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### Computational Finance

Zvi Wiener

02-588-3049

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

Bank Hapoalim

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

CF1

Linear Algebra

Matrix can operate on a vector

How does zero matrix operate?

How does unit matrix operate?

CF1

Linear Algebra

Transposition of a matrix

A symmetric matrix is A=AT

for example a variance-covariance matrix.

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Determinant

Determinant is 0 if the operator maps

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

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Linear Algebra

- Matrix multiplication corresponds to a consecutive application of each operator.
- Note that it is not commutative! ABBA.
- Unit matrix does not change a vector.
- An inverse matrix is such that AA-1=I.

CF1

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 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.

CF1

Simple Exercises

- Show an example of ABBA.
- 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

- 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

- 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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Taylor series

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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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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

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.

CF1

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

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?

CF1

Stochastic (transition) Matrix

Theorem: P0 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

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

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

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Generating Random Samples

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

Sample n independent Gaussian variables x1…xn.

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ODE

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ODE

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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

- Gradient method
- Solve a system of equations(both derivatives)

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Gradient method

CF1

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];

CF1

ContourPlot[ x^2+y^3, {x,-2,2}, {y,- 2,2}], Contours->{1 ,-0.5}, ContourShading->False];

CF1

Example ,-0.5}, ContourShading->False];

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.

CF1

Contour Levels of VaR (static) ,-0.5}, ContourShading->False];

q - duration

mismatch

a (% of spread)

CF1

VaR=3 bp ,-0.5}, ContourShading->False];

position

VaR=2 bp

q - duration

mismatch

a (% of spread)

CF1

In order to reduce ,-0.5}, ContourShading->False];

risk one can increase duration

(in this case).

q - duration

mismatch

a (% of spread)

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duration mismatch (yr) ,-0.5}, ContourShading->False];

Position 2M, and

10% spread

5% weekly VaR=2.2 bp

weekly VaR limit 3 bp

spread %

CF1

Splines ,-0.5}, ContourShading->False];

<<Graphics`Spline`

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

Show[

Graphics[

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

CF1

Splines ,-0.5}, ContourShading->False];

pts = Table[{i, i + i^2 + (Random[] - 0.5)}, {i, 0, 1, .05}];

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

CF1

Fitting data ,-0.5}, ContourShading->False];

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 data ,-0.5}, ContourShading->False];

CF1

Fitting data ,-0.5}, ContourShading->False];

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

Numerical Differentiation ,-0.5}, ContourShading->False];

CF1

Numerical Differentiation ,-0.5}, ContourShading->False];

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S ,-0.5}, ContourShading->False];

time

Finite DifferencesFollowing P. Wilmott, “Derivatives”

Typically equal time and S (or logS) steps.

CF1

Finite Differences ,-0.5}, ContourShading->False];

Time step t

asset step S

(i,k) node of the grid is t = T - kt, iS

0 i I, 0 k K

assets value at each node is

note the direction of time!

CF1

The Black-Scholes equation ,-0.5}, ContourShading->False];

Linear parabolic PDE

Final conditions

Boundary conditions ...

CF1

Transformation of BS ,-0.5}, ContourShading->False];

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Approximating ,-0.5}, ContourShading->False];

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Approximating ,-0.5}, ContourShading->False];

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Approximating ,-0.5}, ContourShading->False];

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Bilinear Interpolation ,-0.5}, ContourShading->False];

V1

V2

A4

A3

A1

A2

Area of the rectangle

V3

V4

CF1

Boundary conditions Call ,-0.5}, ContourShading->False];

For example a Call option

For large S the Call value asymptotes to

S-Ee-r(T-t)

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Boundary conditions S=0 ,-0.5}, ContourShading->False];

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Explicit scheme ,-0.5}, ContourShading->False];

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Explicit scheme ,-0.5}, ContourShading->False];

Value here is calculated

S

These values are

already known

time

CF1

Explicit scheme ,-0.5}, ContourShading->False];

This equation is defined for 1i I-1,

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

CF1

Explicit scheme ,-0.5}, ContourShading->False];

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.

CF1

Explicit scheme ,-0.5}, ContourShading->False];

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 scheme ,-0.5}, ContourShading->False];

CF1

Implicit scheme ,-0.5}, ContourShading->False];

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Implicit scheme ,-0.5}, ContourShading->False];

Values here are calculated

S

This value is used

time

CF1

Crank-Nicolson scheme ,-0.5}, ContourShading->False];

CF1

Crank-Nicolson scheme ,-0.5}, ContourShading->False];

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Crank-Nicolson scheme ,-0.5}, ContourShading->False];

Values here are calculated

S

These values are used

time

CF1

Crank-Nicolson scheme ,-0.5}, ContourShading->False];

A general form of the linear equation is:

Note that M are tridiagonal!

CF1

Crank-Nicolson scheme ,-0.5}, ContourShading->False];

Theoretically this equation can be solved as

In practice this is inefficient!

CF1

LU decomposition ,-0.5}, ContourShading->False];

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 decomposition ,-0.5}, ContourShading->False];

Then in order to solve

Mv=q

or

LUv=q

We will solve

Lw=q

first, and then

Uv=w.

CF1

LU decomposition ,-0.5}, ContourShading->False];

Very fast, especially when M is time independent.

Disadvantages:

Needs a big modification for American options

CF1

Other methods ,-0.5}, ContourShading->False];

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

CF1

Multidimensional case ,-0.5}, ContourShading->False];

Fixed t layer

S

These values are used

to calculate space derivatives

r

Note additional boundary conditions.

CF1

Geometrical Brownian Motion ,-0.5}, ContourShading->False];

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Lognormal process ,-0.5}, ContourShading->False];

CF1

Euler Scheme ,-0.5}, ContourShading->False];

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Milstein Scheme ,-0.5}, ContourShading->False];

CF1

CF1 ,-0.5}, ContourShading->False];

Siegel’s paradox ,-0.5}, ContourShading->False];

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 paradox ,-0.5}, ContourShading->False];

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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