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3 Merton Problem with Stochastic Volatility: Model Coefficient Polynomial Expansions

# 3 Merton Problem with Stochastic Volatility: Model Coefficient Polynomial Expansions

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Financial Signal Processing and Machine Learning

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of the following SDE:

dSt = 𝜇(Yt )St dt + 𝜎(Yt )St dBSt ,

dYt = c(Yt ) dt + 𝛽(Yt ) dBYt ,

d⟨BS , BY ⟩t = 𝜌dt,

where BS and BY are standard Brownian motions with correlation 𝜌. Let W denote the wealth

process of an investor who holds 𝜋t units worth of currency in S at time t, and has (Wt − 𝜋t )

units of currency in a bond. For simplicity, we assume the risk-free rate of interest is zero. As

such, the wealth process W satisfies

dWt =

𝜋t

dSt = 𝜋t 𝜇(Yt ) dt + 𝜋t 𝜎(Yt ) dBSt .

St

Observe that S does not appear in the dynamics of the wealth process W.

An investor chooses 𝜋t to maximize his expected utility of wealth at a time T in the future,

where utility is measured by a smooth, increasing, and strictly concave function U ∶ ℝ+ → ℝ,

and the objective to maximize is 𝔼 U(WT ). Increasing describes a preference for more wealth

than less, whereas concavity captures risk aversion, with more concave being more risk averse.

The analysis is illustrated with power utility functions in Section 7.3.3.

We define the investor’s value function u by

u(t, y, 𝑤) ∶= sup 𝔼[U(WT )|Yt = y, Wt = 𝑤],

𝜋∈Π

where Π is the set of admissible strategies:

}

{

T

𝜋t2 𝜎 2 (Yt ) dt < ∞ and Wt ≥ 0 a.s. ,

∫0

where adapted means adapted to the filtration generated by (BS , BY ).

Assuming that u ∈ C1,2 ([0, T], ℝ, ℝ+ ), the value function solves the Hamilton–Jacobi–

Bellman partial differential equation (HJB-PDE) problem:

(𝜕t + AY )u + max A𝜋 u = 0,

𝜋∈ℝ

u(T, y, 𝑤) = U(𝑤),

(7.27)

where (AY + A𝜋 ) is the generator of (Y, W), assuming a Markov investment strategy

𝜋t = 𝜋(t, Yt , Wt ). Specifically, the operators AY and A𝜋 are given by

1

AY = c(y)𝜕y + 𝛽 2 (y)𝜕y2 ,

2

1

2

+ 𝜋(t, y, 𝑤)𝜌𝜎(y)𝛽(y)𝜕y 𝜕𝑤 .

A𝜋 = 𝜋(t, y, 𝑤)𝜇(y)𝜕𝑤 + 𝜋 2 (t, y, 𝑤)𝜎 2 (y)𝜕𝑤

2

The optimal strategy 𝜋 ∗ is given by

𝜋 ∗ = arg maxA𝜋 u = −

𝜋∈ℝ

𝜇(𝜕𝑤 u) + 𝜌𝛽𝜎(𝜕y 𝜕𝑤 u)

2 u)

𝜎 2 (𝜕𝑤

,

(7.28)

where, for simplicity (and from now on), we have omitted the arguments (t, y, 𝑤).

Inserting the optimal strategy 𝜋 ∗ into the HJB-PDE (7.27) yields

(𝜕t + AY )u + N (u) = 0,

(7.29)

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157

where N (u) is a nonlinear term:

(𝜕𝑤 u)(𝜕y 𝜕𝑤 u) 1 2 2 (𝜕y 𝜕𝑤 u)2

1 (𝜕 u)2

− 𝜌𝛽𝜆

− 𝜌 𝛽

.

N (u) = − 𝜆2 𝑤2

2u

2u

2

2

𝜕𝑤 u

𝜕𝑤

𝜕𝑤

Here, we have introduced the Sharpe ratio 𝜆(y) ∶= 𝜇(y)∕𝜎(y).

