2 Sampling Correlation from a Distribution (Hull and White 2010)
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pﬃﬃﬃﬃﬃﬃﬃ
Replacing r(t) = a(t) in equation (8.3) to allow for negative correlation
between M(t) and z(t), we derive
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dzi (t) = ai (t)dM(t) + 1 − a2 (t)dZi (t)
(8.4)
Hull and White now introduce stochastic correlation by sampling a(t)
from a beta distribution. This sample serves to create a dependency between
a(t) and dM(t). To match empirical
credit default index (CDX) prices, they
pﬃﬃﬃﬃﬃﬃﬃﬃ
choose the dependency to be − 0:5: This creates a positive relationship
between a and default probability: If a increases, the market environment dM
decreases. This increases the default probability of company i.
The asset correlation between companies i and j is ai aj (see Chapter 12,
section 12.6 for details). A higher ai or aj means a higher asset correlation
between i and j, and, as derived above, it means a higher joint default probability.
This relationship of higher asset correlation and higher default probability was
empirically veriﬁed by Servigny and Renault (2002) and Das et al. (2006).
Therefore it is not surprising that the approach with the stochastic
correlation sample a(t) in the model of equation (8.4) is able to match
empirical CDX prices in most cases signiﬁcantly better than is the case
without stochastic correlation; see Hull and White (2010) for details.
8.3 DYNAMIC CONDITIONAL CORRELATIONS
(DCCs) (ENGLE 2002)
In equation (3.3) we had deﬁned the Pearson correlation coefﬁcient for a
random variable as
E(XY) − E(X)E(Y)
r(X; Y) = qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
E(X2 ) − (E(X))2 E(Y 2 ) − (E(Y))2
(3.3)
Assuming the variables X and Y have a mean of zero [i.e., E(X) = E(Y) = 0],
equation (3.3) reduces to
E(XY)
r(X; Y) = pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
E(X2 )E(Y 2 )
(8.5)
In 2002 Robert Engle introduced dynamic conditional correlations
(DCCs) in a model developed by Tim Bollerslev in 1990. The correlation
at time t, rt, is conditioned on the information given in the previous period
t−1. Hence equation (8.5) changes to
Et−1 (r1;t r2;t )
rt (r1 ; r2 ) = qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Et−1 (r21;t )Et−1 (r22;t )
(8.6)
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where r is the variable of interest. r1,t may be the return of asset 1, and r2,t
may be the return of asset 2 at time t. See Chapter 1, section 1.3.1
on returns.
Conditional correlation is a concept within the Nobel Prize-rewarded
autoregressive conditional heteroscedasticity (ARCH) framework (Engle
1982), which was extended to generalized autoregressive conditional heteroscedasticity (GARCH) by Bollerslev (1986).
In the ARCH framework, the variable of interest, the return r, is deﬁned
as the product of its standard deviation and an error term.
ri;t = si;t ei;t
(8.7)
where
ri,t: return of asset i at time t
si,t: standard deviation of the return of asset i at time t (also called
volatility)
ei,t: random drawing of a standard normal distribution for asset i and
time t, e = n ∼ (0,1)
The variance s2 or the standard deviation s in equation (8.7) is modeled
with an ARCH process (or one of many extensions such as GARCH,
NGARCH, EGARCH, TGARCH,1 and more) of the form
s2t = a0 + a1 s2t − 1 e2t − 1 + ::: + aq s2t − q e2t − q
(8.8)
where a0 > 0, a1 ³ 0, so that s2 is positive and q ∈ N.
From equation (8.8) we can observe that the variance is a function of
past error terms e. The error term e is typically derived from a linear
regression of the underlying variable of interest, which in equation (8.7) is
the return of asset i. The critical idea in equation (8.8) is that if the past error
terms et–x are high, so will be the future variance at time t, s2t : The model of
equation (8.8) and extensions of the model have been successfully tested;
see for example Enders (1995) and Hacker and Hatemi-J (2005). The main
contribution of the ARCH and GARCH approach is that the empirical
persistence or clustering of volatility can be modeled: In reality, high
volatility often persists for a certain period of time, and low volatility
often persists for a certain period of time.
