 1)
EuCallBS(1, 0.2, 1, 0.06, 0, 1)

Calculating the price of Quanto American put

Description

The function calculates the price of Quanto American put with Least Squares Monte Carlo method. The Quanto option is cash-settled option, whose pay-off is converted into a third currency/asset at exercise at a pre-specified rate/price (Wystup, 2011), and can also be considered as a usual option but settled in a "wrong" asset (Vecer, 2011). The regression model included in the algorithm is quadratic polynomial (Longstaff & Schwartz, 2000).

Usage

QuantoAmerPutLSM(Spot = 1, sigma = 0.2, n = 1000, m = 365, Strike = 1.1, r = 0.06, 
dr = 0, mT = 1, Spot2 = 1, sigma2 = 0.2, r2 = 0, dr2 = 0, rho = 0)

## S3 method for class 'QuantoAmerPut'
print(x, ...)
## S3 method for class 'QuantoAmerPut'
summary(object, ...)

Arguments

Value

The function returns an object of the class QuantoAmerPut that is a list comprising the price calculated, option type, and the entry parameters. Class-specific print function gives the option type information and the price. The price as a single number can be derived using the price function. An overview of the entire object can be seen using the summary function.

Note

The function rmvnorm included in the pricing algorithm is a part of the mnormt package. Please, load that package before the use of the QuantoAmerPutLSM function.

Author(s)

Mikhail A. Beketov

References

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

Vecer, J. 2011. Stochastic Finance: A Numeraire Approach. CRC Press.

Wystup, U. 2011. Quanto Options. MathFinance AG.

See Also

Functions: price, QuantoAmerPutLSM_AV, AmerPutLSM, AsianAmerPutLSM, and AmerPutLSM_AV.

Examples

QuantoAmerPutLSM(n=200, m=50)
put<-QuantoAmerPutLSM(Spot=14.2, Strike=16.5, n=200, m=50)
put
summary(put)
price(put)

Deriving a table of Quanto American put prices at different volatilities and strikes

Description

The function calculates the prices at different volatilities and strikes using the QuantoAmerPutLSM function.

Usage

QuantoAmerPutLSMPriceSurf(Spot = 1, vols = (seq(0.1, 2, 0.1)), n = 1000, m = 365, 
strikes = (seq(0.5, 2.5, 0.1)), r = 0.06, dr = 0, mT = 1, Spot2 = 1, sigma2 = 0.2, 
r2 = 0, dr2 = 0, rho = 0)

Arguments

Value

The function returns an object of the class PriceSurface that is a matrix of prices at different volatilities and strikes. Class-specific summary function gives the sequences of volatilities and strikes used, as well as maximum, minimum, and average prices. Class-specific plot function constructs a 3-D plot of the price surface.

Note

The function rmvnorm included in the pricing algorithm is a part of the mnormt package. Please, load that package before the use of the QuantoAmerPutLSMPriceSurf function. Using the function plot with default pallet requires package fBasics, if you do not want to load this package, you can set color=NULL or other palette).

Author(s)

Mikhail A. Beketov

See Also

Functions: QuantoAmerPutLSM, summary.PriceSurface, plot.PriceSurface, AmerPutLSMPriceSurf, and AsianAmerPutLSMPriceSurf.

Examples

surface<-QuantoAmerPutLSMPriceSurf(vols = (seq(0.1, 1.7, 0.2)), n=100, m=5, 
strikes = (seq(0.7, 1.7, 0.2)))
summary(surface)
plot(surface, color = divPalette(150, "RdBu"))

Pricing Quanto American put with Antithetic Variates

Description

The function calculates the price of Quanto American put with Least Squares Monte Carlo method with Antithetic Variates (Glasserman, 2004). The Quanto option is cash-settled option, whose pay-off is converted into a third currency/asset at exercise at a pre-specified rate/price (Wystup, 2011), and can also be considered as a usual option but settled in a "wrong" asset (Vecer, 2011). The regression model included in the algorithm is quadratic polynomial (Longstaff & Schwartz, 2000).

Usage

QuantoAmerPutLSM_AV(Spot = 1, sigma = 0.2, n = 1000, m = 365, Strike = 1.1, 
r = 0.06, dr = 0, mT = 1, Spot2 = 1, sigma2 = 0.2, r2 = 0, dr2 = 0, rho = 0)

## S3 method for class 'QuantoAmerPut_AV'
print(x, ...)
## S3 method for class 'QuantoAmerPut_AV'
summary(object, ...)

Arguments

Value

The function returns an object of the class QuantoAmerPut_AV that is a list comprising the price calculated, option type, and the entry parameters. Class-specific print function gives the option type information and the price. The price as a single number can be derived using the price function. An overview of the entire object can be seen using the summary function.

Note

The function rmvnorm included in the pricing algorithm is a part of the mnormt package. Please, load that package before the use of the QuantoAmerPutLSM_AV function.

Author(s)

Mikhail A. Beketov

References

Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. Springer.

Longstaff, F.A., and E.S. Schwartz. 2000. Valuing american option by simulation: A simple least-squared approach. The Review of Financial Studies. 14:113-147.

Vecer, J. 2011. Stochastic Finance: A Numeraire Approach. CRC Press.

Wystup, U. 2011. Quanto Options. MathFinance AG.

See Also

Functions: price, QuantoAmerPutLSM, AmerPutLSM, AsianAmerPutLSM, and AmerPutLSM_AV.

Examples

QuantoAmerPutLSM_AV(n=200, m=50)
put<-QuantoAmerPutLSM_AV(Spot=14.2, Strike=16.5, n=200, m=50)
put
summary(put)
price(put)

Generating Geometric Brownian motion

Description

Quick Generating Geometric Brownian motion avoiding unnecessary loops using the cumsum function. Technical function implemented in the pricing functions of the package.

Usage

fastGBM(Spot = 1, sigma = 0.2, n = 1000, m = 365, r = 0.06, dr = 0, mT = 1)

Arguments

Value

Table with paths generated (each row is a path, class "matrix")

Author(s)

Mikhail A. Beketov

See Also

Functions: AmerPutLSM, AmerPutLSM_AV, AmerPutLSM_CV, AsianAmerPutLSM, QuantoAmerPutLSM, and QuantoAmerPutLSM_AV.

Examples

fastGBM(n=10, m=5)
matplot(t(fastGBM(n=100, m=100)), type="l") # matrix transpose by "t()"

Returning the first >0 value in each row of a matrix

Description

Technical function implemented in the pricing functions of the package. It returns the first >0 value in each row of a matrix and assign zero to all subsequent values.

Usage

firstValueRow(x)

Arguments

Value

A matrix.

Author(s)

Mikhail A. Beketov

See Also

Functions: AmerPutLSM, AmerPutLSM_AV, AmerPutLSM_CV, AsianAmerPutLSM, QuantoAmerPutLSM, and QuantoAmerPutLSM_AV.

Examples

mat<-matrix(c(0,0,2,0,4,0,3,0,1,9,8,7), ncol=4)
mat
firstValueRow(mat)

Extracting price from the pricing functions outputs

Description

The function is nothing else, but the object$price action, with the object returned by the pricing functions in the package.

Usage

price(x)

Arguments

Value

The function returns the price as a single number (class "numeric").

See Also

Functions: AmerPutLSM, AmerPutLSM_AV, AmerPutLSM_CV, AsianAmerPutLSM, QuantoAmerPutLSM, and QuantoAmerPutLSM_AV.

Examples

put<-AmerPutLSM()
price(put)
put$price