7.3.2

Asymptotic Approximation

For general {𝛽, c, 𝜆}, there is no closed-form solution of (7.29). Hence, we seek an asymptotic

approximation for u. To this end, using equation (7.22) from Section 7.2.5.1 as a guide, we

expand the coefficients in (7.29) in a Taylor series about an arbitrary point ȳ . Specifically, for

any function 𝜒 ∶ ℝ → ℝ, we may formally write

𝜒(y) =

𝜀n 𝜒 n (y),

𝜒 n (y) ∶=

n=0

1 n

𝜕 𝜒(̄y)⋅(y − ȳ )n ,

n! y

𝜀 = 1,

(7.30)

where we have once again introduced 𝜀 for purposes of accounting. We also expand the function u as a power series in 𝜀

u=

𝜀n un ,

𝜀 = 1.

(7.31)

n=0

Now, for each group of coefficients appearing in (7.29), we insert an expansion of the form

(7.30), and we define

An ∶= cn 𝜕y + ( 12 𝛽 2 )n 𝜕y2 ,

n ∈ {0} ∪ ℕ.

(7.32)

We also insert into (7.29) our expansion (7.31) for u.

Next, collecting terms of like powers of 𝜀, we obtain at lowest order

(𝜕𝑤 u0 )(𝜕y 𝜕𝑤 u0 )

(𝜕y 𝜕𝑤 u0 )

(𝜕 u )2

1

1

− ( 𝜌2 𝛽 2 )0

= 0,

(𝜕t + A0 )u0 − ( 𝜆2 )0 𝑤2 0 − (𝜌𝛽𝜆)0

2

2u

2

2

𝜕𝑤 u0

𝜕𝑤 u0

𝜕𝑤

0

2

with u0 (T, y, 𝑤) = U(𝑤). We can look for a solution u0 = u(t, 𝑤) that is independent of y, and

then we have

(𝜕 u )2

1

𝜕t u0 − ( 𝜆2 )0 𝑤2 0 = 0,

2

𝜕𝑤 u0

u0 (T, 𝑤) = U(𝑤).

(7.33)

We observe that (7.33) is the same nonlinear PDE problem that arises when one considers an

underlying that has a constant drift 𝜇0 = 𝜇(̄y), diffusion coefficient 𝜎0 = 𝜎(̄y), and Sharpe ratio

𝜆0 = 𝜆(̄y) = 𝜇(̄y)∕𝜎(̄y).

It is convenient to define the risk-tolerance function

R0 ∶=

−𝜕𝑤 u0

2u

𝜕𝑤

0

,

and the operators

k

,

Dk = Rk0 𝜕𝑤

k = 1, 2,. · · · .

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We now proceed to the order O(𝜀) terms. Using u0 = u0 (t, 𝑤) and (7.33), we obtain

1

1

(𝜕t + A0 )u1 + ( 𝜆20 )D2 u1 + 𝜆20 D1 u1 + (𝜌𝛽𝜆)0 D1 𝜕y u1 = −( 𝜆2 )1 D1 u0 ,

2

2

u1 (T, y, 𝑤) = 0,

(7.34)

(7.35)

which is a linear PDE problem for u1 .

We can rewrite equations (7.34)–(7.35) more compactly as

(𝜕t + A0 + B 0 (t))u1 + H1 = 0,

u1 (T, y, 𝑤) = 0,

(7.36)

where the linear operator B(t) and the source term H1 are given by

B 0 (t) = 12 𝜆20 D2 + 𝜆20 D1 + (𝜌𝛽𝜆)0 D1 𝜕y ,

H1 = ( 12 𝜆2 )1 R0 𝜕𝑤 u0 .

Observe that (7.36) is a linear PDE for u1 .

The following change of variables (see Fouque et al., 2013) will be useful for solving the

PDE problem (7.36). Define

u1 (t, y, 𝑤) = q1 (t, y, z(t, 𝑤)),

z(t, 𝑤) = − log 𝜕𝑤 u0 (t, 𝑤) + 12 𝜆20 (T − t).

