simPricesAndMinimumFromGBM: Simulation of the joint finite-dimensional distribution of...

In fExpressCertificates: Structured Products Valuation for ExpressCertificates/Autocallables

Description

Simulates from the joint distribution of finite-dimensional distribution (S(t_1),…,S(t_n)) and the minimum m(t_n) of a Geometric Brownian motion by either using simple grid approach or using the multivariate normal distribution of the returns and the conditional distribution of a minimum of a Brownian Bridge given the returns.

Usage

simPricesAndMinimumFromGBM(N = 100, S, T, mu, sigma, log = FALSE, m=Inf)

simPricesAndMinimumFromGBM2(N = 10000, S, T, mu, sigma, mc.steps = 1000)

Arguments

N	number of samples to draw
S	start value of the Arithmetic/Geometric Brownian Motion, i.e. S(0)=S0 or B(0) = S0
T	Numeric vector of valuation times (length n). T = (t_1,...,t_n)'
mu	the drift parameter of the Geometric Brownian Motion
sigma	volatility p.a., e.g. 0.2 for 20%
log	logical, if true the returns instead of prices are returned
m	Possible prior minimum value.
mc.steps	Number of gridpoints

Details

grid-approach
The simPricesAndMinimumFromGBM2 method uses the Monte Carlo Euler Scheme, the stepsize is δ t = t_n/mc.steps. The method is quite slow.

multivariate-normal distribution approach

The method simPricesAndMinimumFromGBM draws from the multivariate normal distribution of returns. 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 Geometric Brownian motion. This is done by using the multivariate normal distribution of the returns of a GBM and the conditional distribution of a minimum of a Brownian Bridge (i.e. in-between valuation dates).

First we simulate (S(t_1),…,S(t_n)) from a 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}

Next, we simulate the period minimum m(t_{i-1},t_i) = \min_{t_{i-1} ≤ t ≤ t_i}S(t) between two times t_{i-1} and t_i for all i=1,…,n. This minimum m(t_{i-1},t_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(t_n) = \min(m_{(0,1)},m_{(1,2)},…, m_{(n-1,n)}) = \min m(t_{i-1},t_i) for all i=1,…,n.

Value

A matrix (N x 2*n) with rows (S(t_1),…,S(t_n), m(t_1),…,m(t_n))

Note

Since we are considering a specific path for the prices and are interested in the minimum given the specific trajectory (i.e. m(t_n) | S(t_1),…,S(t_n)), it is not sufficient to sample from the bivariate density (S(t_n), m(t_n)), for which formulae is given by Karatzas/Shreve and others. Otherwise we could face the problem that some of the S(t_1),...,S(t_{n-1}) are smaller than the simulated m(t_n). However, both approaches yield the same marginal density for m(t_n).

Author(s)

Stefan Wilhelm [e-mail]

References

Beskos, A.; Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective Exact Simulation of Diffusion Sample Paths with Applications Bernoulli, 12, 1077–1098

Karatzas/Shreve (2008). Brownian Motion and Stochastic Calculus, Springer

See Also

The method simPricesAndMinimumFromGBM2 returns the same, but using the Euler Scheme.

See also calcGBMProbability for the CDF of the minimum m_t (i.e. Type="P(m_t <= B)")

Examples

# Comparison of sampling of GBM Minimum m_t via finite dimensional approach +
# Brownian Bridges vs. crude Monte Carlo

# naive grid-based approach
X0 <- simPricesAndMinimumFromGBM2(N=5000, S=100, T=c(1,2,3), mu = 0.05, sigma=0.3, 
  mc.steps=1000)

# Simulation of minimums m_t via prices at valuation dates 
# (S(t_1),S(t_2),...,S(t_n)) and Brownian Bridges in-between	
X1 <- simPricesAndMinimumFromGBM(N=5000, S=100, T=c(1,2,3), mu=0.05, sigma=0.3)
m1 <- X1[,4]
  
# Monte Carlo simulation of m_t via gridpoints (m2)
mc.loops <- 5000
mc.steps <- 2000
S <- matrix(NA, mc.loops, mc.steps + 1)
for (j in 1:mc.loops) {
 S[j,] <- GBM(S0=100, mu=0.05, sigma=0.3, T=3, N=mc.steps)
}	
m2 <- apply(S, 1, min) # minimum for each price path
  
# Compare probability density function and CDF for m_t against each other
# and against theoretical CDF.
par(mfrow=c(2,2))
# a) pdf of GBM minimum m_t at maturity for both approaches
plot(density(m1, to=100), col="black")
lines(density(m2, to=100), col="blue")
  
# b) compare empirical CDFs for m_t with theoretical probability P(m_t <= B)
B  <- seq(0, 100, by=1)
p3 <- calcGBMProbability(Type="P(m_t <= B)", 
  S0=100, B=B, t=3, mu=0.05, sigma=0.3)
  
plot(ecdf(m1), col="black", main="Sampling of GBM minimum m_t")
lines(ecdf(m2), col="blue")
lines(B, p3, col="red")
legend("topleft", legend=c("Finite-dimensions and Brownian Bridge", 
   "MC Euler scheme", "Theoretical value"), 
   col=c("black","blue","red"), lwd=2) 

fExpressCertificates documentation built on May 2, 2019, 11:02 a.m.