Monte Carlo valuation methods for Express Classic Certificates using the Euler scheme or sampling from conditional densities
MonteCarlo.ExpressCertificate.Classic(S, X, T, K, r, r_d,
sigma, ratio = 1, mc.steps = 1000, mc.loops = 20)
Conditional.MonteCarlo.ExpressCertificate.Classic(S, X, T, K, r, r_d,
sigma, ratio = 1, mc.loops = 20, conditional.random.generator = "rnorm")
MonteCarlo.ExpressCertificate(S, X, T, K, B,
r, r_d, sigma, mc.steps = 1000, mc.loops = 20, payoff.function)the asset price, a numeric value
a vector of early exercise prices ("Bewertungsgrenzen"), , vector of length (n-1)
a vector of evaluation times measured in years ("Bewertungstage"), vector of length n
vector of fixed early cash rebates in case of early exercise, length (n-1)
barrier level
the annualized rate of interest, a numeric value; e.g. 0.25 means 25% pa.
the annualized dividend yield, a numeric value; e.g. 0.25 means 25% pa.
the annualized volatility of the underlying security, a numeric value; e.g. 0.3 means 30% volatility pa.
ratio, number of underlyings one certificate refers to, a numeric value; e.g. 0.25 means 4 certificates refer to 1 share of the underlying asset
Monte Carlo steps in one path
Monte Carlo Loops (iterations)
A pseudo-random or quasi-random (Halton-Sequence, Sobol-Sequence)
generator for the conditional distributions, one of "rnorm","rnorm.halton","rnorm.sobol"
payoff function
returns a list of
stops
vector of prices, length mc.loops
Monte Carlo estimate of the price = mean(prices)
vector of underlying prices at maturity
The conventional Monte Carlo uses the Euler scheme with mc.steps steps in order
to approximate the continuous-time stochastic process.
The conditional Monte Carlo samples from conditional densities \(f(x_{i+1}|x_i)\) for \(i=0,\dots,(n-1))\),
which are univariate normal
distributions for the log returns of the Geometric Brownian Motion and Jump-diffusion model:
\(f(x_1,x_2,..,x_n) = f(x_n|x_{n-1}) \cdot \dots \cdots f(x_2|x_1) \cdot f(x_1|x_0)\)
The conditional Monte Carlo does not need the mc.steps points
in between and has a much better performance.