(7.37)

Inserting (7.37) into (7.36), we find that q1 satisfies

0 = (𝜕t + A0 + C 0 )q1 + Q1 ,

q1 (T, y, z) = 0,

(7.38)

where the operator C 0 is given by

C0 =

1 2 2

𝜆 𝜕 + (𝜌𝛽𝜆)0 𝜕y 𝜕z ,

2 0 z

(7.39)

and the function Q1 satisfies H1 (t, y, 𝑤) = Q1 (t, y, z(t, 𝑤)).

Now, from (7.32) and (7.39), we observe that the operator (A0 + C 0 ) is the infinitesimal

generator of a diffusion in ℝ2 whose drift vector and covariance matrix are constant. The

semigroup P 0 (t, t′ ) generated by (A0 + C 0 ) is given by

P 0 (t, T)G(y, z) ∶=

∫ℝ2

d𝜂 d𝜁 Γ0 (t, y, z; T, 𝜂, 𝜁 )G(𝜂, 𝜁 ),

where Γ0 , the fundamental solution corresponding to (𝜕t + A0 + C 0 ), is a Gaussian kernel:

(

)

1

1

exp − mT C−1 m ,

Γ0 (t, y, z; T, 𝜂, 𝜁 ) = √

2

(2𝜋)3 |C|

with covariance matrix C and vector m given by

( 2

)

(𝛽 )0 (𝜌𝛽𝜆)0

C = (T − t)

,

(𝜌𝛽𝜆)0 (𝜆2 )0

)

𝜂 − y − (T − t)c0

.

𝜁 −z

(

m=

By Duhamel’s principle, the unique classical solution to (7.38) is given by

T

q1 (t) =

∫t

ds P 0 (t, s)Q1 (s),

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159

In the case of a general utility function, (7.33) is easily solved numerically, for instance by

solving the fast diffusion (or Black’s) equation for the risk tolerance function R0 (see Fouque

et al., 2013). Then, u1 can also be computed numerically using the formulas above. In the case

of power utility, there are explicit formulas, as given in Section 7.3.3.

Having obtained an approximation for the value function u ≈ u0 + u1 , we now seek an

expansion for the optimal control 𝜋 ∗ ≈ 𝜋0∗ + 𝜋1∗ . Inserting the expansion (7.30) of the coefficients and the expansion (7.31) for u into (7.28), and collecting terms of like powers of 𝜀, we

obtain

𝜇0 (𝜕𝑤 u0 )

O(1) ∶

𝜋0 = −

O(𝜀) ∶

𝜋1 = −𝜋0

2u )

(𝜎 2 )0 (𝜕𝑤

0

,

(7.40)

2

(𝜕𝑤 u0 )

(𝜕𝑤

u1 )

(𝜎 2 )1

− 𝜇1

𝜋

0

2

2

2u )

2

(𝜎 )0

(𝜕𝑤 u0 )

(𝜎 )0 (𝜕𝑤

0

− 𝜇0

(𝜕𝑤 u1 )

(𝜎 2 )

2

0 (𝜕𝑤 u0 )

− (𝜌𝛽𝜎)0

(𝜕y 𝜕𝑤 u1 )

2u )

(𝜎 2 )0 (𝜕𝑤

0

.

(7.41)

Higher order terms for both the value function u and the optimal control 𝜋 ∗ can be obtained

in the same manner as u1 and 𝜋1 . Analysis of the asymptotic formulas for different utility functions and stochastic volatility models is presented in more detail in Lorig and Sircar (2015).

7.3.3

Power Utility

Finally, we consider a utility function U from the constant relative risk aversion (CRRA), or

power family:

CRRA utility:

U(𝑤) ∶=

𝑤1−𝛾

, 𝑤 > 0,

1−𝛾

𝛾 > 0, 𝛾 ≠ 1,

where 𝛾 is called the risk aversion coefficient. Here, all the quantities above can be computed

explicitly.

