Geometric Brownian Motion Karl Sigman

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)...

Other Essays Like Geometric Brownian Motion - Karl Sigman

The French And Indian War: The "Real" First World War

1955 words - 8 pages The Seven Years War, or more commonly referred to as “The French and Indian War”, has been called the true First World War. In this book The French and Indian War: Deciding the Fate of North America, the author and historian Walter R. Borneman paints a detailed and elaborate picture that justifies the claim of it being the first true war of global proportions. If ever there truly was a climax to the never ending feud of the European powers

Is The Use Of Animals In Medical Research A Necessary Measure?

1513 words - 7 pages Throughout history, animals have been used in experiments to test product safety and obtain medical knowledge that benefits both humans and animals alike. Every year there are numerous medical breakthroughs, such as medications and surgical instruments, which are tested on animals to insure their safety before they are deemed acceptable for human use. Even though the results of the experiments saved millions of human lives, they are also

Education And The Evolving Job Market

2363 words - 10 pages The lives of students today are changing. They are preparing for lives and jobs that have never before existed. If teachers hope to have a significant and worthwhile impact on these quickly changing lives, they must change the way they think, prepare, and instruct our future generations. Children cannot afford to have teachers who remain stagnant in their methods and ideals. Students crave instructors that are willing to allow them to tap

Young And Relentless

1737 words - 7 pages There are numerous influences that can be responsible of teenager’s behaviors and attitude as they develop. One factor that is important to these behaviors is parental figures being over involved or uninvolved in their children’s lives. Many of these effects include illegal substance abuse, rising sexual activity, underage alcohol consumption, and tobacco use. Studies show parental participation plays a key role in the characteristics developed

The Natural Law Theory

1231 words - 5 pages arguments of all. Aquinas first looks at motion which is a similar development to whatever is here derived from something else, this leads to Aquinas’ first proof the proof from motion, and it is the most obvious case to prove: We observe motion all around us. Whatever is in motion now was at rest until moved by something else, and that by something else, and so on. But if there were an infinite series of movers, all waiting to be moved by

Resolved: Presidential Signing Statements Threaten To Undermine The Rule Of Law And The Separation Of Powers

1811 words - 8 pages The subject of signing statements has created much debate among the houses of Congress, government officials, and the public alike. These signing statements fall under the categories of constitutional and legislative history signing statements. Constitutional signing statements are those in which the president deems certain provisions of the legislation as unconstitutional, therefore they should not be enforced (Bradley & Posner, 2006

Oppressive Systems Of Government In Egypt And Animal Farm

1529 words - 7 pages As in Egypt, Orwell demonstrates through his allegorical novel “Animal Farm” that leaders are able to establish and maintain power over a people, and in turn create an oppressive and corrupt government system. Orwell shows the significant difference in the education and levels of knowledge in the animals, and how the government takes advantage of this difference. The split between the levels of intelligence is portrayed in the first chapter when

The Pathway To Psychosis

1415 words - 6 pages “How all occasions do inform against me” is a line from act IIII, scene IIII of William Shakespeare’s Hamlet. This line, spoken by Hamlet, expresses his emotional state as he is currently overwhelmed by the death of his father, the king of Denmark, and the situation surrounding it. After Hamlet learns of his father’s death he finds out that his mother has married Claudius, Hamlet’s uncle. On top of all of that, Hamlet soon after

Rated “M” For “More Censorship Not Needed”

1241 words - 5 pages rated movie are far worse than the content of an M rated game, the Motion Picture association Of America states that an R rated movie may contain “adult themes, Adult activity, hard language, intense or persistent violence, sexually-oriented nudity, drug abuse, or other elements”. All of the possible content for an R rated movie is far worse than an M rated video game but yet games get all the flak for inducing violence and unruly behavior in

Four Components Of A Legally Astute Social Media Marketing Manager

1914 words - 8 pages With the advances in technology constantly changing companies have to find better ways to market their products to consumers. The explosion of Facebook has given companies a way to market products to consumers. Examine how a legally astute manager can use social media to his advantage. Review different forms of dispute resolution and determine which one works the best. What is the best course of action, the government can use to

Obama's Values

1217 words - 5 pages As individuals, we don’t tend to take action unless it’s for self-interest; however, in Dreams from My Father, Obama spends three years, after college, as a community organizer in Chicago. Obama goes through tribulations, disappointments, and even complete failures organizing meaningful events, decisive meetings, and humble gatherings but he keeps working toward achieving any possible change in the community. Even though results give him every

Related Papers

The Separation Of Capital Ownership And Control

1577 words - 7 pages The argument of whether the separation of capital ownership and control is an efficient form of organization has constantly been a controversial issue. The criticism whether the controllers’ act is in the best interest of the owners’ wills never end as long as hired managers operate management. As the number of public companies has been increasing over the course of this century, meanwhile the American style of contact based corporation has

The Versatility And Flexibility Of Oled's

1014 words - 5 pages In April 1, 2002, organic light emitting diodes gain rise in the scientific community with their published, more practical form at Ames Laboratory. “Scientists at the U.S. Department of Energy's Ames Laboratory, in collaboration with scientists at the University of Michigan, Ann Arbor, have developed and demonstrated a novel, fluorescence-based chemical sensor that is more compact, versatile and less expensive than existing technology of its

Comparing The Moral Virtues Of Antony And Julian The Apostate

1103 words - 5 pages Roman emperor Julian the Apostate and Christian leader Antony both exhibited many qualities of character during their existence. Both of them led very distinctive lives although shared several ethical values. Book 25 of “The Later Roman Empire” and the book “Early Christian Lives” show concrete evidence of this. In the following essay, I will argue how both leaders’ lives were devoted to their religious beliefs and their mutual cardinal virtues

Living In A Cashless Society Essay

1637 words - 7 pages Money in a traditional sense no longer exists. Money is becoming much of a concept than a physical material, and most ordinary bitter have not see the reality of the switch. People today are using credit and debit cards on a regular basis and in everyday situations such as meal purchased at fast food, highway tolls, clothing, groceries, gas stations, etc. all of these means of systems could be regarded as a cashless society or world. The question