1. The N in NGARCH stands for nonlinear, the E in EGARCH stands for exponential, and the T in TGARCH stands for truncated. See Bollerslev (2008) for a nice
overview of all ARCH extensions.
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Et 1 (r1;t r2;t )
The correlation at time t in equation rt (r1 ; r2 ) = pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
depends
Et 1 (r21;t )Et 1 (r22;t )
on information given at time t − 1. Given certain assumptions, the correlation
can be expressed purely in error terms e. Let’s show this. We express the
variance of returns as a function of the return, given the information in the
previous period.
s2i;t = Et−1 (r2i;t )
(8.9)
Assuming the return r is standard normal, we have Et−1 (r2i;t ) = 1, and
from equation (8.9) s2i;t = 1: Hence from equation (8.7) we derive that the
return r is just the error term e, and together with Et−1 (r2i;t ) = 1, equation (8.6)
reduces to
rt (r1 ; r2 ) = Et−1 (e1;t ; e2;t )
(8.10)
The conditional correlation expressed in equations (8.6) and (8.10)
correlates just two assets, 1 and 2. The model can be generalized to include
multiple assets. In this case, we derive the conditional correlation matrix R,
which contains the pairwise conditional correlations rt(rij) between the n asset
returns. Formally, from equation (8.10) we have
Et−1 (ei;t ; ej;t ) = Rij
(8.11)
where R: conditional correlation matrix containing the pairwise conditional
correlations of the returns of the assets i = 1,:::, n.
In equation (8.11), the correlation matrix Rij is constant. The approach
can be made dynamic; that is, Rij can be time varying, Rij(t). This constitutes
dynamic conditional correlations (DCCs), as suggested by Engle (2002).
Parameterization of the dynamic conditional correlation matrix Rij(t) in a
GARCH framework can be achieved by exponential smoothing, with certain
parameter constellations allowing mean reversion of the matrix process. See
Engle (2002) for details.
8.4 STOCHASTIC CORRELATION—
STANDARD MODELS
In this section, we introduce three approaches that model stochastic correlation. The three models are quite closely related.
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8.4.1 The Geometric Brownian Motion (GBM)
The geometric Brownian motion, whose basic idea was derived by the
biologist Robert Brown in 1827, has already been mentioned several
times in this book; see Chapter 4, equation (4.1), and in this Chapter 8,
equation (8.2).2 In ﬁnance, variables such as stocks, bonds, commodities,
interest rates, and volatility are often modeled with the GBM. When
modeling correlation with the GBM, we derive
pﬃﬃﬃﬃﬃ
dr
= m dt + se dt
r
(8.2)
where
r: correlation between two or more variables
m: expected growth rate of r
s: expected volatility of r
e: random drawing from a standard normal distribution; formally,
e = n ∼ (0,1)
We can compute e as =normsinv(rand()) in Excel/VBA and norminv(rand) in
MATLAB.
Düllmann, Küll, and Kunisch (2008) model correlation with equation (8.2). They study whether stock prices or default rates can better
estimate asset correlations. Applying stochastic asset correlation in equation (8.2) rather than constant asset correlation, they ﬁnd that the
stochastic correlation model weakens but does not reject the result that
stock prices are superior for estimating asset correlations compared to
default rates.
Is the GBM in equation (8.2) a good approach to model correlation? It
actually has two limitations:
1. Equation (8.2) is not bounded, meaning correlation r can take values
bigger than 1 and smaller than –1. From equation (8.2) we see that a
value of r > 1 is more likely to happen when the growth rate m is high, if
the volatility s is high, and if we have a high value of e in a simulation.
2. It was actually the Dutch biologist Jan Ingenhousz who ﬁrst published papers in
German and French in 1784 and 1785 on the dispersion of charcoal particles in
alcohol. Therefore, he should be credited for what is known today as the Brownian
motion.