The explicit solution u0 to (7.33) is

(

(

)

)

1 2 1−𝛾

𝜆0

u0 (t, 𝑤) = U(𝑤) exp

(T − t) .

2

𝛾

The risk-tolerance function is R0 =

𝑤

𝛾,

and the transformation in (7.37) is then

(

z(t, 𝑤) = 𝛾𝑤 + (T − t)

2𝛾 − 1

𝛾

)

1 2

𝜆 .

2 0

An explicit computation reveals that u1 is given by

u1 (t, y, 𝑤) = q1 (t, y, z(t, 𝑤))

(

)′ (

(

))

1−𝛾

1 2

1−𝛾

1

2

u0 (t, 𝑤) 𝜆 (̄y)

𝜌𝛽0 𝜆0

=

(T − t)(y − ȳ ) + (T − t) c0 +

.

𝛾

2

2

𝛾

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For the specific case ȳ = y, the above expression simplifies to

(

)′ (

(

))

1−𝛾

1 2

1−𝛾

1

2

u0 (t, 𝑤) 𝜆 (y)

(T − t) c0 +

𝜌𝛽0 𝜆0

u1 (t, y, 𝑤) =

.

𝛾

2

2

𝛾

Using these explicit representations of u0 and u1 , the expressions (7.40) and (7.41) for the

optimal stockholding approximations become

𝜇

𝜋0∗ = 02 ,

𝛾𝜎0

(

)

)′ )

(

(

𝜇0 ( 2 )′

𝜇′ (̄y)

(1 − 𝛾)(T − t)

1 1 2

𝜋1 (t, y) = (y − ȳ )

𝜆 (̄y)

− 4 𝜎 (̄y)

+

𝜌𝛽0

.

𝛾𝜎0

𝛾 2

𝛾𝜎 2

𝛾𝜎

0

0

For the specific case ȳ = y, the formula for 𝜋1∗ simplifies to

)′

(

(1 − 𝛾)(T − t)

1 1 2

𝜆 (y) .

𝜋1∗ (t, y) =

𝜌𝛽0

𝛾𝜎0

𝛾 2

7.4 Conclusions

Asymptotic methods can be used to analyze and simplify pricing and portfolio optimization

problems, and we have presented some examples and methodologies. A key insight is to perturb problems with stochastic volatility around their constant volatility counterparts to obtain

the principle effect of volatility uncertainty.

These approaches reduce the dimension of the effective problems that have to be solved, and

often lead to explicit formulas that can be analyzed for intuition. In the context of portfolio

problems accounting for stochastic volatility, recent progress has been made in cases where

there are transaction costs (Bichuch and Sircar, 2014, Kallsen and Muhle-Karbe, 2013) or

stochastic risk aversion that varies with market conditions (Dong and Sircar, 2014), and under

more complex local-stochastic volatility models (Lorig and Sircar, 2015).

Acknowledgements

RL’s work is partially supported by NSF grant DMS-0739195. RS’s work is partially supported

by NSF grant DMS-1211906.

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8

Statistical Measures of Dependence

for Financial Data

David S. Matteson, Nicholas A. James and William B. Nicholson

Cornell University, USA

8.1 Introduction

The analysis of financial and econometric data is typified by non-Gaussian multivariate

observations that exhibit complex dependencies: heavy-tailed and skewed marginal distributions are commonly encountered; serial dependence, such as autocorrelation and conditional

heteroscedasticity, appear in time-ordered sequences; and nonlinear, higher-order, and tail

dependence are widespread. Illustrations of serial dependence, nonnormality, and nonlinear

dependence are shown in Figure 8.1.

When data are assumed to be jointly Gaussian, all dependence is linear, and therefore only

pairwise among the variables. In this setting, Pearson’s product-moment correlation coefficient

uniquely characterizes the sign and strength of any such dependence.

