calcBMProbability: Calculates probabilities for the Arithmetic Brownian Motion

In fExpressCertificates: Structured Products Valuation for ExpressCertificates/Autocallables

Description

This method is a compilation of formulas for some (joint) probabilities for the Arithmetic Brownian Motion B_t=B(t) with drift parameter μ and volatility σ and its minimum m_t=m(t) or maximum M_t=M(t).

Usage

 calcBMProbability(
  Type = c(
    "P(M_t >= a)",
    "P(M_t <= a)",
    "P(m_t <= a)",
    "P(m_t >= a)",
    "P(M_t >= a, B_t <= z)",
    "P(m_t <= a, B_t >= z)",
    "P(a <= m_t, M_t <= b)",
    "P(M_s >= a, B_t <= z | s < t)",
    "P(m_s <= a, B_t >= z | s < t)",
    "P(M_s >= a, B_t <= z | s > t)",
    "P(m_s <= a, B_t >= z | s > t)"), 
  a, z=0, t = 1, mu = 0, sigma = 1, s = 0)

Arguments

Type	Type of probability to be calculated, see details.
a	level
z	level
t	point in time, t > 0
mu	Brownian Motion drift term μ
sigma	Brownian Motion volatility σ
s	Second point in time, used by some probabilities like P(M_s >= a, B_t <= z | s < t)

Details

Let \code{M_t = max(B_t)} and \code{m_t = min(B_t)} for t > 0 be the running maximum/minimum of the Brownian Motion up to time \code{t} respectively.

• P(M_t >= a) (P(M_t <= a)) is the probability of the maximum M_t exceeding (staying below) a level \code{a} up to time \code{t}. See Chuang (1996), equation (2.3).

• P(m_t <= a) (P(m_t >= a)) is the probability of the minimum m_t to fall below (rise above) a level \code{a} up to time \code{t}.

• P(M_t >= a, B_t <= z) is the joint probability of the maximum, exceeding level \code{a}, while the Brownian Motion is below level \code{z} at time \code{t}. See Chuang (1996), equation (2.1), p.82.

• P(m_t <= a, B_t >= z) is the joint probability of the minimum to be below level \code{a}, while the Brownian Motion is above level z at time t.

• P(M_s >= a, B_t <= z | s < t) See Chuang (1996), equation (2.7), p.84 for the joint probability (M_s,B_t) of the maximum M_s and the Brownian Motion B_t at different times s < t

• P(m_s <= a, B_t >= z | s < t) See Chuang (1996), equation (2.7), p.84 for the joint probability of (M_s,B_t) s < t. Changed formula to work for the minimum.

• P(M_s >= a, B_t <= z | s > t) See Chuang (1996), equation (2.9), p.85 for the joint probability (M_s,B_t) of the maximum M_s and the Brownian Motion B_t at different times s > t

• P(m_s <= a, B_t >= z | s > t) See Chuang (1996), equation (2.9), p.85 for the joint probability (M_s,B_t) of the maximum M_s and the Brownian Motion B_t at different times s > t. Adapted this formula for the minimum (m_s,B_t) by P(M_s >= a, B_t <= z) = P(m_s <= -a, B^{*}_t >= -z).

Some identities:
For s < t:

P(M_s <= a , M_t >= a , B_t <= z ) = P(M_t >= a , B_t <= z) - P(M_s >= a , B_t <= z)

P(M_s >= a, B_t <= z) = P(M_s >= a)- P(M_s >= a, B_t >= z)

P(X <= -x,Y <= -y) = P(-X >= x, -Y >= y) = 1 - P(-X <= x) - P(-Y <= y) + P(-X <= x, - Y <= y)

Changing from maximum M_t of B_t to minimum m^*_t of B^*_t = -B_t:
P(M_t >= z) becomes P(m^*_t <= -z).

Value

The method returns a vector of probabilities, if used with vector inputs.

Author(s)

Stefan Wilhelm [e-mail]

References

Chuang (1996). Joint distribution of Brownian motion and its maximum, with a generalization to correlated BM and applications to barrier options Statistics & Probability Letters 28, 81–90

Examples

################################################################################
#
# Example 1: Maximum M_t of Brownian motion
#
################################################################################

# simulate 1000 discretized paths from Brownian Motion B_t
B <- matrix(NA,1000,101)
for (i in 1:1000) {
  B[i,] <- BrownianMotion(S0=100, mu=0.05, sigma=1, T=1, N=100)
}

# get empirical Maximum M_t
M_t <- apply(B, 1, max, na.rm=TRUE)
plot(density(M_t, from=100))

# empirical CDF of M_t
plot(ecdf(M_t))
a <- seq(100, 103, by=0.1)
# P(M_t <= a)
# 1-cdf.M_t(a-100, t=1, mu=0.05, sigma=1)
p <- calcBMProbability(Type = "P(M_t <= a)", a-100, t = 1, 
    mu = 0.05, sigma = 1)
lines(a, p, col="red")

################################################################################
#
# Example 2: Minimum m_t of Brownian motion
#
################################################################################

# Minimum m_t : Drift ändern von 0.05 auf -0.05
m_t <- apply(B, 1, min, na.rm=TRUE)

a <- seq(97, 100, by=0.1)
# cdf.m_t(a-100, t=1, mu=0.05, sigma=1)
p <- calcBMProbability(Type = "P(m_t <= a)", a-100, t = 1, mu = 0.05, sigma = 1)

plot(ecdf(m_t))
lines(a, p, col="blue")

Example output

Loading required package: tmvtnorm
Loading required package: mvtnorm
Loading required package: Matrix
Loading required package: stats4
Loading required package: gmm
Loading required package: sandwich
Loading required package: fCertificates
Loading required package: fBasics
Loading required package: timeDate
Loading required package: timeSeries
Loading required package: fOptions
Loading required package: fExoticOptions

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