MTH3251 Exam Notes .pdf

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F INANCIAL M ATHEMATICS
E .MnC1 j Fn / D Mn

EjXn j < 1;

Martingales

E .M / D E .M0 /

 is determined by Fn (stopping time);

Optional Stopping Theorem

Xn D X0 CY1 C  CYn D X0 C

Random Walk

n
X

Yi

i D1

UnC1 D Un Cc XnC1 ;

Insurance Model

C U0

P .T  n/  e
„ ƒ‚ …

C D

;

2.c

X /
2

ruin probability

8
ˆ
ˆ
B t Bs  N.0; t
<Gaussian Increments
Independent Increments B t Bs ?
? Bs
ˆ
ˆ
:Path Continuity
B t continuous

Brownian Motion

s/

Stochastic Calculus
For finite variance:

RT
0

8
R
<Zero Mean, E. T X t dB t / D 0
R0
2 R
T
T
:Isometry,
E 0 X t dB t D 0 E.X t2 / dt
8
2
ˆ
ˆ
<.dt / D dt dt D 0
.dB t /2 D dB t dB t D dt
ˆ
ˆ
:dt dB t D dB t dt D 0

E.X t2 / dt < 1;

Stochastic differentials

df .X t / D f 0 .X t / dX t C 12 f 00 .X t /.dX t /2

Itô formula

d.X t Y t / D X t dY t CY t dX t CdX t dY t

Product Rule
df .X t ; Y t / D

Bivariate Itô

@f
@X t

@f
dX t C @Y
dY t C 12 @
t

2f

2f

@X t2

.dX t /2 C 21 @

@Y t2

2f

.dY t /2 C @X@

t @Y t

.dX t /.dY t /

8R
RT
t
2
ˆ
ˆ
< 0 Xs dBs is a martingale if 0 EX t dt < 1
A martingale differential dM t cannot contain a dt term.
ˆ
ˆ
:A stochastic integral of a martingale is also a martingale.

Stochastic Martingales

Stochastic Differential Equations
Type 1:

dX t D X t dt CX t dB t

Type 2:

dX t D X t dt C dB t

calculate d.ln X t /
calculate d.e

t

Xt /

Options

uS
B INOMIAL M ODEL

(
C1 D

S
dS

(
Portfolio, aS1 Cbr D

K/C

.ST

Option price at maturity

auS C br D Cu
adS C br D Cd

Cu D .uS
Cd D .dS

price goes up
; implies
price goes down
d
d

aD

Alternatively, under a new probability: p D

r
u

n-step Binomial

!
T
1 X T
C D T
p i .1 p/T
i
r

K/C
K/C

Cu Cd
;
.u d /r

C D 1r E .C1 / D

i D0

i

price goes up
price goes down

.ui d T

1
r

i

uCd dCu
;
.u d /r

pCu C .1

S K/C

p/Cd

C D aS Cb


C t D S t .h t / Ke

Black-Scholes Model

where

8
R
2
ˆ
ˆˆ.h/ D h1 p1 e x =2 dx
ˆ
ˆ
2
ˆ
ˆ
<r D continuous interest rate

p
ˆ.h t  T

r.T t /

t/

S t D stock price at time t

 D volatility
T D maturity time of the call T t D remaining time
ˆ
ˆ
ˆ
2
S
ˆ
ˆ
ln. t /C.rC  /.T t /
ˆK D exercise price
:
ht D K p 2


dˇ t D rˇ t dt ) ˇ t D ˇ0 e rt

Riskless Rate

dS t D S t dt CS t dB t

BS Stock Dynamics
SDE Solution
BS PDE

(type 1 SDE)

1 2
2  /t CB t

S t D S0 e .
8
@C
@C
1 2 2 @2 C
ˆ
ˆ
rC D 0
< 2  x @x 2 C rx @x C @t
C.x; T / D payoff.x/
ˆ
ˆ
:C.0; t / D e r.T t / payoff.0/
C De

Option Price

rT

K/C

EQ .ST

Ct D Vt

Replicating Portfolio,

V t D a t Cb t ˇ t

Self-Financing Portfolio
Self-Financing Condition
Z
Solution

T t

V t D V0 C

t

0

dV t D a t dS t Cb t dˇ t
Z t
au dSu C
bu dˇu
0

Girsanov’s theorem: Let B t be a Brownian Motion under P and c be a constant. There exists an equivalent measure Q
such that BO t D B t C ct is a Q Brownian motion.
Equivalent Measure: A probability Q is equivalent P if they agree on what is possible and impossible.
Q.A/ D 0 () P .A/ D 0
Equivalent Martingale Measure:
Completeness: A market model is called complete if any option can be replicated by a self-financing portfolio.
1st Fundamental Theorem: A model does not have arbitrage strategies if there is an equivalent martingale probability
measure Q such that S t e rt is a martingale.
nd
2 Fundamental Theorem: A market model is complete iff there is a unique EMM.
Ct D e

r.T t /

EQ .X j F t /
ˇt D e

Random Savings Account
Bond Price

P .t; T / D e

Forward Rate

f .t; T / D

RT
t

Rt
0

rs ds

r.s/ ds

@ ln P .t; T /
@T

General
Moment Generating Function of X  N.;  2 /;
Indicator Expectation
Minimum Stopping Time Expectation

Brownian Covariance Matrix

Common Martingales

1 2

2

m.t / D E.e tX / D e tC 2 t 

P .A/ D E 1fAg


E .MT ^n / D E MT 1fT ng CE Mn 1fT >ng


 E MT 1fT ng  E 1fT ng D P .T  n/
0
1
t1    t1
B: :
C
:
: : ::: C
†DB
@:
A
t1    tn
8
ˆ
ˆ
<M t D B t
M t D B t2 t
ˆ
ˆ
:M t D e B t 12  2


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