Definition 8.1 For random variables X and Y with joint density fXY , Pearson’s correlation

coefficient is defined as

(x − 𝜇X ) (y − 𝜇Y )

⎢ (X − 𝜇X ) (Y − 𝜇Y ) ⎥

fXY (x, y) dx dy,

𝜌P (X, Y) = E ⎢ √

⎥=∫ ∫ √

𝜎X2

𝜎Y2 ⎥

𝜎Y2

𝜎X2

where 𝜇X = E(X) = ∫ ∫ xfXY (x, y) dx dy, and 𝜎X2 = E[(X − 𝜇X )2 ] = ∫ ∫ (x − 𝜇X )2 fXY (x, y) dx dy,

are the mean and variance of X, respectively, and 𝜇Y and 𝜎Y2 are defined similarly.

This conventional measure of pairwise linear association is well-defined provided 𝜎X2 and 𝜎Y2

are positive and finite, in which case 𝜌P (X, Y) ∈ [−1, +1]. A value of −1 or +1 indicates

Financial Signal Processing and Machine Learning, First Edition.

Edited by Ali N. Akansu, Sanjeev R. Kulkarni and Dmitry Malioutov.

© 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

Statistical Measures of Dependence for Financial Data

−4

−2

−0.010 −0.005

0

0.000

2

0.005

4

0.010

163

0

50

100

150

200

0

100

200

Time

300

400

500

Time

4

2

0

−2

Sample Quantiles

−4

Density

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Normal Q−Q Plot

1

2

3

4

5

−3

6

−2

−1

0

1

2

3

−10

0.0

−15

−10

0.2

−5

0.4

0

0.6

0.8

5

1.0

Theoretical quantiles

−5

0

5

10

15

0.0

0.2

0.4

0.6

0.8

1.0

Figure 8.1 Top left: Strong and persistent positive autocorrelation, that is, persistence in local level;

top right: moderate volatility clustering, that is, i.e., persistence in local variation. Middle left: Right

tail density estimates of Gaussian versus heavy- or thick-tailed data; middle right: sample quantiles of

heavy-tailed data versus the corresponding quantiles of the Gaussian distribution. Bottom left: Linear

regression line fit to non-Gaussian data; right: corresponding estimated density contours of the normalized sample ranks, which show a positive association that is stronger in the lower left quadrant compared

to the upper right.

Financial Signal Processing and Machine Learning

164

perfect negative or positive linear dependence, respectively, and in either case X and Y have

an exact linear relationship. Negative and positive values indicate negative and positive linear

associations, respectively, while a value of 0 indicates no linear dependence.

A sample estimator 𝜌̂P is typically defined by replacing expectations with empirical expectations in the above definition. For a paired random sample of n observations (X1∶n , Y 1∶n ) =

{(xi , yi )}ni=1 , define

1 ∑ (xi − 𝜇̂ X ) (yi − 𝜇̂ Y )

1∑

1 ∑

∶ 𝜇̂ X =

xi ; 𝜎̂ X2 =

(xi − 𝜇̂ X )2 ,

n − 1 i=1

n

n

1

i=1

i=1

𝜎̂ X2

𝜎̂ Y2

n

𝜌̂P (X1∶n , Y 1∶n ) =

n

n

where 𝜇̂ Y and 𝜎̂ Y2 are defined similarly. For jointly Gaussian variables (X, Y), zero correlation is

equivalent to independence. For an independent and identically distributed (i.i.d.) sample from

the bivariate normal distribution, inference regarding 𝜌P can be conducted using the following

asymptotic approximation:

D

n(𝜌̂P − 𝜌P ) → N [0, (1 − 𝜌2P )2 ],

D

in which → denotes convergence in distribution. Under the null hypothesis of zero correla√

D

tion, this expression simplifies to n𝜌̂P → N (0, 1). More generally, for an i.i.d sample from

an arbitrary distribution with finite fourth moments, E(X 4 ) and E(Y 4 ), a variance-stabilizing

transformation may be applied (Fisher’s transformation) to obtain the alternative asymptotic

approximation (cf. Ferguson, 1996)

√ (

)

n

1 + 𝜌P D

1 + 𝜌̂P

− log

→ N (0, 1).

log

2

1 − 𝜌̂P

1 − 𝜌P

Although the previous approximation is quite general, assuming finite fourth moments may

be unreasonable in numerous financial applications where extreme events are common. Furthermore, Pearson’s correlation coefficient measures the strength of linear relationships only.

