calcBMProbability: Calculates probabilities for the Arithmetic Brownian Motion

Description

This method is a compilation of formulas for some (joint) probabilities for the Arithmetic Brownian Motion $B_t=B(t)$ with drift parameter $\mu$ and volatility $\sigma$ 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)",">= 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="" 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.

level

point in time, $t > 0$

Brownian Motion drift term $\mu$

sigma

Brownian Motion volatility $\sigma$

Second point in time, used by some probabilities like P(M_s >= a, B_t <= z | s < t)

Value

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

Details

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

$P(M_t \ge a)$ ($P(M_t \le a)$) is the probability of the maximum $M_t$ exceeding (staying below) a level $a$ up to time $t$. See Chuang (1996), equation (2.3).
$P(m_t \le a)$ ($P(m_t \ge a)$) is the probability of the minimum $m_t$ to fall below (rise above) a level $a$ up to time $t$.
$P(M_t \ge a, B_t \le z)$ is the joint probability of the maximum, exceeding level $a$, while the Brownian Motion is below level $z$ at time $t$. See Chuang (1996), equation (2.1), p.82.
$P(m_t \le a, B_t \ge z)$ is the joint probability of the minimum to be below level $a$, while the Brownian Motion is above level z at time t.
$P(M_s \ge a, B_t \le z | s < t)$ 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 $s < t$
$P(m_s \le a, B_t \ge z | s < t)$ 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.
$P(M_s \ge a, B_t \le z | s > t)$ 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 $s > t$
$P(m_s \le a, B_t \ge z | s > t)$ 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 $P(M_s \ge a, B_t \le z) = P(m_s \le -a, B^{*}_t \ge -z)$.

Some identities: For $s < t$: $$P(M_s \le a , M_t \ge a , B_t \le z ) = P(M_t \ge a , B_t \le z) - P(M_s \ge a , B_t \le z)$$

$$P(M_s \ge a, B_t \le z) = P(M_s \ge a)- P(M_s \ge a, B_t \ge z)$$

$$P(X \le -x,Y \le -y) = P(-X \ge x, -Y \ge y) = 1 - P(-X \le x) - P(-Y \le y) + P(-X \le x, - Y \le y)$$

Changing from maximum $M_t$ of $B_t$ to minimum $m^{*}_t$ of $B^{*}_t = -B_t$: $P(M_t \ge z)$ becomes $P(m^{*}_t \le -z)$.

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

Run this code

# NOT RUN {
################################################################################
#
# 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 <e4>ndern 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")
# }

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