By Darrell Duffie and Peter Glynn, Stanford University, 1995
Paper Review by:
Niv Nayman
Technion- Israel Institute of Technology
July 2015
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
Definition. We say that a random process,  X t : t  0, is a Brownian motion
with parameters (µ,σ) if
1. For 0 < t1 < t2 < …< tn-1 < tn
X
t2



 X t1 , X t3  X t2 , ... , X tn  X tn1

are mutually independent.
2. For s>0,  X s t  X t 
N   s,  2 s 
3. Xt is a continuous function of t.
We say that Xt is a B (µ,σ) Brownian motion with drift µ and volatility σ
Remark. Bachelier (1900) – Modeling in Finance
Einstein (1905) – Modeling in Physics
Wiener (1920’s) – Mathematical formulation
 When µ=0 and σ=1 we have a Wiener Process
or a standard Brownian motion (SBM) .

We will use Bt to denote a SBM and we always assume that B0=0
Then,

Bt   Bt  B0 
N  0, t 
Note that if Xt ~B (µ,σ) and X0=x then we can write
X t  x  t   Bt
Where Bt is an SBM.
Therefore see that
Xt
N  x   t ,  2t 
Since,
E  Xt 
 x  t    E  Bt   x  t
Var  X t    2 Var  Bt    2t
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
Definition. A partition of an interval [0,T] is a finite sequence of numbers πT
of the form
 T : [(t1 , t2 , , tn1 , tn ) | 0  t1  t2   tn1  tn  T ]
Definition. The norm or mesh of a partition πT is the length of the longest of it’s
subintervals, that is
h :  T  max  ti  ti 1 
i 1,...,n
Recall the Riemann integral of a real function µ
n
   s, X  ds  lim    t
t
s
0
h 0
i 1
i 1
, X i 1  ti  ti 1 
And the Riemann-Stieltjes integral of a real function µ w.r.t a real function gt=g(t)

t
0
  s, X s  dg s  lim    ti 1 , X i 1   gt  gt
n
h 0
i
i 1
i 1

Then the Itō integral of a random process σ w.r.t a SBM Bt
   s, X  dB
t
0
s
s
n

 lim    ti 1 , X i 1  Bti  Bti1
h 0
i 1

It can be shown that this limit converges in probability.
A stochastic differential equation (SDE) is of the form
dX t    t , X t  dt   t , X t  dBt
Which is a short-hand of the integral equation
X t  X 0     s, X s  ds     s, X s  dBs
t
t
0
0
Where

Bt is a Wiener Process (SBM).

   s, X  ds

   s, X  dB
t
0
is a Riemann integral.
s
t
0
s
s
is an Itō integral.
An Itō process in n-dimensional Euclidean space
X : 0,T  
is a process
Xt 
n
defined on a probability space  , , P 
n
and
satisfying a stochastic differential equation of the form
dX t    t , X t  dt   t , X t  dBt
Where

T>0

Bt is a m-dimensional SBM

 : 0,T 

 : 0,T 
Such that

n
n

C , D 
n
satisfies the Lipschitz continuity and polynomial growth conditions.
nm
satisfies the Lipschitz continuity and polynomial growth conditions.
: t   0, T  ; x, y 
n
 t, x    t, y    t, x    t, y   C x  y
Where
 :  i
2
i
2
and
 :   ij
2
i, j
2
;   t , x     t , x   D 1  x

.
These conditions ensure the existence of a unique strong solution Xt to the SDE (Øksendal 2003)
A time-homogeneous Itō Diffusion in n-dimensional Euclidean space
X : 
is a process
Xt 
n
defined on a probability space  , , P 
n
and satisfying a stochastic differential equation of the form
dX t    X t  dt    X t  dBt
Where

Bt is a m-dimensional SBM

:
n

n

:
n

nm
Meaning
Where
is Lipschitz continuous.
is Lipschitz continuous.
C 
 :  i
2
i
2
s.t. x, y 
and
  x    y    x   y  C x  y
n
 :   ij
2
i, j
2
.
This condition ensures the existence of a unique strong solution Xt to the SDE
(Øksendal 2003).
Suppose Xt is an Itō process satisfying the SDE dX t    t , X t  dt   t , X t  dBt
And f(t,x) is a twice- differentiable scalar function.
Then
 f
f  2  2 f 
f
df     
dt


