Download Management of Financial Risk

Computational Finance
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
CF-1
Bank Hapoalim
Jun-2001
Plan
1. Introduction, deterministic methods.
2. Stochastic methods.
3. Monte Carlo I.
4. Monte Carlo II.
5. Advanced methods for derivatives.
Other topics:
queuing theory, floaters, binomial trees,
numeraire, ESPP, convertible bond, DAC,
ML-CHKP.
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slide 2
Linear Algebra
1
-2
Vectors
rows or
columns
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1
{1, 1}, {-2, 1}
1
-2
1
1
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slide 3
Linear Algebra
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slide 4
Basic Operations
1
2
-2
1
3 2
2
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+
2
-1
1
=
3
6
6
=
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3
1
-1
slide 5
Linear Algebra
 x1 
 
vector x   x 2 
x 
 3
Vectors form a linear space.
 x1 
 x1  y1 




x  x 2  x  y   x2  y 2 
x  y 
 x 
3
 3
 3
Zero vector
Scalar multiplication
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slide 6
Linear Algebra
matrix
 a11

A   a 21
a
 31
Zero matrix
a12
a 22
a32
a13  Matrices also

a 23  form a linear
a33  space.
 0 0 0


 0 0 0
 0 0 0


1 0 0


Unit matrix  0 1 0 
0 0 1


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slide 7
Linear Algebra
Matrix can operate on a vector
 a11

Ax   a 21
a
 31
a12
a 22
a32
a13  x1   a11 x1  a12 x2  a13 x3 
  

a 23  x2    a 21 x1  a 22 x 2  a 23 x3 
a33  x3   a31 x1  a32 x 2  a33 x3 
How does zero matrix operate?
How does unit matrix operate?
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slide 8
Linear Algebra
Transposition of a matrix
 a11

A   a 21
a
 31
a12
a 22
a32
 a11

T
A   a12
a
 13
a13 

a 23 
a33 
a21
a 22
a 23
a31 

a32 
a33 
A symmetric matrix is A=AT
for example a variance-covariance matrix.
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slide 9
Linear Algebra
Matrix multiplication
 a11 a12 a13  b11 b12 b13 



AB   a 21 a 22 a 23  b21 b22 b23 
a
 b

a
a
b
b
32
33  31
32
33 
 31
 a11b11  a12b21  a13b31 a11b12  a12b22  a13b32

  a 21b11  a 22b21  a 23b31 a 21b12  a 22b22  a 23b32
a b  a b  a b
 31 11 32 21 33 31 a31b12  a32b22  a33b32
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a11b13  a12b23  a13b33 

a 21b13  a 22b23  a 23b33 
a31b13  a32b23  a33b33 
slide 10
Scalar Product
a  b   ai bi
i
 b1 
 
a1 , a2 , a3  b2   a1b1  a2b2  a3b3
b 
 3
a is orthogonal to b if ab = 0
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slide 11
Linear Algebra
Scalar product of two vectors
 y1 
 
T
x y   x1 x2 x3  y 2   x1 y1  x2 y 2  x3 y3
y 
 3
Euclidean norm
x
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2
x xx x x
T
2
1
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2
2
2
3
slide 12
Determinant
 a11 a12
Det 
 a21 a22

  a11a22  a12a21

Determinant is 0 if the operator maps
some vectors to zero (and can not be
inverted).
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slide 13
Linear Algebra
• Matrix multiplication corresponds to a
consecutive application of each operator.
• Note that it is not commutative! ABBA.
• Unit matrix does not change a vector.
• An inverse matrix is such that AA-1=I.
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slide 14
Linear Algebra
• Determinant of a matrix ...
• A matrix can be inverted if det(A)0
• Rank of a matrix
• Matrix as a system of linear equations Ax=b.
• Uniqueness and existence of a solution.
• Trace tr(A) – sum of diagonal elements.
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slide 15
Linear Algebra
• Change of coordinates C-1AC.
• Jordan decomposition.
• Matrix power Ak.
• Matrix as a quadratic form (metric) xTAx.
• Markov process.
• Eigenvectors, eigenvalues Ax=x,
optimization.
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slide 16
Problems
Check how the following matrices act on vectors:
1 0 


