MATH 6910: REVIEW
I expect that you are familiar with the following basic concepts from undergraduate probability.
These are covered quite nicely in the Rice text (Mathematical Statistics and Data Analysis), chapters 15, say.
Basic probability: axioms,
@
"Xn W Q J i
rs
L
k
a X in f "
cxne
xYl
+0
@
n3m.i

XI 7 I'a
and
Xn,
t t
\ID
(ind~pac~deult
icb&cd$
Jz = ELXJ ~ k f 3
i, if
X n 3
X
fix,)
a  ~ +la&
3
q)
$
aS.
.
1fy
2
Xn
Y
3 X a,
+b X n t L 3 X t Y
as.
NB.
 & b~
ass.
~
J
"R" J
6 STOc ~ A ~7r PROCES
L, i F a so\+'n a i b k
, ~lneu, it's
0 0
a u&
9/
SJ,=
~lk'n
q t t aBt z~ & t
L
e
sop
1 d & , d 1 
O L ~ L , ~
G
FIN A N c ~ SWFF :
C
c=uallle
4
q0t;bn
ort h i t
GENERATING SAMPLE PATHS OF A BROWNIAN MOTION
To do this part of the problem set you will need to download and install the FREE program R from
http:/cran.rproject.org/ (you want the base package).
1. Open R. You should see a small window called the R Cons
MATH 6910: PROBLEM SET X  MONDAY, MAY 4th
No solutions will be provided to this problem set.
1. Exercise (4) from the review problem set.
2. Read Section 6.6 in the text, and especially Example 6.6.1  pricing an Asian option. Why do
we need the multidi
MATH 6910: PROBLEM SET IX  FRIDAY, 19th
RiskNeutral Pricing:
The goal of this section (together with the same section of Problem set XIII) is to work through
the ideas presented in class, but using the market model with the more general formulas. That
i
MATH 6910: PROBLEM SET VII  FRIDAY, MARCH 5th
Please hand in questions 4.13 and 4.18 for next lecture.
1. Consider the SDE
dXt = Xt dt + dBt , X0 = x.
(a).
(b).
(c).
(d).
Find a solution to this SDE.
Is your solution weak or strong?
Is your solution uniq
MATH 6910: PROBLEM SET III  MONDAY, MARCH 23rd
Taking Limits
p
1. Prove that if Xn X in L2 , then Xn X . Hint: look at the proof we did in class of the LLN.
2. Suppose that X1 , . . . , Xn are IID Cauchy random variables (recall the pdf and characteristi
MATH 6910: PROBLEM SET IV  FRIDAY, JANUARY 29
2
1. (1) Show that Bt t is a martingale with respect to the canonical ltration of Brownian motion.
(2) Show that (Nt t)2 t is a martingale with respect to the canonical ltration of the
Poisson process.
HINT:
MATH 6910: PROBLEM SET II  MONDAY, MARCH 16th
More General Probability
1. For the normal random variable X , show that MX (0) = 1, MX (0) = and MX (0) = 2 + 2 .
Recall that this property is true in general, and is the origin of the name moment generating
MATH 6910: PROBLEM SET IX  MONDAY, APRIL 27th
Itos formula (even more practice):
1. In each of the following problems, nd the differential satised by Xt by using Itos formula. In
each case, take both of two approaches: (1) solve the problem directly, (2)
MATH 6910: PROBLEM SET IX  FRIDAY, 19th
RiskNeutral Pricing:
Suppose that Dt = exp
t
0
Rs ds is the discount process and that the stock model is a generalized geometric Brownian motion dSt = t St dt + t St dBt , where t , t are adapted processes.
1. Wh
MATH 6910: PROBLEM SET IV  MONDAY, MARCH 30th
Brownian Motion
1. Write down the distribution of cfw_Bt1 , . . . , Btk , where t1 < t2 , . . . , < tk . Note that you have
actually described the nite dimensional distributions of Brownian motion.
2. Let > 0
MATH 6910: ASSIGNMENT ONE
This assignment is due on Monday, March 30th at 9am. No late assignments will be accepted.
1. Give (your own) heuristic denitions of the following concepts
(i). independence
(ii). martingale
(iii). algebra
(iv). conditional expe
MATH 6910: PROBLEM SET VIII
Itos formula (extra practice):
1. In each of the following processes, use Itos formula to write the process as the sum of a
drift and a martingale. Do not present your nal answer in the shorthand differential notation
(1)
(2)
MATH 6910: ASSIGNMENT 1
DUE THURSDAY FEBRUARY 5TH (see details on website)
N.B. Only three randomly selected questions will be marked.
1. In the binomial model, show that arbitrage exists if (1 + r) d for the oneperiod setting.
2. Let = [0, 1) and let F
MATH 6910: PROBLEM SET VIII  SATURDAY, APRIL 25th
Itos formula (extra practice):
1. In each of the following processes, use Itos formula to write the process as the sum of a
drift and a martingale. Do not present your nal answer in the shorthand differe
York University
GS/MATH 6910 3.0MW, Winter 2007
STOCHASTIC CALCULUS IN FINANCE
Course Outline
Prerequisites:
Calculus and some basic probability. In particular, Measure Theory is not assumed.
This course is designed (and required) for theDiploma in Financ
MATH 6910: STOCHASTIC CALCULUS IN FINANCE  WINTER 2009
This course will introduce the basic ideas and methods of stochastic calculus and apply these
methods to nancial models, particularly the pricing and hedging of derivative securities. We
start by int
=
[3tf
4
qf d t
\
t
idB,
f
GGBM
ts
la
[it\
dx,  X
,
C
d1,
t
a
Xt (dlT)
2
"9
G8M
dSt= qStdt
I
i
&dBt
So =