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GBM
1.00
0.36
Correlation ρ
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0.13
0.17
0.20
t2
t3
0.22
0.25
0.28
0.27
0.14
0.13
0.00
–1.00
t1
t4
t5
t6
t7
t8
t9
t10
FIGURE 8.1 Sample Correlation Path for 10 Time Steps for Equation (8.12)
The parameter values are m = 1%, s = 30%, and Dt = 1. Starting value in t0 is 0.1.
Conversely, values of r < –1 are more likely to occur for low values of m
and high values of s and e.
2. Mean reversion (i.e., the tendency of correlation to revert back to its
mean) is not modeled with equation (8.2). In the empirical Chapter 2, we
derived that ﬁnancial correlations exhibit strong mean reversion.
For computational purposes, we discretize equation (8.11). With dr =
rt+1 – rt, we derive
pﬃﬃﬃﬃﬃﬃ
rt+1 = rt + rt mDt + rt set Dt
(8.12)
Figure 8.1 shows a sample path of the GBM.
In Figure 8.1, at each time step, equation (8.12) is applied. The
different values for correlation at each time step occur since the random
drawing e is different at each t.3 For details, see the spreadsheet “Stochastic correlation.xlsx” at www.wiley.com/go/correlationriskmodeling, under
“Chapter 8.”
3. Although the correlation values at t1 and t10 are both 0.13, e in t1 and t10 are
different since r increases from t0 to t1 from 0.1 to 0.13 and decreases from t9 to t10
from 0.14 to 0.13.
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8.4.2 The Vasicek 1977 Model
Another approach to model stochastic correlation is what is known as the
Vasicek 1977 model, which, however, should be credited to Uhlenbeck and
Ornstein (1930). The model is
pﬃﬃﬃﬃﬃ
dr = a(mr − rt )dt + sr et dt
(8.13)
where a is the mean reversion speed (gravity) (i.e., degree with which the
correlation at time t, rt, is pulled back to its long-term mean mr); a can take the
values 0 £ a £ 1. mr is the long-term mean of the correlation r. Other
variables are deﬁned as in equation (8.2).
Equation (8.13) is an improvement to the GBM in equation (8.2) since it
includes mean reversion, the tendency of a variable to be pulled back to its
long-term mean. We derived in Chapter 2 that ﬁnancial correlations exhibit
strong mean reversion.
The limitation of the Vasicek 1977 model with respect to modeling
correlation is that the model is not bounded; correlation values bigger than 1
and smaller than –1 can occur. These values are more likely to occur when
mean reversion a is low and volatility sP is high.
For computational reasons, we again discretize. With dr = rt+1 – rt,
equation (8.13) then becomes:
pﬃﬃﬃﬃﬃﬃ
(8.14)
rt+1 = rt + a(mr − rt )Dt + sr et Dt
Figure 8.2 shows a sample path of the Vasicek model.
Comparing Figures 8.1 and 8.2, we observe the higher volatility in
Figure 8.2. This is mainly because the relative change dr/r is modeled in
Figure 8.1, whereas the absolute change dr is modeled in Figure 8.2;
compare equations (8.2) and (8.13).
8.4.3 The Bounded Jacobi Process
The two approaches that we have introduced so far, the geometric Brownian
motion and the Vasicek model, both have the limitation that correlation
values can become bigger than 1 and smaller than –1. This is an undesired
property if the correlation is modeled in the Pearson correlation framework,
where the correlation coefﬁcient is bounded between –1 and +1.
A model that can comply with correlation bounds is the bounded Jacobi
process.4 Applying the bounded Jacobi process to correlation, we derive
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ
(8.15)
dr = a(mr − rt )dt + sr (h − rt )(rt − f )et dt
4. For a nice paper on the Jacobi process, see Gourieroux and Valery (2004).
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Vasicek 1977 Model
1.00
0.84
0.58
0.61
0.59
0.55
0.40
0.35
Correlation ρ
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0.25
0.00
–0.29
–1.00
t1
t2
t3
t4
t5
t6
t7
t8
t9
–0.24
t10
FIGURE 8.2 Sample Correlation Path for 10 Time Steps for Equation (8.14)
The mean reversion parameter a = 30%, and the long-term mean mr = 10%. The
other parameter values are the same as in Figure 8.1: Volatility s = 30% and Dt = 1.