 0 1
 0 1


1 0 
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 1 0 


 0 1
 0 1


 1 0 
1 0 


0 0
 cos  sin  


  sin  cos  
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slide 17
Simple Exercises
• Show an example of ABBA.
• Construct a matrix that inverts each vector.
• Construct a matrix that rotates a two
dimensional vector by an angle .
• Construct a covariance matrix, show that it is
symmetric.
• What is mean and variance of a portfolio in
matrix terms?
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slide 18
Examples
• Credit rating and credit dynamics.
• Variance-covariance model of VaR.
• Can the var-covar matrix be inverted
• VaR isolines (the ovals model).
• Prepayment model based on types of clients.
• Finding a minimum of a function.
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slide 19
Calculus
• Function of one and many variables.
• Continuity in one and many directions.
• Derivative and partial derivative.
• Gradient and Hessian.
• Singularities, optimization, ODE, PDE.
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slide 20
Linear and quadratic terms
$
x
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slide 21
Taylor series
x 2
f ( x  x)  f ( x)  f ' ( x)x  f " ( x)
 o x 2
2
 
g ( x  x, y  y )  g ( x, y )  g x x  g y y
 g xx
x 2
y 2
 g xy xy  g yy
 o x 2  y 2
2
2



1 T
F ( x  x)  F ( x)  F ' ( x)x  x F " ( x)x  o x
2
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2

slide 22
Variance-Covariance
V ( x)  V ( x1 , x2 ,..., xn )
  11  12   1n 


 1 
 
 
  21  22
   ,   



 


 n


 nn 
 n1 
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slide 23
Variance-Covariance
Gradient vector:
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 V 


 x1 
V ' ( x)   


 V 
 x 
 n
slide 24
Variance-Covariance
1
2
V ( x  x)  V ( x)  V ' ( x)x  V " ( x)x 
2
1
2
V ( x)  V ' ( x)x  V " ( x)x 
2
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slide 25
Variance-Covariance
V " ( x)
2
E V   E x V ' ( x) 
E x 
2
1
E V    V ' trV " 
2
T
 V   V '  V '
T
Zvi Wiener
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slide 26
Variance-Covariance
For a short time period , the changes in the value
are distributed approximately normal with the
following mean and variance:
1
 T

E V     V ' trV " 
2


2
T
 V   V '  V '
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slide 27
Variance-Covariance
Then VaR can be found as:
VaR 1%,  V    (V )  2.33 (V )
 T 1

T
   V ' trV "   2.33 V '  V '
2


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slide 28
Weighted Variance covariance
Volatility estimate on day i based on last M days.
1

t ( M  1)
1 

t
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i
i
 (R
j i  M 1

i j
j  
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j
 R)
(R j  R )
2
2
slide 29
Weighted Variance covariance
Covariance on day i based on last M days.
1
 12 
t ( M  1)
i
 (R
j i  M 1
1j
 R1 )( R2 j  R2 )
It is important to check that the resulting
matrix is positive definite!
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slide 30
Positive Quadratic Form
For every vector x a we have x.A.x > 0
Only such a matrix can be used to define a
norm.
For example, this matrix can not have
negative diagonal elements. Any variance-
covariance matrix must be positive.
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slide 31
Positive Quadratic Form
Needs["LinearAlgebra`MatrixManipulation`"];
ClearAll[ positiveForm ];
positiveForm[ a_?MatrixQ ] := Module[{aa, i},
aa = Table[
Det[ TakeMatrix[ a, {1, 1}, {i, i}] ],
{i, Length[a]}];
{ aa, If[ Count[ aa, t_ /; t < 0] > 0, False, True]}
];
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slide 32
Stochastic (transition) Matrix
Used to define a Markov chain (only the last
state matters).
A matrix P is stochastic if it is non-negative
and sum of elements in each line is 1.
One can easily see that 1 is an eigenvalue of
any stochastic matrix.
What is the eigenvector?
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slide 33
Markov chain
• credit migration
• prepayment and freezing of a program
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slide 34
Stochastic (transition) Matrix
Theorem: P0 is stochastic iff (1,1,…1) is an
eigenvector with an eigenvalue 1 and this is
the maximal eigenvalue.
If both P and PT are stochastic, then P is called
double stochastic.
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slide 35
Cholesky decomposition
The Cholesky decomposition writes a
symmetric positive definite matrix as the
product of an uppertriangular matrix and its
transpose.
In MMA
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CholeskyDecomposition[m]
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slide 36
Generating Random Samples
We need to sample two normally distributed
variables with correlation .
If we can sample two independent Gaussian
variables x1 and x2 then the required variables
can be expressed as
y1  x1
y2  x1  x2 1  
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2
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slide 37
Generating Random Samples
We need to sample n normally distributed
variables with correlation matrix ij, ( >0).
Sample n independent Gaussian variables
x1…xn.
n
yi   aik xk
y  A.x
k 1
n
1   a , and A A   , A  
k 1
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2
ik
T
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slide 38
Function of a matrix
1
A  T JT
1
1
1
A  T JTT JT  T J T
2
1
2
A T J T
n
n
2
3
A
A
e  E  A


