Delay geometric Brownian motion in financial option valuation

Functionals of Brownian Motion in Path-Dependent Option Valuation

Path-dependent options have become increasingly popular over the last few years, in particular in FX markets, because of the greater precision with which they allow investors to choose or avoid exposure to well-defined sources of risk. The goal of the paper is to exhibit the power of stochastic time changes and Laplace transform techniques in the evaluation and hedging of path-dependent options...

Geometric Brownian Motion with Delay: Mean Square Characterisation

A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condit...

1 Geometric Brownian motion

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...

Simulating Brownian motion ( BM ) and geometric Brownian

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...

Sigman 1 Geometric Brownian motion