Excess Distribution of Renewal Process
1. Definition
Now we are going to derive for a random .
Interpretation: You show up “at random”. What is the probability that you wait more than x for the next event?
2. Derivation of
- if
- otherwise
Now we are going to derive for a random .
Interpretation: You show up “at random”. What is the probability that you wait more than x for the next event?
3 partition http://en.wikipedia.org/wiki/3-partition_problem
Choose such that
Then .
Thus the expected present value of the stock at t equals the present value of the stock when it is purchased at time s. That is, the discount rate exactly equals to the expected rate of the stock returns.
First, we drop the subscript i to simplify notation.
Expected present value of the return on this bet is
=
Setting this equal to zero implies that
=
has value 0 when , i.e., when
Thus the integral becomes
=
Now, apply a change of variables:
, i.e.,
1. Introduction
Definition: the simultaneous buying and selling of securities, currency, or commodities in different markets or in derivative forms in order to take advantage of differing prices for the same asset.
2. Option Pricing
Question:
How to choose x and y:
How to maximize profit:
Key Assumption: There is no limit to buying or selling of options. In practice,
you may only be able to buy, but no sell, for example.
3. Aibitrage Theorem
Definition:
Consider n possible wagers on m possible outcomes: .
Let to be the outcome of wagers i if outcome j occurs.
If is bet on wager , then is earned if outcome j occurs.
Arbitrage Theorem:
## TODO: explain more about these two theorem
1. Motivation
Definition: Let be Brownian motion with drift coefficient and variance parameter . Let . Then is geometric Brownian motion.
Motivation: Let be the price of a stock at time (where n is distance); Let be the fractional increase/decrease in the price of the stock from time n-1 to time n.
We suppose that are i.i.d. Then
The process looks like a random walk.
2. Property of Geometric Brownian Motion
Answer: Let , where is standard Brownian Motion.
Let where is standard Brownian motion, is Brownian motion with drift, and is geometric Brownian motion.
=
=
=
=
=
Let denote the state of Markov Chain after n jumps, then
The denote the continuous Markov Chain after n jumps, then
Then we have
and .
Let , then
.
3. Standard Brownian Motion (SBM)
Definition 1: Let be the standard Brownian motion. Let , then is Brownian motion with drift .
Definition 2: . is Brownian Motion with drift and variance parameter if
Example: Let be Brownian motion with and drift . What is Pr{X(30) >0 | X(10) = -3}Pr{X(30) – X(20) >3 | X(10) =3 }Pr{X(30) – X(10) >3 }Pr{X(20) – X(0) >3 }Pr{X(20)>3 }Pr{N(2,80) > 3} = Pr{X(0,1) > frac{3-2}{sqrt{80}}1-Phi(frac{1}{4sqrt{5}})X(s) = x | X(t) = Bfrac{s}{t} cdot Bfrac{s}{t} cdot (t-s)s = t/2X(t)N(0, sigma^2 t)10 after 6 hours, what is the probability that the stack was above its starting value after 3 hours?
Example 2: If a bicycle race between two competitors, Let denote the amount of time (in seconds) by which the racer that started in the insider position is ahead when 100t percent of the race has been completed, and suppose that , can be effectively modeled as a Brownian Motion process with variance parameter .
6. First Passage Time
Let denote the first time that standard Brownian motion hits level a (starting at X(0) = 0), assuming a >0, then we have
Change variables: , we have .
By symmetry, we get .