dBt
2 

t

x
2

x

x


In more detail
2
 f

f
1
f
2  f
df  t , X t     t , X t     t , X t   t , X t     t , X t 
t
,
X
dt


t
,
X




t , X t  dBt

t
t
2

t

x
2

x

x


Sketch of proof
The Taylor series expansion of f(t,x) is
Recall that
dBt   Bt  dt  Bt  ~ N  0, dt   O
Substituting x  X t and

dx   dt   dBt
f
f
1 2 f 2
df  dt  dx 
dx  ...
t
x
2 x 2
dt

gives
f
f
1 2 f
df 
dt    dt   dBt  
 2 dt 2  2 dtdBt   2 dBt 2   ...
2 
t
x
2 x
As dt  0 the terms dt 2 and dtdBt  O  dt1.5  tend to zero faster.
Setting dBt 2  dt and we are done.
Suppose Xt satisfies following SDE dX t    t , X t  dt   t , X t  dBt
With   t , X t    X t and   t , X t    X t
Thus the SDE turns into dX t   X t dt   X t dBt
Since f  t , X t   log  X t  is a twice- differentiable scalar function in (0,∞)
By Itō Lemma we have
Then
 f
f  2  2 f 
f
df     
dt  
dBt
2 

t

x
2

x

x


d log  X t 

 1
  0   X t 
 Xt

The integral equation
Finally
log  X t 
  2 Xt2  1  
 1


dt


X




t 
2  X t 2  

 Xt


2 
 dBt    
 dt   dBt
2



t


2 
2 
 log  X 0      
 ds  0  dBs  log  X 0     
 t   Bt
0
2
2




X t  X 0e
t
 2 
   t  Bt
2 

If X0 >0 then Xt > 0 for all t> 0
Definition. We say that a random process,  X t : t  0 , is a geometric
Brownian motion (GBM) if for all t≥0
X t  X 0e
 2 
   t  Bt
2 

Where Bt is an SBM.
We say that Xt ~GBM (µ,σ) with drift µ and volatility σ
Note that
X t  s  X 0e
 X 0e
 Xte
 2 
    t  s   Bt  s
2 

 2 
 2 
   t  Bt      s   Bs t  Bt 
2 
2 


 2 
    s   Bs t  Bt 
2 

-a representation which we will later see that is useful for simulating security prices.
Define
Et  : Et  | t 
, where
t
denotes the information available at time t.
Recall the moment generating function of
Y
M Y  s   E e   e
sY
N   , 2 
1
2
 s   2 s2
Suppose Xt ~GBM (µ,σ) , then
 2 

    s   Bs t  Bt  
2 

Et  X t  s   Et  X t e




 Xte
 Xte
 2 
    s
2 

 ~ N 0,s  
 B B
Et e  st t  




 2  2
    s  s
2 
2

 X t es
- So the expected growth rate of Xt is µ
Note that,
X t s
e
Xt
 2 
    s   Bt  s  Bt 
2 


 X t s  
2 
log 
   
 s    Bt  s  Bt 
X
2

 t  
Thus
1. Fix t1 < t2 < …< tn-1 < tn . Then
X tn
X t2 X t3
,
,...,
X t1 X t2
X tn1
are mutually independent.
2. Paths of Xt are continuous as a function of t, i.e. they do not jump.
3. For s>0,
X
log  t  s
 Xt



2 
2
~
N


s
,

s




2




- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
The Black-Scholes equation is a partial differential equation (PDE) satisfied by a
derivative (financial) of a stock price process that follows a deterministic volatility
GBM, under the no-arbitrage condition and risk neutrality settings.
Starting with a stock price process St : t  0 that follows GBM
dSt   St dt   St dBt
 St  S0e
 2 
   t  Bt
2 


We represent the price of the derivative of the stock price process as f  t , St   ert  St  K 
Where f : 2 
a twice- differentiable scalar function at St>K.
r – interest rate
K- strick price
We apply the Itō lemma
 f
f  2 St 2  2 f 
f
df     St

dt   St
dBt
2 

t

s
2

s

s


The arbitrage-free condition serves to eliminate the stochastic BM component,
leaving only a deterministic PDE.
 f
f  2 St 2  2 f 
df     St

