Sigman 1 Geometric Brownian motion
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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...
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متن کامل1 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...
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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...
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