simPricesAndMinimumFromTruncatedGBM: Simulation of the joint finite-dimensional distribution of a...

In fExpressCertificates: fExpressCertificates - Structured Products Valuation for ExpressCertificates/Autocallables

Description

Simulates from the joint distribution of finite-dimensional distributions (S(t_1),...,S(t_n)) and the minimum m(t_n) of a restricted Geometric Brownian motion by using the truncated multivariate normal distribution of the returns and the conditional distribution of a minimum of a Brownian Bridge given the returns.

Usage

simPricesAndMinimumFromTruncatedGBM(N = 100, S, T, mu, sigma, 
  lowerX = rep(0, length(T)), 
  upperX = rep(+Inf, length(T)), 
  log = FALSE, m=Inf)

Arguments

N	number of samples to draw
S	start value of the Arithmetic/Geometric Brownian Motion, i.e. S(0)=S_0 or B(0) = S_0
T	Numeric vector of n valuation times T = (t_1,...,t_n)'
mu	the drift parameter of the Geometric Brownian Motion
sigma	volatility p.a., e.g. 0.2 for 20%
lowerX	Numeric vector of n lower bounds for the Geometric Brownian Motion, zeros are permitted, default is rep(0,length(T))
upperX	Numeric vector of n upper bounds for the Geometric Brownian Motion, +Inf are permitted, default is rep(+Inf,length(T))
log	logical, if true the returns instead of prices are returned
m	Possible prior minimum value.

Details

For the n valuation times given by T = (t_1,…,t_n)' we simulate from the joint distribution (S(t_1),…,S(t_n),m(t_1),…,m(t_n)) of the finite-dimensional distribution (S(t_1),…,S(t_n)) and the running minimum m(t_i) = \min_{0 ≤ t ≤ t_i}(S_t) of a restricted/truncated Geometric Brownian motion.

The Geometric Brownian Motion is conditioned at the n valuation dates (t_1,...,t_n) on lowerX_i ≤ S(t_i) ≤ upperX_i for all i=1,…,n.

First we simulate (S(t_1),…,S(t_n)) from a truncated multivariate normal distribution of the returns with mean vector

(μ - σ^2/2) * T

and covariance matrix

\code{Sigma = outer(T, T, pmin) * sigma^2}

and lower and upper truncation points lower=log(lowerX/S) and upper=log(upperX/S) respectively.

Given the realized prices (S(t_1),…,S(t_n)) we simulate the global minimum as the minimum of several Brownian Bridges as described in Beskos (2006):

We simulate the period minimum m_{(i-1,i)} between two times t_{i-1} and t_i for all i=1,…,n. This minimum m_{(i-1,i)} | S(t_{i-1}),S(t_i) is the minimum of a Brownian Bridge between t_{i-1} and t_i.

The global minimum is the minimum of all period minima given by
m_n = \min(m_{(0,1)},m_{(1,2)},…,m_{(n-1,n)}) = \min(m_{(i-1,i)}) for all i=1,…,n.

Value

A (N \times 2*n) matrix with N rows and columns (S(t_1),…,S(t_n),m(t_1),…,m(t_n))

Note

This function can be used to determine the barrier risk of express certificates at maturity, i.e. the probability that barrier B has been breached given that we reach maturity: P(m(t_n) ≤ B | \forall_{i < n} S(t_i) < X(t_i))

Author(s)

Stefan Wilhelm [e-mail]

See Also

See the similar method simPricesAndMinimumFromGBM for the unrestricted Geometric Brownian Motion (i.e. lowerX=rep(0,n) and upperX=rep(Inf,n)).

Examples

# 1. Simulation of restricted GBM prices and minimums m_t
# finite-dimensional distribution and Brownian Bridge
X1 <- simPricesAndMinimumFromTruncatedGBM(N=5000, S=100, T=c(1,2,3), 
  upperX=c(100,100,Inf), mu=0.05, sigma=0.3)
m1 <- X1[,4]
  
# 2. Compare to distribution of unrestricted GBM minimums
X2 <- simPricesAndMinimumFromGBM(N=5000, S=100, T=c(1,2,3), 
  mu=0.05, sigma=0.3)
m2 <- X2[,4]
	
plot(density(m1, to=100), col="black", main="Minimum m_t for Express Certificate 
  price paths at maturity")
lines(density(m2, to=100), col="blue")
legend("topleft", legend=c("Restricted GBM minimum","Unrestricted GBM minimum"),
  col=c("black","blue"), lty=1, bty="n")

fExpressCertificates documentation built on May 2, 2019, 4:48 p.m.