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Chapter 15. Project: Covariance Estimation and Scenario Generation in Value-at-Risk

376

C H A P T E R 15

. Project: Covariance estimation and scenario generation in value-at-risk

eigenvalues. To accomplish this, one makes use of the MFLapack library and performs a

singular value decomposition (SVD) on the matrix B,

B = O OT

(15.1)

The matrix O is now an orthonormal matrix whose columns Oi are normalized eigenvectors of B. is the diagonal matrix of eigenvalues i of B. That is, BOi = i Oi , with Oi ·

Oj = ij , i j = 1

N . The Oi are essentially the randomly generated principal components of the covariance matrix. After having performed this SVD, the eigenvalues i are

readjusted (i.e., specifically reassigned) by making the change i → i for a chosen set of i ,

i=1

N . The new diagonal matrix of eigenvalues i is then used to give the desired

covariance matrix:

C = O OT

(15.2)

√ at liberty to choose an eigenvalue set. For example, by setting i =

√One is now

N + 1 − i for some positive constant , one has a slowly decaying spectral density

(i.e., eigenvalue density) as one moves away from the origin of zero eigenvalue.

15.2 Reestimating the Covariance Matrix and the Spectral Shift

As in the previous VaR project, we assume a multivariate normal distribution given by

equation (14.2) for the returns. Scenarios are then generated for returns r using the same

procedure described in detail in the plain Monte Carlo approach of the VaR project. Namely,

one generates a vector y k of independent standard normals and multiplies this vector by

the Cholesky factored form of the foregoing covariance matrix C of equation (15.2). This

gives a scenario r k . Each r k is then used to form the exponentially weighted sum over Ns

scenarios:

Cˆ ij ≈ 1 −

Ns

k−1

k

ri rj

k

(15.3)

k=1

for all i j = 1

N and where is a damping parameter or decay factor strictly less than

unity, 0 < < 1. In particular, the value for is typically chosen between 0 94 to 0 97.

The choice of = 0 97 roughly corresponds to assuming a 1-year time window of trading

days. This parameter, therefore, determines the relative weights given to past observations

(i.e., the return scenarios) and hence the amount of data that is actually used to estimate

the variance-covariance of the return time series. The factor 1 − is a normalization since

n

k

≈ 1 − −1 for large n. Note that equation (15.3) is not applicable for the special

k=0

case = 1. Hence, for zero damping ( = 1) one must replace equation (15.3) by

1

Cˆ ij ≈

Ns

Ns

k

ri rj

k

(15.4)

k=1

Note that the time series considered here are scenario sets, which are quite lengthy,

typically of order 10,000.

Having estimated the covariance matrix using equation (15.3) or (15.4), one can then

compare the Cˆ ij elements with the original matrix elements Cij . A more interesting compariˆ matrices. Earlier

son, however, is obtained by computing the eigenvalues for both C and C

15.2 Reestimating the Covariance Matrix and the Spectral Shift

377

0.60

Covariance spectral density

0.50

Recovered density (lambda damping)

0.40

0.30

0.20

0.10

0.00

0.35

0.95

1.55

2.15

2.75

3.35

3.95

4.55

5.15

5.75

6.35

6.95

7.55

8.15

8.75

Eigenvalue

FIGURE 15.1 Eigenvalue distributions for a 100-dimensional covariance matrix using 10,000 scenarios. The recovered distribution is computed with damping factor = 0 97.

we denoted the eigenvalues of C by 1 2

N . Correspondingly, we will denote the

ˆ by

eigenvalues of C

1

2

N . Note that these eigenvalues can be obtained in a variety

of ways, one of which is the singular value decomposition, as given earlier, of the respective

covariance matrices. The so-called vectors of singular values give the sets of eigenvalues.

The objective is to compare eigenvalues in terms of the density (or distribution) for the i

versus the distribution in the i . The eigenvalue density f

at the point is defined as

the number of eigenvalues lying between and + d for infinitesimal d . The densities

are actually estimated by considering histogram plots of the respective eigenvalue sets. The

density plots should demonstrate a probability increase or shift of distribution toward the

origin in the spectrum of eigenvalues as the decay factor is decreased from 1 0 to 0 94, the

latter case corresponding to more damping of past observations. Figure 15.1 gives a histogram

comparison of actual versus recovered eigenvalue distributions for a covariance matrix with

100 risk factors, as generated in the recov spreadsheet. A simple extension to this project is

ˆ .

to include an analysis of the differences in the principal components of C and C

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CHAPTER

. 16

Project: Interest Rate Trees:

Calibration and Pricing

The purpose of this project is essentially twofold: to calibrate interest rate trees against

market discount (zero-coupon) curves and to subsequently use the calibrated lattices to

price interest rate products, such as bonds, bond options, caplets, floorlets, and swaptions.

The theory and implementation allow for four different stochastic interest rate models:

Black–Derman–Toy and Ho–Lee (within a binomial lattice approach) and the Hull–White

model and Black–Karasinski (within a trinomial lattice approach).

