Sim.DiffProc (version 2.5)

PEBS: Parametric Estimation of Model Black-Scholes (Exact likelihood inference)

Description

Parametric estimation of model Black-Scholes.

Usage

PEBS(X, delta, starts = list(theta= 1, sigma= 1), leve = 0.95)

Arguments

X

a numeric vector of the observed time-series values.

delta

the fraction of the sampling period between successive observations.

starts

named list. Initial values for optimizer.

leve

the confidence level required.

Value

• coef
  Coefficients extracted from the model.
• AIC
  A numeric value with the corresponding AIC.
• vcov
  A matrix of the estimated covariances between the parameter estimates in the linear or non-linear predictor of the model.
• confint
  A matrix (or vector) with columns giving lower and upper confidence limits for each parameter. These will be labelled as (1-level)/2 and 1 - (1-level)/2.

Details

The Black and Scholes, or geometric Brownian motion model solves the stochastic differential equation: $$dX(t) = theta * X(t)* dt + sigma * X(t)* dW(t)$$ The conditional density function p(t,.|x) is log-normal with mean = x * exp(theta*t) and variance = x^2 * exp(2*theta*t)*(exp(sigma^2 *t ) -1 ). R has the [dqpr]lnorm functions to evaluate the density, the quantiles, and the cumulative distribution or generate pseudo random numbers from the lognormal distribution.

See Also

PEABM Parametric Estimation of Arithmetic Brownian Motion, PEOU Parametric Estimation of Ornstein-Uhlenbeck Model, PEOUexp Explicit Estimators of Ornstein-Uhlenbeck Model, PEOUG Parametric Estimation of Hull-White/Vasicek Models.

Examples

## Parametric estimation of model Black-Scholes.
## t0 = 0 ,T = 1
 data(DATA2)
 res <- PEBS(DATA2,delta=0.001,starts=list(theta=2,sigma=1))
 res
 GBMF(N=1000,M=10,T=1,t0=0,x0=DATA2[1],theta=res$coef[1],sigma=res$coef[2])
 points(seq(0,1,length=length(DATA2)),DATA2,type="l",lwd=3,col="blue")