dt
2 

t

s
2

s


f
f  2 St 2  2 f
 rSt

 rf
2
we have the celebrated BS PDE: t
s
2 s
Risk neutrality is achieved by setting µ = r
With
df
 rf
dt
A more elaborated model, dealing with a stochastic volatility stock price process
that follows a GBM
dSt   St dt  vt St dBtS
Where vt , the instantaneous variance, is a Cox–Ingersoll–Ross (CIR) process
dvt     vt  dt   vt St dBtv
S
v
and Bt , Bt
are SBM with correlation ρ, or equivalently, with covariance ρdt.
 μ is the rate of return of the asset.
 θ is the long variance, or long run average price variance:
as t tends to infinity, the expected value of νt tends to θ.
 κ is the rate at which νt reverts to θ.
 σ is the volatility of the volatility.
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
Suppose X t 
n
is an Itō process satisfying the SDE
dX t    t , X t  dt   t , X t  dBt
and one is interested in computing
E  f  X T  
l
Sometimes it is hard to analytically solve the SDE or to determine its distribution.
Although numerical computations methods are usually available
(i.e. the Kolmogorov backward equation (KBE) via some finite-difference algorithm),
in some cases it is convenient to obtain a Monte Carlo approximation
l
This requires simulation of the system.
1
N
 f  Xˆ   
N
i 1
i
T
The Monte Carlo realizations are done by simulating a discrete-time approximation of
the continuous-time SDE.
The time interval [0,T] is sampled at periods of length h>0.
h
Denote Xˆ as Xt evaluated at t=kh.
k
Thus Xˆ Thh is the discrete evaluation of XT.
Denote the approximation of
l
E  f  X T  
by lh E  f  Xˆ h  .
T
h
And the approximation error by eh lh  l .
eh  0 .
It can be shown, under some technical conditions, that lim
h 0
Definition. A sequence eh  has order-k convergence if eh h  k is bounded in h.
The integral equation form of the SDE
X t h  X t  
t h
t
t h
  s, X s  ds     s, X s  dBs
t
The Euler method approximates the integrals using the left point rule.

t h
t
  s, X s  ds    t , X t   
  t, X t   h
t h
t

ds
t h
t
  s, X s  dBs    t , X t   
t h
t
dBs
   t , X t  Bt  h  Bt 
   t , X t  h
With ε~N(0,1).
h
Thus, one can evaluate XT with Xˆ Thh by starting at Xˆ 0  x and proceeding by



Xˆ kh1  Xˆ kh   kh, Xˆ kh  h   kh, Xˆ kh

h k 1
For k=0,…, T/h-1.
Where ε1,ε2,… are N(0,1) i.i.d.
h
Note. The processes Xt and Xˆ T h are not necessarily defined on the same
probability space (Ω,Σ,Ρ).
Is said to be a first-order discretizaion scheme



Xˆ kh1  Xˆ kh   kh, Xˆ kh  h   kh, Xˆ kh

h k 1
In which eh has order-1 convergence.
There is an error coefficient  h 
s.t. eh   h h has a order-2 convergence.
The error coefficient gives a notion of bias in the approximation.
Although usually unknown it can be approximated to first order by
h 
2  l2 h  lh 
h
Under the polynomial growth condition of f (Talay and Tubaro, 1990).
The integral equation form of the SDE
X t h  X t  
t h
t
t h
  s, X s  ds     s, X s  dBs
t
  t , X t     X t  : t
The Milshtein method apply the Itō’s lemma on   t , X t     X t  :  t .

d t  


d t  

t
t  t 2  2 t 
t
1

2
 t

dt


dB

'



''


t
t
t
t  dt  t ' t dBt
 t t
t
x
2 x 2 
x
2


 t
 t  t 2  2 t 
 t
1


 t

dt   t
dBt   t ' t   t '' t 2  dt   t ' t dBt
2 
t
x
2 x 
x
2


The integral form
s
1


t
t
2


s
s
1

 s   t     u ' u   u '' u 2  du    u ' u dBu
t
t
2


s
 s  t    u ' u  u '' u 2  du   u ' u dBu
Then the original SDE turns into
X t h  X t  
t h
t

t h
t
s
s

1

2



'



''

du


'

dB
u
u
u  ds
 t t  u u 2 u u 

t




s
s

1

2



'