Starting value in t0 is 0.1.
where h is the upper boundary level, and f is the lower boundary level (i.e.,
h ³ r ³ f). Other variables are deﬁned as in equations (8.2) and (8.13).
With equation (8.15) the user can choose speciﬁc upper and lower
boundaries. For correlation modeling in the Pearson framework, these
boundaries are h = +1 and f = –1. In this case equation (8.15) reduces to
dr = a(mr − rt )dt + sr
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ
(1 − r2t )et dt
(8.16)
Equation (8.15) requires correlation
values withinﬃ a lower bound f and
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
an upper bound h (otherwise the term (h − rt )(rt − f ) cannot be evaluated).
Equation (8.16) requires
correlation
values within the bounds –1 to +1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
(otherwise the term (1 − r2t ) cannot be evaluated). However, for low mean
reversion levels a and high volatility sP, it can happen that the model
generates correlation levels smaller than –1 and higher than +1. Therefore
we have to introduce boundary conditions. These boundary conditions for
equation (8.15) are
a³
s2 (h − f )=2
(mr − f )
(8.17)
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for the lower bound f and
a³
s2 (h − f )=2
(h − m r )
(8.18)
for the higher bound h.
Applying the boundary levels f = –1 and h = +1, we derive the boundary
levels for equation (8.16) as
a³
s2
(mr + 1)
(8.19)
a³
s2
(1 − mr )
(8.20)
for the lower bound f and
for the higher bound h.
From equations (8.17) to (8.20) we observe the intuitive feature that the
bounds are more likely to be satisﬁed for high values of mean reversion a and
low values of volatility s. See Emmerich (2006) and Wilmott (1998) for the
derivation of the boundaries.
Ma (2009) applies the bounded Jacobi process to value correlation
dependent quanto options and multi-asset options. This inclusion of
stochastic correlation in the Black-Scholes-Merton model improves the
valuation of these options compared to the standard Black-Scholes-Merton
model.
Discretizing equation (8.16), again applying dr = rt+1 – rt, we derive
rt+1 = rt + a(mr − rt )Dt + sr
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
(1 − r2t )et Dt
(8.21)
Figure 8.3 shows a sample path of the equation (8.21).
Comparing Figures 8.2 and 8.3, we observe somewhat minor differences between the correlation modeling with Vasicek in equation (8.14)
and the bounded Jacobi process in equation (8.21). The correlation models
introduced so far in this chapter can be found in the spreadsheet
“Stochastic correlation.xlsx” at www.wiley.com/go/correlationriskmodeling, under “Chapter 8.”
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Bounded Jacobi Process
1.00
0.75
0.56
0.56
0.53
0.50
0.41
0.35
Correlation ρ
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0.26
0.00
–0.26
–1.00
t1
t2
t3
t4
t5
t6
t7
t8
t9
–0.22
t10
FIGURE 8.3 Sample Correlation Path for 10 Time Steps for Equation (8.21)
As in Figures 8.1 and 8.2, the mean reversion parameter a = 30%, volatility
s = 30%, the long-term mean mr = 10%, and Dt = 1. Starting value in t0 is 0.1.
8.5 EXTENDING THE HESTON MODEL WITH
STOCHASTIC CORRELATION (BURASCHI ET AL.
2010; DA FONSECA ET AL. 2008)
In Chapter 4, section 4.1, we had analyzed and praised the Heston 1993
correlation model. It is a mathematically rigorous, dynamic correlation
model, which is widely applied in ﬁnance.
Slightly rewriting the equations in Chapter 4.1, the Heston model
consists of three main equations.
dS
= m dt + st dz1
S
(8.22)
ds2t = a(m2s − s2t )dt + xst dz2
(8.23)
where
S: variable of interest e.g., a stock price
m: growth rate of S
s: volatility of S
dz: Brownian motion or Wiener process with e: random drawing from of
standard normal distribution with a mean of 0 and a standard
deviation of 1. Formally, e = n ∼ (0,1)
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a: mean reversion rate (gravity), i.e., degree with which s2 at time t, s2t , is
pulled back to its long-term mean m2r . a can take the values 0 £ a £ 1
m2s : long-term mean of the variance rate s2
x: volatility of the volatility s.