2! 3!
A
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slide 39
ODE
dx
 Ax
dt
x(t )  e
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x(t0 )  x0
A ( t t 0 )
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x0
slide 40
ODE
dx
 Ax  f (t )
dt
x(t )  e
A ( t t 0 )
x(t0 )  x0
t
x0   e
A( t  )
f ( )dt
t0
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slide 41
Bisection method
f
x
If the function is monotonic, e.g. implied vol.
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slide 42
Newton’s method
f
x3 x2
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x1
x
slide 43
Solve and FindRoot
Solve[ 0 = = x2- 0.8x3- 0.3, {x}]
{{x -> -0.467297}, {x ->0.858648 -0.255363*I},
{x -> 0.858648 + 0.255363*I}}
FindRoot[ x2 + Sin[x] - 0.8x3 - 0.3, {x, 0,1}]
{x -> 0.251968}
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slide 44
Max, min of a multidimensional
function
• Gradient method
• Solve a system of equations (both derivatives)
1
1
0.5
0.5
0
-1
0
-0.5
-0.5
0
0.5
1
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-1
slide 45
Gradient method
1
0.5
0
-0.5
-1
-1
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-0.5
0
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0.5
1
slide 46
Level curve of a multivariate
function
ContourPlot[ x^2+y^3, {x,-2,2}, {y,- 2,2}]
ContourPlot[ x^2+y^3, {x,-2,2}, {y,- 2,2}],
Contours->{1 ,-0.5}, ContourShading->False];
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slide 47
ContourPlot[ x^2+y^3, {x,-2,2}, {y,- 2,2}]
2
1
0
-1
-2
-2
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-1
0
CF1
1
2
slide 48
ContourPlot[ x^2+y^3, {x,-2,2}, {y,- 2,2}],
Contours->{1 ,-0.5}, ContourShading->False];
2
1
0
-1
-2
-2
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-1
0
CF1
1
2
slide 49
Example
Consider a portfolio with two risk factors and
benchmark duration of 6M.
The VaR limit is 3 bp. and you have to make two
decisions:
a – % of assets kept in spread products
q – duration mismatch
we assume that all instruments (both treasuries
and spread) have the same duration T+q months.
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slide 50
Contour Levels of VaR (static)
6
4
2
q - duration
mismatch
0
-2
-4
-6
0
0.2
0.4
0.6
0.8
1
a (% of spread)
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slide 51
6
VaR=3 bp
4
2
q - duration
mismatch
0
position
-2
VaR=2 bp
-4
-6
0
0.2
0.4
0.6
0.8
1
a (% of spread)
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slide 52
6
4
In order to reduce
risk one can
increase duration
(in this case).
2
0
-2
-4
-6
0
q - duration
mismatch
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0.2
0.4
0.6
0.8
1
a (% of spread)
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slide 53
What we can do using limits
6
4
2
0
-2
-4
VaR = 6 bp
-6
0
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0.2
0.4
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0.6
0.8
1
slide 54
duration mismatch (yr)
Position 2M, and
10% spread
5% weekly VaR=2.2 bp
0.2
0.1
0
-0.1
weekly VaR limit 3 bp
-0.2
spread %
0
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0.2
0.4
CF1
0.6
0.8
1
slide 55
Splines
x1
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x2
x3
…
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xn
slide 56
Splines
<<Graphics`Spline`
pts = {{0, 0}, {1, 2}, {2, 3}, {3, 1}, {4, 0}}
Show[
Graphics[
Spline[pts, Cubic, SplineDots -> Automatic]]]
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slide 57
Splines
pts = Table[{i, i + i^2 + (Random[] - 0.5)}, {i, 0, 1, .05}];
Show[Graphics[Spline[pts,Cubic,SplineDots ->Automatic]]]
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slide 58
Fitting data
data = Table[7*x + 3 + 10*Random[], {x, 10}];
f[x_] := Evaluate[Fit[data, {1, x}, x]]
Needs["Graphics`Graphics`"]
DisplayTogether[
ListPlot[data, PlotStyle -> {AbsolutePointSize[3],
RGBColor[1, 0, 0]}],
Plot[f[x], {x, 0, 10}, PlotStyle -> RGBColor[0, 0, 1]]
];
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slide 59
Fitting data
60
40
20
2
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4
6
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8
10
slide 60
Fitting data
data = {{1.0, 1.0, .126}, {2.0, 1.0, .219},
{1.0, 2.0, .076}, {2.0, 2.0, .126}, {.1, .0, .186}};
ff[x_, y_] = NonlinearFit[data,
a*c*x/(1 + a*x + b*y), {x, y}, {a, b, c}];
ff[x, y]
nonlinear, multidimensional
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slide 61
Orly - Nelson Siegel
0.145
MAKAM
0.14
0.135
0.13
0.125
50
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100
150
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200
250
300
350
slide 62
Numerical Differentiation
x
f ( x  x)  f ( x)  f ' ( x)x  f " ( x)
 o x 2
2
x 2
2
f ( x  x)  f ( x)  f ' ( x)x  f " ( x)
 o x
2
f ( x  x)  f ( x  x)  2 f ' ( x)x  o x 2
2
 
