6 Stochastic Correlation, Stochastic Volatility, and Asset Modeling (Lu and Meissner 2012)
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FIGURE 8.4 (a) Empirical Relationship between Implied Volatility (VIX) of the S&P
500 and the Correlation between Chevron Corporation and the S&P 500; (b) Time
Series Plot of the Empirical VIX and Empirical Correlation between CVX and the
S&P 500; (c) Histogram of VIX; (d) Histogram of the Correlation Coefﬁcient
Data: 01/03/2000 to 07/27/2011.
Equation (8.33) models stochastic volatility with the Cox-Ingersoll-Ross
(CIR) (1981) model. Equation (8.34)
models
ﬃ stochastic correlation with a
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
modiﬁed Jacobi process. The term 1 − r2 bounds correlation between –1
and +1. However, as discussed in section 8.4, for low mean reversion levels ar
and high volatility nr, 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 are as displayed in equations (8.19) and (8.20).
The model of equations (8.33) to (8.35) intends to replicate real-world
volatility-correlation properties. Figure 8.4 shows some real-world correlationvolatility relationships.
The real-world relationships displayed in Figure 8.4 can be replicated
well by the model of equations (8.33) to (8.35), as displayed in Figure 8.5.
Of special interest is the relationship between volatility s (of the S&P 500
in the example, modeled by the VIX) and the correlation r (between a
particular stock, Chevron, and the S&P 500 in the example), displayed in
the top left chart of Figures 8.4 and 8.5. We observe that the relationship is
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FIGURE 8.5 Simulation Results of the Model of Equations (8.33) to (8.35)
somewhat triangular; that is, (1) it is positive, and (2) the correlation volatility
nr decreases if the volatility s (represented by the VIX) increases.
The positive relationship between correlation r and s in Figure 8.4 is
replicated if rw in equation (8.35) is positive. In addition, the decreasing
correlation volatility nr as a function of the increase in s (the VIX) is
incorporated in the
model: If s increases, r increases (if rw is positive).
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
From the term ur 1 r2 it follows that if r increases, the volatility of r,
nr, decreases. Hence it follows that if s (represented by the VIX) increases, nr
decreases as displayed in Figure 8.5, top left.
8.6.1 Asset Modeling
The model of equations (8.33) and (8.35) can be applied to model assets. Asset
modeling is often done with geometric Brownian motion (GBM), which we
discussed in equations (4.1), (8.2), and (8.22). Here is the GBM once again:
dSi
= mi dt + si dz
Si
where
Si: asset price of a particular asset i
mi: drift of Si,
si: volatility of Si
(8.36)
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FIGURE 8.6 Probability Density Function (PDF) for Coca-Cola Corporation (KO)
Model data are derived with equations (8.33) to (8.35) and equation (8.37).
Lu and Meissner now expand the GBM and model
dSi
= mi dt + si dz + bi rsdw
Si
(8.37)
where bi is constant with 0 £ bi £ 1, and r is the correlation between an
individual stock and the market, represented by the S&P 500; r is modeled as
a stochastic process by equation (8.34). s is the volatility of the market,
represented by the VIX of the S&P 500, which is modeled by equation (8.33).
The Brownian motions of r and s are correlated via equation (8.35). dw is the
Brownian motion of the market component.
Equation (8.37) has a capital asset pricing model (CAPM) interpretation.
The ﬁrst two terms on the right side of equation (8.37) represent the
idiosyncratic stock component. The term s dw represents the systematic
market risk factor, which is shared by all stocks. The impact magnitude of the
systematic component on the stock is bi r.
In Figure 8.6, the performance of the model of equations (8.33) to (8.35)
and equation (8.37) is compared to the standard GBM of equation (8.35).
From Figure 8.6 we observe that the model of equations (8.33) to (8.35)
and equation (8.37) outperforms the standard GBM in equation (8.36).
This is veriﬁed by standard statistics. The chi-square goodness-of-ﬁt test
shows a p-value of 0.8164 (chi2 = 5.986) between the model distribution and
the empirical distribution, while the p-value is 0.054 (chi2 = 19.411) between
the GBM-normal distribution and the empirical distribution. The model gives
similar results for other stocks that were tested.
Lu and Meissner (2012) extend the model to include correlation between
individual stocks in a portfolio approach, applying the conditionally independent default (CID) correlation concept, which we discussed in Chapter 6.
This improves the performance of the model further.
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8.7 CONCLUSION: SHOULD WE MODEL FINANCIAL
CORRELATIONS WITH A STOCHASTIC PROCESS?
