Lecture 23 Option Pricing II
Prof. Mark Brown
April 25, 2016
Prof. Mark Brown
Lecture 23 Option Pricing II
April 25, 2016
1 / 15
Back to BlackScholes
Back to BlackScholes
For a call on a stock with a strike price K, volatility , time to
expiration, T an
Journal of Quantitative Analysis in
Sports
Volume 6, Issue 3 2010' Article 4
An Improved LRMC Method for NCAA
Basketball Prediction
Mark Brown, City College, City Universily of New York
Joel Sokol, Georgia Institute of Technology
Recommended Citation:
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STAT W4606: Elementary Stochastic Processes
Lecture 1
January 20th, 2015
A quick review of probability theory
I
Random Variable: Chapter 2
I
Conditional probability: Chapter 3
I
Some useful probability formulas:
P(E [ F ) =
P(E ) + P(F )
P(EF )
P(E F ) =
STAT W4606: Elementary Stochastic Processes
Lecture 2
January 22nd, 2015
Recurrence/Transience: class property
Corollary 4.2 If i $ j, then either both i and j are recurrent or
both are transient.
Proof of recurrence:
If i $ j, then 9 k, m 2 Z , s.t. Pijk
Stochastic IntegralOptions pricing notes
lecture 2325
Stochastic Integration  Option Pricing
We will give a heuristic presentation and later fill in more mathematical detail. Let cfw_B(t), t 0
be standard Brownian motion. The notation, dB(s) = Bs+ds Bs
STAT W4606: Elementary Stochastic Processes
Lecture 21
April 14th, 2015
Maximum of Brownian Motion
For a standard Brownian motion, define
M(t) = sup X (s).
0st
We will show that,
2yp x
x
p
(i) P(M(t) y , X (t) x) =
+
t
t
for x y , y > 0;
y
p
(ii) P(M(
Lecture 17
Prof. Mark Brown
March 30, 2016
Prof. Mark Brown
Lecture 17
March 30, 2016
1 / 17
Martingales
Martingales
Consider an increasing sequence cfw_Fn , n 1 of algebras. In
applications, Fn will be the information contained in a collection of
random
Lecture 16
Prof. Mark Brown
March 28, 2016
Prof. Mark Brown
Lecture 16
March 28, 2016
1 / 22
Conditional Expectations
Conditional Expectations
Consider random variables X, Y with joint pmf(probability mass
function), fX,Y (x, y) = P r(X = x, Y = y). For e
Lecture 15
Prof. Mark Brown
March 23, 2016
Prof. Mark Brown
Lecture 15
March 23, 2016
1 / 17
More on Ergodicity
More on Ergodicity
For an irreducible, positive recurrent Markov chain in discrete time,
1) j =
1
> 0, where
ENj,j
ENj,j = cfw_min : k 1, such
W4606 HW3 Chap6
CG2908
Chenying Gao
6. 4. Potential customers arrive at a singleserver station in accordance with a Poisson process with
rate . However, if the arrival finds n customers already in the station, then he will enter the system
with probabili
Lecture 17
Prof. Mark Brown
March 30, 2016
Prof. Mark Brown
Lecture 17
March 30, 2016
1 / 17
Martingales
Martingales
Consider an increasing sequence cfw_Fn , n 1 of algebras. In
applications, Fn will be the information contained in a collection of
random
Lecture 16
Prof. Mark Brown
March 28, 2016
Prof. Mark Brown
Lecture 16
March 28, 2016
1 / 22
Conditional Expectations
Conditional Expectations
Consider random variables X, Y with joint pmf(probability mass
function), fX,Y (x, y) = P r(X = x, Y = y). For e
Lecture 20 Martingales and Brownian Motion
Prof. Mark Brown
April 13, 2016
Prof. Mark Brown
Lecture 20 Martingales and Brownian Motion
April 13, 2016
1 / 14
Useful Martingales
Let cfw_X(s), s 0 be standard Brownian motion, and define, cfw_Ft =
(X(s), 0 s
Lecture 19
Prof. Mark Brown
April 11, 2016
Prof. Mark Brown
Lecture 19
April 11, 2016
1 / 19
Brownian motion I
Brownian motion I
A standard Brownian motion process, cfw_X(t), t 0, is defined by,
(i) X(0) = 0
(ii) Stationary increments
(iii) independent in
Lecture 25 Option Related Topics
Prof. Mark Brown
May 2, 2016
Prof. Mark Brown
Lecture 25 Option Related Topics
May 2, 2016
1 / 16
Product rule
We first briefly discuss some stochastic calculus issues relevant to
finiancial math.
Product rule
Suppose that
Preface
This text is intended as an introduction to elementary probability theory and stochastic
processes. It is particularly well suited for those wanting to see how probability theory
can be applied to the study of phenomena in fields such as engineeri
Lecture 24
Prof. Mark Brown
April 27, 2016
Prof. Mark Brown
Lecture 24
April 27, 2016
1 / 12
Original BlackScholes derivation
Original BlackScholes derivation
The famous BlackScholes paper used a bit of stochastic calculus,
combined with pde theory, an
STAT W4606: Elementary Stochastic Processes
Lecture 15
March 24th, 2015
Consider an ergodic continuous time Markov chain. Denote by
Ti,j , the waiting time starting in state i to reach state j (Tii = 0,
as we are already in state i). Suppose that
P the ch
STAT W4606: Elementary Stochastic Processes
Lecture 22
April 16th, 2015
We now give several Brownian motion results derived via
martingales.
1) For standard Brownian motion define T to be the first time
that either b or a is hit, whichever comes first,
(a
STAT W4606: Elementary Stochastic Processes
Lecture 4
February 3rd, 2015
Variation of above
Let B S, C S be subsets of the state space with B \ C =
and suppose that P(TB[C < 1) = 1 for i 2 B c \ C c .
We wish to compute, i = P(Hit B before CX0 = i).
P
e
STAT W4606: Elementary Stochastic Processes
Lecture 13
March 5th, 2015
More General Chains
Suppose that, in the Markov Chain,
(h)
pij = P(X (t + h) = j X (t) = i) = qij h + o(h), j 6= i.
(h)
pii
= P(X (t + h) = i X (t) = i) = 1
h
X
qji + o(h),
j6=i
where
STAT W4606: Elementary Stochastic Processes
Lecture 16
March 26th, 2015
Time Reversing
Consider an ergodic continuous
time Markov chain with
P
infinitesimal matrix Q(
q
= Vi ). Note that in steady
ij
j6=i
state (X (t) , for all t),
(j)Pji (t)
Pij (t) , P(