 
 
f ( x  x)  f ( x  x)
 f ' ( x)  ox 
2x
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slide 63
Numerical Differentiation
x
f ( x  x)  f ( x)  f ' ( x)x  f " ( x)
 o x 2
2
x 2
2
f ( x  x)  f ( x)  f ' ( x)x  f " ( x)
 o x
2
 
2
 
 
f ( x  x)  2 f ( x)  f ( x  x)  f " ( x)x  o x
2
2
f ( x  x)  2 f ( x)  f ( x  x)
2
 f " ( x)  O x
2
x
 
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CF1
slide 64
Finite Differences
S
Following P. Wilmott, “Derivatives”
time
Typically equal time and S (or logS) steps.
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CF1
slide 65
Finite Differences
Time step t
asset step S
(i,k) node of the grid is t = T - kt, iS
0  i  I, 0  k  K
assets value at each node is
Vi  V (iS , T  kt )
k
note the direction of time!
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CF1
slide 66
The Black-Scholes equation
V 1 2 2  V
V
  S
 (r  rf ) S
 rV  0
2
t 2
S
S
2
Linear parabolic PDE
Final conditions V ( S , T )  Payoff ( S )
Boundary conditions ...
Zvi Wiener
CF1
slide 67
Transformation of BS
V ( S , t )  ex   U ( x,  )
1  2r

    2  1,
2 

 U U

2
x

2
2
1  2r

    2  1 ,
4 

S  ex ,
2r
t T  2 .

Zvi Wiener
CF1
slide 68
Approximating 
V
V ( S , t  h)  V ( S , t )
 lim
h 0
t
h
Vi  Vi
V
(S , t ) 
t
t
k
Zvi Wiener
CF1
k 1
 Ot 
slide 69
Approximating 
V V
V
(S , t ) 
S
2S
k
i 1
Zvi Wiener
CF1
k
i 1

 O S
2

slide 70
Approximating 
V
V
(S , t ) 
2
S
2
Zvi Wiener
k
i 1
 2Vi  V
2
S
k
CF1
k
i 1

 O S
2
slide 71

Bilinear Interpolation
V1
V2
A3
A4
4
V
AV
j 1
4
A
j 1
A2
A1
j
j
Area of the rectangle
V3
V4
Zvi Wiener
j
CF1
slide 72
Final conditions and payoffs
V ( S , T )  Payoff ( S )
Vi  Payoff (iS )
0
For example a European Call option
Vi  Max (iS  E ,0)
0
Zvi Wiener
CF1
slide 73
Boundary conditions Call
For example a Call option
k
0
V
0
For large S the Call value asymptotes to
S-Ee-r(T-t)
V  IS  Ee
k
I
Zvi Wiener
CF1
 rkt
slide 74
Boundary conditions Put
For example a Put option
k
0
V
 Ee
 rkt
For large S
VI  0
k
Zvi Wiener
CF1
slide 75
Boundary conditions S=0
V
(0, t )  rV (0, t )  0
t
k 1
0
V
V
 rV0k 1
t
k
0
V  (1  rt )V
k
0
Zvi Wiener
k 1
0
CF1
slide 76
Explicit scheme
V
V
V
 a ( S , t ) 2  b( S , t )
 c( S , t )V  0
t
S
S
2
k 1
Vi  Vi
 2Vi  V
k V
 ai