''

du


'

dB
dBs
t
u
u
u
u
u
u
u






t
t
2




As h  0 the terms dsdu  O  h 2  and dsdBu  O  h1.5  are ignored.
Thus
X t h  X t  
t h
t h
t h
t
t
t ds    t dBs  
t
s

t
u
' u dBu dBs
Applying the Euler approximation to the last term we obtain
t h
 
t
s
t
 u ' u dBu dBs   t ' t 
t h
t

  t ' t

t
t h
t
s
dBu dBs   t ' t 
t h
t
 Bs  Bt  dBs

Bs dBs  Bt  Bt  h  Bt    t ' t

t h
t
Bs dBs  Bt Bt  h  Bt 2

1 2
 Bt  t 
2
.
Define dYt  at t , Yt  dt  bt t , Yt  dBt  Bt dBt and suggest a solution
Indeed,
Yt 
 Yt
Yt bt 2  2Yt 
Y
1
 1

dYt  
 at

dt  bt t dBt     0  11 dt  1 Bt dBt  Bt dBt
2 
B 2 B 
B
2
 2

 t
Thus,

t h
t
and
t h
s
 
t
t
u
Bs dBs  
t h
t
dYs  Yt  h  Yt  Yt 
1
1
Bt  h 2  Bt 2   h

2
2
1
1
2
1
 1
' u dBu dBs   t ' t   Bt  h 2  Bt 2   h  Bt Bt  h  Bt 2    t ' t  Bt  h  Bt   h    t ' t h  2  1

 2
2
2
 2
Thus
t h
 
t
t
s
1
2
1
 1
Bt  h 2  Bt 2   h  Bt Bt  h  Bt 2    t ' t  Bt h  Bt   h 



2
2
 2
 u ' u dBu dBs   t ' t 
Then
X t h  X t  
t h
t
t h
t h
t
t
t ds    t dBs  
s

t
u
' u dBu dBs
Turns into the Milshtein discretization
X t  h  X t  t h   t
With ε~N(0,1).
1
h   t ' t h   2  1
2
Thus, one can evaluate XT with Xˆ h by starting at Xˆ 0h  x and proceeding by
T
 
h
 
Xˆ kh1  Xˆ kh   Xˆ kh  h   Xˆ kh
For k=0,…, T/h-1.
Where ε1,ε2,… are N(0,1) i.i.d.
   
1
h k 1   ' Xˆ kh  Xˆ kh h   k 12  1
2
If the terms dsdu  O  h 2  and dsdBu  O  h
3
2

are to be visible.
We have
Where
And
Remark. Euler and Milshtein deal with first and second-order discretization
schemes respectively for 1-D SDE with X t  .
as Talay (1984,1986) provides second-order discretization schemes for
n
multidimensional SDE with X t  : n  1 .
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
The Monte Carlo realizations are done by simulating a discrete-time approximation of
the continuous-time SDE.
Denote,
h (i )
 Yi Xˆ T h as the i’th discrete-time simulation output.
 l  h, N 
 e  h, N 
1
N
N
 f Y 
i 1
i
as a discrete-time crude monte carlo approximation of l
l  h, N   l as the approximation error .
N
 .
h
 
The time required to compute l  h, N  is roughly proportional to n 2T 
Where n and T are fixed for a given problem.
This paper pursues the optimal tradeoff between N and h
given limited computation time.
E  f  X T  
.
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
This paper provides an asymptotically efficient algorithm for the
allocation of computing resources to the problem of Monte Carlo
integration of continuous-time security prices. The tradeoff between
increasing the number of time intervals per unit of time and increasing
the number of simulations, given a limited budget of computer time, is
resolved for first-order discretization schemes (such as Euler) as well as
second- and higher order schemes (such as those of Milshtein or Talay).
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
f is C  and satisfies the polynomial growth condition.
 
ˆh
i. f X T h  f  X T 
 
h

ii. E  f Xˆ T h

2
as h
 E f X 
T


0 .
2
p
p
iii. lh  l   h  O  h  as h
iv.
   as h
0
(i.e.


f Xˆ Thh

2
 is uniformly integrable ).
h0
0 , where β≠0 and p>0.
h
The (computer) time required to generate f  Xˆ  is given the deterministic
T
 h   h  q  O  h  q  as h
h
0 , where γ>0 and q>0.
Given t units of computer time.
1
i.
If ht t
 q2 p 
  or ht t
1
 q2 p 
 0 as t   then
p
t
1
ii.
If ht t
p
t
 q2 p 
 q2 p 
 q2 p 
e  ht , Nt   
 c , where 0<c<∞, as t   then ht t
 c
e  ht , Nt    
q
1
2
  cp
as t   .
1
 q2 p 
0
as t   , where ε is N(0,1) and
 2  Var  f  X T  
Assuming the discritization error alone has order-p convergence with h lh  l  O  h p    h p
As the computation budget t gets large t  
1
the period ht should have order-1/(q+2p) convergence with t. ht t  q2 p   c
Then the estimation error has order-p/(q+2p) convergence in probability with t.
p
t
1
 q2 p 
 c
e  ht , Nt    
1
q
2
  cp
If it the above does not hold, ht t    0 /  the estimation error does not converge in
distribution to zero “as quickly” (i.e. not at this order t   e  ht , Nt    ).
q2 p
p
q2 p
It follows that this rule is “Asymptotically Optimal”.
1
Informally, if ht t
q2 p
 t  c , where t  Nt   ht   N t   h  q  O  h  q  .
 c then ht
 q2 p 
ht 2 p N t  Const
Thus,
Let
T 
 