Worksheets: ir and ycc

Required Libraries: MFioxl, MFFuncs, MFBlas, MFLapack, MFFit

16.1 Background Theory

In developing interest rate trees we consider a subdivision of calendar time t ∈ 0 T into

TM with time spacing t = Ti − Ti−1 . Throughout we

M subintervals T0 = 0 T1 T2

shall assume equal time steps, although the lattice methods we present can be extended to

the more general case of unequal time steps. Discount bond prices at current calendar time

t maturing at calendar time T are denoted by Zt T (≡ Zt rt T ) (see Chapter 2). Consider,

then, a generic (European-style) security with payoff function rT T depending only on the

value attained at maturity time T for the short rate rT . If one assumes market completeness,

the arbitrage-free price of such a security, at current time t = 0, is given by the expectation

P0 r0 T = P0 T = E0 exp −

T

0

rs ds

rT T

(16.1)

t

Here, the numeraire is chosen as the rolled-over money market account Bt ≡ e 0 rs ds , as

discussed in Chapter 2. In more explicit terms, this expectation (which is conditional on the

short rate’s having value r0 at time t = 0) has the form of an infinite product of conditional

integrations for every incremental time t → 0. In particular, if we denote the risk-neutral

379

380

C H A P T E R 16

. Project: Interest rate trees: Calibration and pricing

conditional probability density that the short rate will attain a value ri at time Ti , given ri−1

at time Ti−1 , by p ri ri−1 t , then (for a generally stochastic rt diffusion process) the price

can be accurately approximated by an M-dimensional integral,

M

P0 r0 T =

0

···

0

t e−ri−1

p ri ri−1

t

rM T drM

dr1

(16.2)

i=1

where TM = T is the terminal time. In the limit t → 0 (or M → since T = M t) this

gives an exact path integral representation of the price. Lattice methods arise by choosing a

finite number M of time slices and evaluating equation (16.2) by using efficiently recombining

lattice point integral approximations. For zero-coupon bonds we have a pay-off of one dollar

with certainty ( rT T = 1); hence

Z0 T = Z0 r0 T = E0 exp −

T

0

rs ds

(16.3)

Of interest are the Arrow–Debreu prices, denoted by G r0 0 r T and given by

G r0 0 r T = E0 exp −

T

0

rs ds

rT − r rt=0 = r0

(16.4)

which is the expectation of an infinitely narrow butterfly spread pay-off (i.e., the Dirac delta

function) conditional on the short rate’s starting at r0 at time t = 0. These correspond to the

worth at time t = 0, given (i.e., conditional on) current state r0 , of a riskless security that

pays one dollar if state rT = r is attained at any later time T > 0. The zero-coupon bonds are

expressed in terms of the Arrow–Debreu values as follows:

Z0 T =

0

G r0 0 r T dr

(16.5)

An important consistency requirement is the continuity relation

G r0 0 ri Ti =

0

G r0 0 ri−1 Ti−1 G ri−1 Ti−1 ri Ti dri−1

(16.6)

This formula is the basis for a discrete version that is used in the sections that follow

to generate a forward induction procedure for propagating the Arrow–Debreu prices. The

function G ri−1 Ti−1 ri Ti is the Arrow–Debreu value conditional on the short rate’s having

value ri−1 at time Ti−1 and attaining a value of ri at a later time Ti > Ti−1 . We conclude this

section by noting that the quantity Z r t t + t = Zt r t + t defined by the conditional

expectation

Z r t t + t = Et exp −

=

0

t+ t

t

rs ds

rt = r

drT G r t rT T = t + t

(16.7)

gives the price of a discount bond at time t ≥ 0 (any time later than current time), with

time to maturity of t, conditional on the short rate’s having value r at time t. Note that

here we have explicitly denoted the conditional nature of the expectation. This formula, in

conjunction with concatenating equation (16.6) for every time step Ti − Ti−1 , forms the basis

for producing lattice pricing formulas of derivatives, such as caplets, floorlets, and swaptions

dealt with later.

## Advanced derivative pricing risk management, albanese campolieti

## 4 Brownian Motion, Martingales, and Stochastic Integrals

## 1 Bonds, Futures, Forwards, and Swaps

## Chapter 3. Advanced Topics in Pricing Theory: Exotic Options and State-Dependent Models

## 6 Green’s Functions Method for Diffusion Kernels

## 8 New Families of Analytical Pricing Formulas: “From x-Space to F-Space”

## 11 Appendix C: Some Properties of Bessel Functions

## Chapter 4. Numerical Methods for Value-at-Risk

## 1 Basic Terminology and Concepts: Asset Prices, States, Returns, and Pay-Offs

## 3 An Example of Single-Period Asset Pricing: Risk-Neutral Probabilities and Arbitrage

## Chapter 6. Project: The Black–Scholes (Lognormal) Model

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Chapter 15. Project: Covariance Estimation and Scenario Generation in Value-at-Risk