Fundamental to many applications in financial engineering is the normal (Gaussian) distribution. It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion. In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometr...

Fundamental to many applications in financial engineering is the normal (Gaussian) distribution. It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion. In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometr...

where X(t) = σB(t) + μt is BM with drift and S(0) = S0 > 0 is the intial value. We view S(t) as the price per share at time t of a risky asset such as stock. Taking logarithms yields back the BM; X(t) = ln(S(t)/S0) = ln(S(t))− ln(S0). ln(S(t)) = ln(S0) +X(t) is normal with mean μt + ln(S0), and variance σ2t; thus, for each t, S(t) has a lognormal distribution. As we will see in Section 1.4: let...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [5] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [7] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

2) and 3) together can be summarized by: If t0 = 0 < t1 < t2 < · · · < tk, then the increment rvs B(ti) − B(ti−1), i ∈ {1, . . . k}, are independent with B(ti) − B(ti−1) ∼ N(0, ti − ti−1) (normal with mean 0 and variance ti − ti−1). In particular, B(ti) − B(ti−1) is independent of B(ti−1) = B(ti−1)−B(0). If we only wish to simulate B(t) at one fixed value t, then we need only generate a unit no...

This paper has four goals: (a) relate ladder height distributions to option values; (b) show how Laguerre expansions may be used in the computation of densities, distribution functions and option prices; (c) derive some new results on the integral of geometric Brownian motion over a finite interval; (d) apply the preceding results to the determination of the distribution of the integral of geom...