There are many situations in which correlations are zero but a strong nonlinear relationship

exists, such that variables are highly dependent. In Section 8.2, we discuss several robust

measures of correlation and pairwise association, and illustrate their application in measuring

serial dependence in time-ordered data. In Section 8.3, we consider multivariate extensions and

Granger causality, and introduce measures of mutual independence. Finally, in Section 8.4 we

explore copulas and their financial applications.

8.2 Robust Measures of Correlation and Autocorrelation

Financial data are often time-ordered, and intertemporal dependence is commonplace. The

autocovariance and autocorrelation functions are extensions of covariance and Pearson’s correlation to a time-dependent setting, respectively.

Definition 8.2 For a univariate ordered sequence of random variables {Xt }, the autocovariance 𝛾 and autocorrelation 𝜌 at indices q and r (Shumway and Stoffer, 2011) are defined as

𝛾(Xq , Xr ) = 𝛾(q, r) = E[(Xq − 𝜇q )(Xr − 𝜇r )]

Statistical Measures of Dependence for Financial Data

and

𝜌P (Xq , Xr ) = 𝜌P (q, r) = √

165

𝛾(q, r)

,

𝛾(q, q) 𝛾(r, r)

respectively, in which 𝜇q = E(Xq ) and 𝜇r = E(Xr ). The above quantities are well-defined provided E(Xt2 ) is finite for all t; however, estimating these quantities from an observed sequence

X1∶n = {xt }nt=1 requires either multiple i.i.d. realizations of the entire sequence (uncommon in

finance), or some additional assumptions. The first basic assumption is that the observations are

equally spaced, and t denotes their discrete-time index. We will refer to such sequences generically as time-series. In finance, this assumption may only hold approximately. For example, a

sequence of daily market closing asset prices may only be available for weekdays, with additional gaps on holidays, or intraday asset transaction prices may be reported every hundredth of

a second, but there may be no transaction at many of these times. In either case, the consecutive

observations are commonly regarded as equally spaced, for simplicity.

The next basic assumption is some form of distributional invariance over time, such as stationarity.

Definition 8.3 A univariate sequence of random variables {Xt } is weakly (or covariance)

stationary if and only if

E(Xt ) = E(Xt−h ) = 𝜇

and

𝛾(t, t − h) = 𝛾(|h|) = 𝛾h

∀t, h, and

𝛾0 < ∞.

This implies that the means and variances are finite and constant, and the autocovariance is

constant with respect to t, and only depends on the relative time lag h between observations.

For any k-tuple of indices t1 , t2 , … , tk , let Ft1 ,t2 ,…,tk (⋅) denote the joint distribution function

of (Xt1 , Xt2 , … , Xtk ). Then, the sequence is strictly stationary if and only if

Ft1 ,t2 ,…,tk (⋅) = Ft1 −h,t2 −h,…,tk −h (⋅) ∀h, k, and ∀t1 , t2 , … , tk .

This implies that the joint distributions of all k-tuples are invariant to a common time shift h

such that their relative time lags remain constant.

Strict stationary implies weak stationarity provided the variance is also finite.

