Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is not appropriate for modeling stock prices. Instead, we introduce here a non-negative variation of BM called geometric Brownian motion, S(t), which is defined by S(t) = S0 eX(t) , (1)
where X(t) = σB(t) + µt is BM with drift and S(0) = S0 > 0 is the intial value. 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 σ 2 t; thus, for each t, S(t) has a lognormal distribution. 2 As we will see in Section 1.4: letting r = µ + σ , 2 E(S(t)) = ert S0 the expected price grows like a fixed-income security with continuously compounded interest rate r. In practice, r >> r, the real fixed-income interest rate, that is why one invests in stocks. But unlike a fixed-income investment, the stock price has variability due to the randomness of the underlying Brownian motion and could drop in value causing you to lose money; there is risk involved here. (2)
1.1
Lognormal distributions
If Y ∼ N (µ, σ 2 ), then X = eY is a non-negative r.v. having the lognormal distribution; called so because its natural logarithm Y = ln(X) yields a normal r.v. X has density f (x) = This is derived via computing xσ 1 √
e 2π
−(ln(x)−µ)2 2σ 2
0, d dx F (x)
, if x ≥ 0; if x < 0.
for
F (x) = P (X ≤ x) = P (Y ≤ ln(x)) = Θ((ln(x) − µ)/σ), where Θ(x) denotes the c.d.f. of N (0, 1). Observing that E(X) = E(eY ) and E(X 2 ) = E(e2Y ) are simply the moment generating function (MGF) MY (s) = E(esY ) of Y ∼ N (µ, σ 2 ) evaluated at s = 1 and s = 2 respectively yields E(X) = eµ+ σ2 2 2 2
E(X 2 ) = e2µ+2σ
2
V ar(X) = e2µ+σ (eσ − 1). As with the normal distribution, the c.d.f.