The stochastic process of S in equation (8.22) and the stochastic variance
rate of S, s2 in equation (8.23) are correlated with the identity
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
dz1 (t ) = r1 dz2 (t ) + 1 − r1 dz3 (t )
(8.24)
where dz2(t) and dz3(t) are independent, and dz(t) and dz(tʹ ) are independent,
t ≠ tʹ .
Buraschi, Porchia, and Trojani (2010) (the ﬁrst version of the paper
appeared in 2006) and Da Fonseca, Grasselli, and Ielpo (2008) extend the
Heston 1993 model with a more rigorous correlation structure. The model is
based on the Wishart afﬁne stochastic correlation (WASC) model, introduced
by Bru (1991) and extended by Gourieroux and Sufana (2010).
The model is presented as an n-dimensional stochastic process of
covariance matrices.5 For ease of exposition, we will concentrate on n =
2 assets. In this case, S in equation (8.22) expands to a price vector of two
assets, S1 and S2, formally S = (S1, S2)T, where T stands for transpose. The
stochastic process for S is
pﬃﬃﬃﬃﬃ
dSt = I s m dt + St dZt
(8.25)
where
IS = Diag[S1, S2], i.e. a diagonal 2 ´ 2 matrix, with entries of equation
(8.25) on the diagonal and zero entries otherwise
m: growth rate of the two-dimensional vector S
dZt: 2-dimensional Brownian motion
St: covariance matrix of the returns of asset S1 and S2
In our two-asset case, the covariance matrix St takes the form
2
St = 4
12
S11
t St
22
S21
t St
3
5
(8.26)
5. We will use some matrix algebra in the following. See “Matrix primer.xlsx”
at www.wiley.com/go/correlationriskmodeling, under “Chapter 1,” for some basic
matrix operations.
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22
where S11
t and St are the variances of the returns of asset 1 and asset 2,
21
respectively, and S12
t and St are the covariances of the returns of assets 1 and
2. Note that the covariance of a single asset is equal to its variance; that is,
Covariance(i,i) = Variance(i). Therefore, a covariance matrix has variances
on its main diagonal and is therefore also called variance-covariance matrix.
Also note that the covariance of two assets is commutative; that is,
21
Covariance(ij) = Covariance(ji); hence in the matrix (8.26) S12
t = St .
At the core of the model, the covariance matrix (8.26) follows a
stochastic process of the form
dSt = (WWT + MSt + St MT )dt +
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
St dW t Q + QT (dW t )T St
(8.27)
where
pﬃﬃﬃﬃﬃ
Q: volatility of co-volatility matrix St , corresponding to x in equation
(8.23)
M: negative semideﬁnite matrix,6 which controls the degree of mean
reversion of St, corresponding to a in equation (8.23)
W: related to the long-term mean of the covariance matrix St, corresponding to m2s in equation (8.23)
W: two-dimensional Brownian motion
In the original Heston model, the stochastic process for the underlying
asset S and the stochastic process of the variance rate s2 are correlated by
correlating the Brownian motions of these processes; see equation (8.24).
Accordingly, the Brownian motions of equations (8.25) and (8.27) are
correlated:
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dZ(t ) = rdW (t ) + 1 − rT r dB(t )
(8.28)
where dW(t) and dB(t) are independent, and dZ(t) and dZ(tʹ ) are independent, t ≠ tʹ.
Equation (8.28) correlates the Brownian motions of equations (8.25) and
(8.27). Conveniently, the model admits a closed form solution for the
correlation between the underlying return assets S and their variance S.
For asset 1 we have
r1 Q11 + r2 Q21
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Corr(d ln S1 ; dS11 ) = q
Q211 + Q221
(8.29)
6. See “Matrix primer.xlsx” at www.wiley.com/go/correlationriskmodeling, under
“Chapter 1,” for details.