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Computational Finance

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Computational Finance

Zvi Wiener

02-588-3049

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

Bank Hapoalim

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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1

-2 1

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

rows or1-2

columns1 1

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1 2 3

2 +-1= 1

-2 1-1

13

3 2 =6

26

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vector

Vectors form a linear space.

Zero vector

Scalar multiplication

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Matrices also

form a linear

space.

matrix

Zero matrix

Unit matrix

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Matrix can operate on a vector

How does zero matrix operate?

How does unit matrix operate?

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Transposition of a matrix

A symmetric matrix is A=AT

for example a variance-covariance matrix.

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Matrix multiplication

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a is orthogonal to b if ab = 0

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Scalar product of two vectors

Euclidean norm

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Determinant is 0 if the operator maps

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

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- 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.

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- 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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- 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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Check how the following matrices act on vectors:

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- 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?

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- 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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- 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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$

x

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Gradient vector:

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

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Then VaR can be found as:

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Volatility estimate on day i based on last M days.

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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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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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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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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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- credit migration
- prepayment and freezing of a program

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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.

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

In MMACholeskyDecomposition[m]

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

Sample n independent Gaussian variables x1…xn.

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f

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

x

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f

x2

x3

x

x1

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

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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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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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q - duration

mismatch

a (% of spread)

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VaR=3 bp

position

VaR=2 bp

q - duration

mismatch

a (% of spread)

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In order to reduce

risk one can increase duration

(in this case).

q - duration

mismatch

a (% of spread)

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What we can do using limits

VaR = 6 bp

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duration mismatch (yr)

Position 2M, and

10% spread

5% weekly VaR=2.2 bp

weekly VaR limit 3 bp

spread %

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x1x2x3…xn

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<<Graphics`Spline`

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

Show[

Graphics[

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

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

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

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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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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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MAKAM

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S

time

Following P. Wilmott, “Derivatives”

Typically equal time and S (or logS) steps.

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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!

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Linear parabolic PDE

Final conditions

Boundary conditions ...

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V1

V2

A4

A3

A1

A2

Area of the rectangle

V3

V4

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For example a European Call option

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For example a Call option

For large S the Call value asymptotes to

S-Ee-r(T-t)

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For example a Put option

For large S

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Local truncation error

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Value here is calculated

S

These values are

already known

time

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This equation is defined for 1i I-1,

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

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For the BS equation (with dividends)

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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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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.

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Values here are calculated

S

This value is used

time

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Values here are calculated

S

These values are used

time

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A general form of the linear equation is:

Note that M are tridiagonal!

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Theoretically this equation can be solved as

In practice this is inefficient!

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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.

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Then in order to solve

Mv=q

or

LUv=q

We will solve

Lw=q

first, and then

Uv=w.

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Very fast, especially when M is time independent.

Disadvantages:

Needs a big modification for American options

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

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Fixed t layer

S

These values are used

to calculate space derivatives

r

Note additional boundary conditions.

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Standard Normal

Diffusion process

ABM

GBM

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dtdX

dt00

dX0dt

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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

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

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