Many assets in ﬁnance are modeled with a stochastic process. Assets that are
assumed to have little or no mean reversion, such as stocks, exchange rates, or
real estate values, are modeled with a non-mean-reverting stochastic process
such as the GBM, displayed in equations (4.1), (8.2), and (8.22), or they
can be modeled with no-arbitrage, non-mean-reverting models such as the
Ho-Lee (1986) model.
Assets that display a certain degree of mean reversion such as bonds,
interest rates, or default probabilities are typically modeled with a stochastic
process, which includes mean reversion such as the Vasicek 1977 model
displayed in equation (8.13), or with mean-reverting no-arbitrage models
such as Hull and White (1990) or the Black-Derman-Toy (1990) model. The
continuous-time, mean-reverting Heath-Jarrow-Morton (HJM) 1990 model
and its discrete version, the LIBOR market model (LMM) of 1997 (credited to
three groups of authors: Brace, Gatarek, and Musiela; Miltersen, Sandmann,
and Sondermann; and Jamshidian) are generalized models and include the
aforementioned models as special cases.
Since many ﬁnancial assets are successfully modeled with a stochastic
process, should we also model ﬁnancial correlations with a stochastic
process? This is mainly an empirical question: Do ﬁnancial correlations in
the real world behave in a way that can be captured with a certain stochastic
model? The research in this area has just started, but the ﬁrst results are
promising.
We discussed the conditional correlation modeling approach of
Bollerslev (1990), generalized by Engle (2002); sampling correlation values
from a stochastic distribution (Hull and White 2010); the Vasicek model
applied by Duellman et al. (2008); modeling correlation with a modiﬁed
Jacobi process (Ma 2009); the Wishart afﬁne stochastic correlation (WASC)
model applied by Buraschi et al. (2010; ﬁrst version 2006); and an extension
by Da Fonseca et al. (2008); as well as the approach by Lu and Meissner
(2012). All these approaches show that when applying a certain form of
stochasticity for ﬁnancial correlations, correlation properties in reality can be
replicated quite well. In addition, the valuation of correlation-dependent
structures such as multi-asset options or quanto options (Ma 2009) can be
improved if correlation is modeled with a stochastic process.
In conclusion, while stochastic correlation modeling is still in its infancy,
ﬁrst results are promising. Just as other ﬁnancial variables such as stocks,
bonds, interest rates, commodities, volatility, and more are modeled with a
stochastic process, it can be expected that in the near future ﬁnancial
correlations will also typically be modeled with a stochastic process.
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8.8 SUMMARY
The modeling of ﬁnancial correlations with a stochastic process is fairly new,
but several promising approaches exist. We discussed them in this chapter.
Hull and White (2010) ﬁnd a simple way to address stochastic correlation. They create a dynamic version of the one-factor Gaussian copula
(OFGC) model (see Chapter 6 for details). Hull and White then sample
the critical correlation parameter, which indirectly correlates the variables
with a beta distribution. The model matches empirical CDX prices in most
cases better than a comparable model without stochastic correlation.
In 2002 Robert Engle introduced a dynamic conditional correlation
(DCC) concept within the ARCH and GARCH framework. The correlation
depends on expectations given at a previous point in time. In addition, the
correlation matrix can be made a function of time, constituting a dynamic
stochastic correlation model.
A further way to model ﬁnancial correlations is to use the standard
geometric Brownian motion (GBM), which is often applied to model other
ﬁnancial variables such as stocks, exchange rates, commodities, and more.
However, the standard GBM suffers from two drawbacks: (1) Mean reversion, a critical property of ﬁnancial correlations as we derived in Chapter 2, is
not incorporated in the GBM, and (2) the model is not bounded, meaning
correlation values larger than 1 and smaller than –1 can occur.
An improvement of the GBM for modeling ﬁnancial correlations is the
Vasicek 1977 model, which incorporates mean reversion, or the bounded
Jacobi process, which incorporates mean reversion and is also bounded (i.e., the
correlation values lie between –1 and +1 if boundary conditions are imposed).
A rigorous, mathematically quite intensive approach to model ﬁnancial
correlations is based on the Wishart afﬁne stochastic correlation (WASC)
model. Here a stochastic covariance matrix follows a stochastic process,
which is—as in the Heston 1993 model—correlated with the stochastic
process of the underlying price matrix. The model has numerous parameters
and is able to model several real-world correlation properties well.
In the related stochastic correlation model of Lu and Meissner (2012),
which correlates stochastic correlation with stochastic volatility, it is shown
that asset modeling can be improved compared to the standard GBM.
PRACTICE QUESTIONS AND PROBLEMS
1. What is a deterministic process? Name two examples.
2. What is a stochastic or random process? Name two examples.