 ht 
nt
be the number of time intervals, as ht is the period.
For the Euler scheme (p=1) :
For the Milshtein or Talay schemes (p=2):
nt : 2  nt 
 ht :
1
nt : 2  nt 
 ht :
1
1
For the asymptotically optimal allocation, if ht t
Thus
p
t
Thus,
 q2 p 
e  ht , Nt   ht
p
 q2 p 
h t  e h , N   h h t 
1
t
 q2 p 
p
1
p
t
t
t
t
e  ht , Nt   ht p
As the root-mean-squared estimation error is
For the Euler scheme (p=1) :
For the Milshtein or Talay schemes (p=2):
 q2 p 

e
p
2
 ht 
 Nt : 4  N t
2
 ht 
 N t : 16  N t
 c then
p
t
 q2 p 
 c
e  ht , Nt    
 c
e  ht , Nt   ht c e  ht , N t    
p
c q2 p
p
q
1
2
1
q

and the error bias is β.
c q2 p
nt : 2  nt 
 ht :
1
2
 ht 
 e :
1
nt : 2  nt 
 ht :
1
2
e
2
 ht 
 e :
1
4
e
  cp
  c p
  
ht p
2
Consider the SDE
dX t    t , X t  dt   t , X t  dBt
  t , X t     X t   rX t
with   t , X t     X t    X t     X t  0 .
The diffusion process Xt>0 which satisfies the constant-elasticity-of-variance model (Cox,1975)
dX t  rX t dt   X t  dBt
with
 0.5,1
, can be simulated by the Euler or Milshtein Schemes.
A European call option on the asset with strike price K and expiration date T is


Y  e rT  X T  K  , has we would try to estimate its initial price f  X T   e  rT Y   X T  K  by E  f  X T   .
For γ=1 we obtain the Black-Scholes model.
Technical Details.
For which f is C  and satisfies the polynomial growth conditions everywhere but at x=K.
Then it can be uniformly well approximated for above by
f  x  
xK 
2
x  K  
2
   0,1
.
If we take δ to be of order-2p convergence we preserve the quality of aproximation.
For  0.5,1 we have f  Xˆ h   f  X T  as h 0 (Yamada, 1796 and dominated convergence).
T
h
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
For the particular case:
We take
The following results are obtained,
Comments
 For the Euler scheme (p=1) :
For the Milshtein scheme (p=2):
n : 2  n 
 N : 4  N , e : 1 2  e
n : 2  n 
 N : 16  N , e : 1 4  e
 The assumption hold with other but normal i.i.d increments with zero mean and unit
variance uder technical conditions (Talay, 1984,1986).
- Brownian Motion
-
Wiener Process (SBM)
- Introduction
Discretization Schemes
- Stochastic Calculus (Itō’s Calculus)
-
Stochastic Integral (Itō’s Integral)
Stochastic Differential Equation (SDE)
Itō Process
Diffusion Process
Itō’s Lemma
Example : Geometric Brownian Motion (GBM)
- Security Price Models
-
The Black-Scholes Model (1973)
The Heston Model
(1993)
Euler
Milshtein (1978)
Talay (1984, 1986)
-
Motivation
Abstract
Assumptions
Theorem
-
Proof
Interpretation
- Experiments
- More to go
 More monte carlo methods for asset pricing problems:
 Boyle (1977,1988,1990)
 Jones and Jacobs (1986)
 Boyle, Evnine and Gibbs (1989)
 An Alternative large deviation method for the optimal tradeoff :
 chapter 10 of Duffie (1992).
 Extensions
 Path dependent security payoffs
X
T
T
  X t dt
0


 Stochastic short-term interest rates rt  R  X t  (Duffie,1992)
 A path dependent “lookback” payoff

X T  inft X t
Extension of the general theory of weak convergence for more general path-dependent functionals
 More discretization schemes: Slominski (1993,1994), Liu (1993)
 Adding a memory constraint