St =
_
is . 
._

 
y t t 68 I&
t; z
xe
1
,
MATH 6910 3.0AF (Stochastic Calculus in Finance)
Assignment 2 Solutions
March 2007 Salisbury
1. Recall that if
dUt = At dt + Ht dBt
dVt = Ct dt + Kt dBt
then d[U, V ]t = Ht Kt dt.
(a) Since
1
df (Bt ) = f (Bt ) dBt + f (Bt ) dt,
2
it follows that d[f (B),
NbOhhC.
Mu\& panod %;w&ae
dd.
M
/
Sou
srrwe
0
"'I
%S1c'fl\
$GP@
0s
bpQ*
VK=
 " [cfw_is2
I
E
~ k f ] ~  ~
si,
,sc.
?
(
a
t
4
bs 2 s k u
 I  P@
CL
bd
b  d q 5 dff)
36588b'lK
* * *
365'
g (36Sntl
L"1
adbc 
a
Moment Generating Function
dk
]
m
Mx(t) = ~ ( e " ) zMx(t)lt=, = E [ x ~ ST.: ~ ( e< ~ )
Normal distribution
Normal@, u 2 )
1
e&(xp)zM(t) = e.tt+tzO(t) eipt+tz
=
f (XI =
Linear transformations of normal:
XNn(p,E); Y = AX b, then YN,(Ap
+
Stochastic Calculus in Finance
MATH 6910  Salisbury
Girsanov Transformations  A short introduction
(1) Two probability measures P and P are equivalent on a sigma
eld G if
P (A) = 0 P (A) = 0
for every A G. The RadonNikodym theorem (from measure
theory)
MATH 6910 3.0AF (Stochastic Calculus in Finance)
Practice problems Solutions
March 2007 Salisbury
1. Let f (x) = log x, for x > 0. Then f (x) = 1/x and f (x) = 1/x2 . From the
2
fact that dSt = t St dt + t St dWt we get that d[S, S]t = t St2 dt. So
1
d lo
Stochastic Calculus in Finance
MATH 6910  Salisbury
Introduction to American Options
(1) An American Option is one that the owner can exercise at any time t T ,
rather than only at time T as is the case with a European option. If exercised
at time t the