4298 words - 18 pages

Copyright c 2006 by Karl Sigman

1

Geometric Brownian motion

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 ...view middle of the document...

Thus computations for F (x) are reduced to dealing with Θ(x). 1

We denote a lognormal µ, σ 2 r.v. by X ∼ lognorm(µ, σ 2 ).

1.2

Back to our study of geometric BM, S(t) = S(0)eX(t)

def

For 0 = t0 < t1 < · · · < tn = t, the ratios Li = S(ti )/S(ti−1 ), 1 ≤ i ≤ n, are independent lognormal r.v.s. which reﬂects the fact that it is the percentage of changes of the stock price that are independent, not the actual changes S(ti ) − S(ti−1 ). For example L1 L2

def

=

def

=

S(t1 ) = eX(t1 ) , S(t0 ) S(t2 ) = eX(t2 )−X(t1 ) , S(t1 )

are independent and lognormal due to the normal independent increments property of BM; X(t1 ) and X(t2 ) − X(t1 ) are independent and normally distributed. Note how therefore we can re-write S(t) = S0 L1 L2 · · · Ln , (3)

an independent product of n lognormal r.v.s. For example, suppose we wish to sample the stock prices at the end of each day. Then we could choose ti = i so that Li = S(i)/S(i − 1), the percentage change over one day, and then realize (3) as the independent product of such daily changes. In this case the Li are also identically distributed since ti − ti−1 = 1 for each i: ln(Li ) is normal with mean µ and variance σ 2 . Geometric BM not only removes the negativity problem but can (in a limited and approximate sense) be justiﬁed from basic economic principles as a reasonable model for stock prices in an “ideal” non-arbitrage world. Roughly speaking, no one should be able to make a proﬁt with certainty, by observing the past values {S(u) : 0 ≤ u ≤ t} of the stock, and this forces us to consider non-negative models possessing this property. The idea is to force a “level playing ﬁeld”, in which the evolution of the stock prices must be such that the activity of buying or selling stock oﬀers no arbitrage opportunities.

1.3

Geometric BM is a Markov process

Just as BM is a Markov process, so is geometric BM: the future given the present state is independent of the past. S(t + h) (the future, h time units after time t) is independent of {S(u) : 0 ≤ u < t} (the past before time t) given S(t) (the present state now at time t). To see that this is so we note that S(t + h) = S0 eX(t+h) = S0 eX(t)+X(t+h)−X(t) = S0 eX(t) eX(t+h)−X(t) = S(t)eX(t+h)−X(t) . Thus given S(t), the future S(t + h) only depends on the future increment of the BM, X(t + h) − X(t). But BM has independent increments, so this future is independent of the past; we get the Markov property. 2

Also note that {X(t + h) − X(t) : h ≥ 0} is yet again BM with the same drift and variance. This means that given S(t), the future process {S(t)eX(t+h)−X(t) : h ≥ 0} deﬁnes (in distribution) the same geometric BM but with new initial value S(t). (So the Markov process has time stationary transition probabilities.)

1.4

Computing moments for Geometric BM

Recall that the moment generating function of a normal r.v. X ∼ N (µ, σ 2 ) is given by MX (s) = E(esX ) = eµs+ Thus for BM with drift, since X(t) ∼ N (µt, σ 2 t), MX(t)...

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