2
t
S
k
k
k Vi 1  Vi 1
k k
2
bi
 ci Vi  O(t , S )
2S
k
Zvi Wiener
k
i 1
CF1
k
k
i 1
slide 77
Explicit scheme
k 1
Vi  Vi
 2Vi  V
k V
 ai

2
t
S
k
k
k Vi 1  Vi 1
bi
 cikVi k  O(t , S 2 )
2S
k
Vi
k 1
k
i 1
k
k
i 1
 A V  (1  B )Vi  C V
k k
i i 1
k
i
k
k k
i i 1

2
2
O

t
,

t

S
Local truncation error
Zvi Wiener
CF1

slide 78
Explicit scheme
Vi
k 1
 A V  (1  B )Vi  C V
k k
i i 1
k
i
k
k k
i i 1
Value here is calculated
S
These values are
already known
time
Zvi Wiener
CF1
slide 79
Explicit scheme
Vi
k 1
 A V  (1  B )Vi  C V
k k
i i 1
k
i
k
k k
i i 1
This equation is defined for 1i I-1,
for i=1 and i=I we use boundary conditions.
Zvi Wiener
CF1
slide 80
Explicit scheme
Vi
k 1
 A V  (1  B )Vi  C V
k k
i i 1
k
i
k
k k
i i 1
For the BS equation (with dividends)
A  ( i  (r  D)i )t / 2,
k
i
2 2
B  ( i  r )t ,
k
i
2 2
C  ( i  (r  D)i)t / 2,
k
i
Zvi Wiener
2 2
CF1
slide 81
Explicit scheme
2a
S 2
S 
, t 
b
2a
Stability problems related to step sizes.
These relationships should guarantee stability.
Note that reducing asset step by half we must
reduce the time step by a factor of four.
Zvi Wiener
CF1
slide 82
Explicit scheme
Advantages
easy to program, hard to make a mistake
when unstable it is obvious
coefficients can be S and t dependent
Disadvantages
there are restrictions on time step, which
may cause slowness.
Zvi Wiener
CF1
slide 83
Implicit scheme
V
V
V
 a ( S , t ) 2  b( S , t )
 c( S , t )V  0
t
S
S
2
k 1
k 1
i 1
k 1
k 1
i 1
Vi  Vi
 2Vi  V
k 1 V
 ai

2
t
S
k 1
k 1
k 1 Vi 1  Vi 1
k 1 k 1
2
bi
 ci Vi  O(t , S )
2S
k
Zvi Wiener
CF1
slide 84
Implicit scheme
k 1
k 1
k 1
Vi k  Vi k 1
V

2
V

V
k 1 i 1
i
i 1
 ai

2
t
S
k 1
k 1
k 1 Vi 1  Vi 1
k 1 k 1
2
bi
 ci Vi  O(t , S )
2S
k 1 k 1
i
i 1
Vi  A V
k
Zvi Wiener
k 1
i
 (1  B )Vi
CF1
k 1
k 1 k 1
i
i 1
C V
slide 85
Implicit scheme
k 1 k 1
i
i 1
Vi  A V
k
k 1
i
 (1  B )Vi
k 1
k 1 k 1
i
i 1
C V
Values here are calculated
S
This value is used
time
Zvi Wiener
CF1
slide 86
Crank-Nicolson scheme
k 1
k 1
i
k 1
i 1
k 1
k 1
i 1
Vi  Vi
a V  2Vi  V


2
t
2
S
aik Vi k1  2Vi k  Vi k1 bik 1 Vi k11  Vi k11


2
2
S
2
2S
k
k
k
k 1
k
bi Vi 1  Vi 1 ci
ci k
k 1
2

Vi  Vi  O(t , S )
2
2S
2
2
k
Zvi Wiener
CF1
slide 87
Crank-Nicolson scheme
k 1 k 1
i
i 1
A V
k 1
i
 (1  B )Vi
k 1
k 1 k 1
i
i 1
C V

