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Computational Finance. Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html. Plan. 1. Introduction, deterministic methods. 2. Stochastic methods. 3. Monte Carlo I. 4. Monte Carlo II. 5. Advanced methods for derivatives. Other topics:

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computational finance

Computational Finance

Zvi Wiener

02-588-3049

http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Bank Hapoalim

slide2
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.

CF1

linear algebra
Linear Algebra

1

-2 1

Vectors {1, 1}, {-2, 1}

rows or 1 -2

columns 1 1

CF1

basic operations
Basic Operations

1 2 3

2 + -1 = 1

-2 1 -1

1 3

3 2 = 6

2 6

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linear algebra2
Linear Algebra

vector

Vectors form a linear space.

Zero vector

Scalar multiplication

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linear algebra3
Linear Algebra

Matrices also

form a linear

space.

matrix

Zero matrix

Unit matrix

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linear algebra4
Linear Algebra

Matrix can operate on a vector

How does zero matrix operate?

How does unit matrix operate?

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linear algebra5
Linear Algebra

Transposition of a matrix

A symmetric matrix is A=AT

for example a variance-covariance matrix.

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linear algebra6
Linear Algebra

Matrix multiplication

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scalar product
Scalar Product

a is orthogonal to b if ab = 0

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linear algebra7
Linear Algebra

Scalar product of two vectors

Euclidean norm

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determinant
Determinant

Determinant is 0 if the operator maps

some vectors to zero (and can not be inverted).

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linear algebra8
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.

CF1

linear algebra9
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.

CF1

linear algebra10
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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problems
Problems

Check how the following matrices act on vectors:

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simple exercises
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?

CF1

examples
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.

CF1

calculus
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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variance covariance1
Variance-Covariance

Gradient vector:

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variance covariance4
Variance-Covariance

For a short time period , the changes in the value are distributed approximately normal with the following mean and variance:

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variance covariance5
Variance-Covariance

Then VaR can be found as:

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weighted variance covariance
Weighted Variance covariance

Volatility estimate on day i based on last M days.

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weighted variance covariance1
Weighted Variance covariance

Covariance on day i based on last M days.

It is important to check that the resulting

matrix is positive definite!

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positive quadratic form
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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positive quadratic form1
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]}

];

CF1

stochastic transition matrix
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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markov chain
Markov chain
  • credit migration
  • prepayment and freezing of a program

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stochastic transition matrix1
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.

CF1

cholesky decomposition
Cholesky decomposition

The Cholesky decomposition writes a symmetric positive definite matrix as the product of an upper­triangular matrix and its transpose.

In MMA CholeskyDecomposition[m]

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generating random samples
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

CF1

generating random samples1
Generating Random Samples

We need to sample n normally distributed variables with correlation matrix ij, ( >0).

Sample n independent Gaussian variables x1…xn.

CF1

slide40
ODE

CF1

slide41
ODE

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bisection method
Bisection method

f

If the function is monotonic, e.g. implied vol.

x

CF1

solve and findroot
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}

CF1

max min of a multidimensional function
Max, min of a multidimensional function
  • Gradient method
  • Solve a system of equations(both derivatives)

CF1

level curve of a multivariate function
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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example
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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contour levels of var static
Contour Levels of VaR (static)

q - duration

mismatch

a (% of spread)

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slide52

VaR=3 bp

position

VaR=2 bp

q - duration

mismatch

a (% of spread)

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slide53

In order to reduce

risk one can increase duration

(in this case).

q - duration

mismatch

a (% of spread)

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slide55

duration mismatch (yr)

Position 2M, and

10% spread

5% weekly VaR=2.2 bp

weekly VaR limit 3 bp

spread %

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splines
Splines

x1 x2 x3 … xn

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splines1
Splines

<<Graphics`Spline`

pts = {{0, 0}, {1, 2}, {2, 3}, {3, 1}, {4, 0}}

Show[

Graphics[

Spline[pts, Cubic, SplineDots -> Automatic]]]

CF1

splines2
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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fitting data
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]]

];

CF1

fitting data2
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

CF1

finite differences

S

time

Finite Differences

Following P. Wilmott, “Derivatives”

Typically equal time and S (or logS) steps.

CF1

finite differences1
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

note the direction of time!

CF1

the black scholes equation
The Black-Scholes equation

Linear parabolic PDE

Final conditions

Boundary conditions ...

CF1

bilinear interpolation
Bilinear Interpolation

V1

V2

A4

A3

A1

A2

Area of the rectangle

V3

V4

CF1

final conditions and payoffs
Final conditions and payoffs

For example a European Call option

CF1

boundary conditions call
Boundary conditions Call

For example a Call option

For large S the Call value asymptotes to

S-Ee-r(T-t)

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boundary conditions put
Boundary conditions Put

For example a Put option

For large S

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explicit scheme1
Explicit scheme

Local truncation error

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explicit scheme2
Explicit scheme

Value here is calculated

S

These values are

already known

time

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explicit scheme3
Explicit scheme

This equation is defined for 1i I-1,

for i=1 and i=I we use boundary conditions.

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explicit scheme4
Explicit scheme

For the BS equation (with dividends)

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explicit scheme5
Explicit scheme

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.

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explicit scheme6
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.

CF1

implicit scheme2
Implicit scheme

Values here are calculated

S

This value is used

time

CF1

crank nicolson scheme2
Crank-Nicolson scheme

Values here are calculated

S

These values are used

time

CF1

crank nicolson scheme3
Crank-Nicolson scheme

A general form of the linear equation is:

Note that M are tridiagonal!

CF1

crank nicolson scheme4
Crank-Nicolson scheme

Theoretically this equation can be solved as

In practice this is inefficient!

CF1

lu decomposition
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.

CF1

lu decomposition1
LU decomposition

Then in order to solve

Mv=q

or

LUv=q

We will solve

Lw=q

first, and then

Uv=w.

CF1

lu decomposition2
LU decomposition

Very fast, especially when M is time independent.

Disadvantages:

Needs a big modification for American options

CF1

other methods
Other methods
  • SOR successive over relaxation
  • Douglas scheme
  • Three time-level scheme
  • Alternating direction method
  • Richardson Extrapolation
  • Hopscotch method
  • Multigrid methods

CF1

multidimensional case
Multidimensional case

Fixed t layer

S

These values are used

to calculate space derivatives

r

Note additional boundary conditions.

CF1

stochastic calculus
Stochastic Calculus

Standard Normal

Diffusion process

ABM

GBM

CF1

ito s lemma

dt dX

dt 0 0

dX 0 dt

Ito’s Lemma

CF1

siegel s paradox
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

By Ito’s lemma the process for 1/S is

CF1

siegel s paradox1
Siegel’s paradox

The paradox is that the expected growth rate of 1/S is not ry - rX, but has a correction term.

CF1

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