Now, under the weak stationarity assumption, the parameters 𝛾h = 𝛾(h) for h = 0, 1, 2, …

denote the autocovariance function of {Xt } with respect to the time lag h, and the corresponding autocorrelation function (ACF) is defined as 𝜌P (h) = 𝛾h ∕𝛾0 . Under the weak

stationarity assumption, the joint distribution of the random variables (X1 , … , Xn ) has

mean vector 𝝁X = 𝟏𝜇, where 𝟏 denotes a length n vector of ones, and a symmetric

Toeplitz covariance matrix ΣX , with [ΣX ]i,j = 𝛾(|i − j|). Furthermore, both ΣX and the

corresponding correlation matrix ΩX are positive definite for any stationary sequence. For

an observed stationary time-series X1∶n = {xt }nt=1 , the mean is estimated as before (𝜇̂ X ),

while autocovariances and autocorrelations are commonly estimated as 𝛾̂ (X1∶n ; h) = 𝛾̂ (h) =

1 ∑n

𝛾 (0), respectively. Using

t=h+1 (xt − 𝜇̂ X )(xt−h − 𝜇̂ X ) and 𝜌̂P (X 1∶n ; h) = 𝜌̂P (h) = 𝛾̂ (h)∕̂

n

1

assures that the corresponding estimated covariance and

the scaling 1n as opposed to n−h

̂ X ]i,j = 𝜌̂P (|i − j|) are both positive definite

correlation matrices [Σ̂ X ]i,j = 𝛾̂ (|i − j|) and [Ω

(McLeod and Jimenéz, 1984).

Financial Signal Processing and Machine Learning

166

8.2.1

Transformations and Rank-Based Methods

Pearson’s correlation and the autocorrelation function are commonly interpreted under an

implicit joint normality assumption on the data. The alternative measures discussed in this subsection offer robustness to outlying and extreme observations and consider nonlinear dependencies, all of which are common in financial data.

8.2.1.1

Huber-type Correlations

Transformations may be applied to define a robust correlation measure between pairs of random variables (X, Y). For example, let 𝜇R and 𝜎R denote robust location and scale parameters,

such as the trimmed mean and trimmed standard deviation. And let 𝜓 denote a bounded monotone function, such as 𝜓(x; k) = xI|x|≤k + sgn(x)kI|x|>k (cf. Huber, 1981), where k is some

positive constant and IA is the indicator function of an event A. Then, a robust covariance

and correlation may be defined as

[ (

) (

)]

X − 𝜇R (X)

Y − 𝜇R (Y)

𝛾R (X, Y) = 𝜎R (X)𝜎R (Y)E 𝜓

𝜓

𝜎R (X)

𝜎R (Y)

and

𝛾R (X, Y)

,

𝜌R (X, Y) = √

𝛾R (X, X) 𝛾R (Y, Y)

respectively (cf. Maronna et al., 2006). Although these measures are robust to outlying and

extreme observations, they depend on the choice of transformation 𝜓. They provide an intuitive measure of association, however, for an arbitrary joint distribution on (X, Y), 𝜌R (X, Y) ≠

𝜌P (X, Y), in general. Sample versions are obtained by replacing the expectation, 𝜇R and 𝜎R ,

by their sample estimates in the above expressions. Asymptotic sampling distributions can be

derived, but the setting is more complicated than above. Finally, robust pairwise covariances

and correlations such as these may be used to define corresponding covariance and correlation matrices for random vectors, but the result is not positive definite (or affine equivariant),

in general.

8.2.1.2

Kendall’s Tau

Rank-based methods measure all monotonic relationships and are resistant to outliers.

Kendall’s tau (Kendall, 1938) is a nonparametric measure of dependence, for which the

sample estimate considers the pairwise agreement between two ranked lists.

Definition 8.4 For random variables X and Y, Kendall’s tau is defined as

𝜏(X, Y) = P[(X − X ∗ )(Y − Y ∗ ) > 0] − P[(X − X ∗ )(Y − Y ∗ ) < 0]

= E{sgn[(X − X ∗ )(Y − Y ∗ )]},

in which (X ∗ , Y ∗ ) denotes an i.i.d. copy of (X, Y).

(8.1)

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