A V  (1  B )Vi  C V
k k
i i 1
Zvi Wiener
k
i
CF1
k
k k
i i 1
slide 88
Crank-Nicolson scheme
Values here are calculated
S
These values are used
time
Zvi Wiener
CF1
slide 89
Crank-Nicolson scheme
k 1
1  B1k 1
V1   A1k 1V0k 1 
C1k 1
0

 k 1
 k 1 
k 1

1  B2
 A2
V2   0

 

 0




0

0
 
 




0 1  BIk21
CIk21 VIk21   0

 


k 1
k 1 

k

1


0
AI 1
1  BI 1 VI 1   0


A general form of the linear equation is:
M
k 1 k 1
L
v
 M v r
k k
R
k
Note that M are tridiagonal!
Zvi Wiener
CF1
slide 90
Crank-Nicolson scheme
Theoretically this equation can be solved as
v
k 1
 M
 M
k 1 1
L
v r
k k
R
k

In practice this is inefficient!
Zvi Wiener
CF1
slide 91
LU decomposition
M is tridiagonal, thus M=LU, where L is
lower triangular, and U is upper triangular.
In fact L has 1 on the diagonal and one
subdiagonal only, U has a diagonal and one
superdiagonal.
Zvi Wiener
CF1
slide 92
LU decomposition
Then in order to solve
Mv=q
or
LUv=q
We will solve
Lw=q
first, and then
Uv=w.
Zvi Wiener
CF1
slide 93
LU decomposition
Very fast, especially when M is time
independent.
Disadvantages:
Needs a big modification for American options
Zvi Wiener
CF1
slide 94
Other methods
• SOR successive over relaxation
• Douglas scheme
• Three time-level scheme
• Alternating direction method
• Richardson Extrapolation
• Hopscotch method
• Multigrid methods
Zvi Wiener
CF1
slide 95
Multidimensional case
Fixed t layer
S
These values are used
to calculate space derivatives
r
Note additional boundary conditions.
Zvi Wiener
CF1
slide 96
Geometrical Brownian Motion
dS  rSdt  SdB
option  e
Zvi Wiener
 r (T t )
Eriskneutral payoff (S )
CF1
slide 97
Lognormal process
dS  rSdt  SdB
 
d (log S )   r 
2

2
S (t )  S (0)e
Zvi Wiener
 2
 r

2


dt  dB


 t Z N ( 0 ,1) t


CF1
slide 98
Euler Scheme
dS  rSdt  SdB
S  rSt  SZ N ( 0,1) t  O(t )
Zvi Wiener
CF1
slide 99
Milstein Scheme
dS  rSdt  SdB
S  rSt  SZ N ( 0,1) t 
1 2 2 2
2
 S ( Z N ( 0,1)  1)t  O(t )
2
Zvi Wiener
CF1
slide 100
Stochastic Calculus
Standard Normal
1
N ( x) 
2
x
e
z2

2
dz

Diffusion process dX   ( X , t )dt   ( X , t )dBt
ABM
dX  dt  dBt
GBM
dX  Xdt  XdBt
Zvi Wiener
CF1
slide 101
Ito’s Lemma
dX   ( X , t )dt   ( X , t )dBt
F
F
 F
2
dX 
dF ( X , t ) 
dt 
dX 
2
t
X
X
2
dt
dt 0
dX 0
Zvi Wiener
dX
0
dt
CF1
slide 102
Y  ln( X )
dX  Xdt  XdBt
1
1
2
dY  0dt  dX  2 (dX )
X
X

2 
dY     dt  dBt
2 

 
Y (t )  Y (0)    
2

2
X (t )  X (0)e
Zvi Wiener

t   t Z (0,1)

 2 
    t  t Z ( 0 ,1)

2 

CF1
slide 103
Siegel’s paradox
Consider two currencies X and Y. Define S an
exchange rate (the number of units of
currency Y for a unit of X).
The risk-neutral process for S is
dS  (rY  rX )Sdt   S SdBt
By Ito’s lemma the process for 1/S is


1
1
2 1
d    rY  rX   S dt   S dBt
S
S
S
Zvi Wiener
CF1
slide 104
Siegel’s paradox
The paradox is that the expected growth rate
of 1/S is not ry - rX, but has a correction term.
Zvi Wiener
CF1
slide 105