Modelling stochastic correlation
 Long Teng^{1}Email author,
 Matthias Ehrhardt^{1} and
 Michael Günther^{1}
DOI: 10.1186/s1336201600184
© Teng et al. 2016
Received: 16 November 2015
Accepted: 2 March 2016
Published: 15 March 2016
Abstract
This work deals with the stochastic modelling of correlation in finance. It is well known that the correlation between financial products, financial institutions, e.g., plays an essential role in pricing and evaluation of financial derivatives. Using simply a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that the correlation is hardly a deterministic quantity. For example, we illustrate this issue with the analysis of correlation between daily returns time series of S&P Index and Euro/USD exchange rates.
The approach of modelling the correlation as a hyperbolic function of a stochastic process has been recently proposed. Here, we review this novel concept and generalize this approach to derive stochastic correlation processes (SCP) from a hyperbolic transformation of the modified OrnsteinUhlenbeck process. We determine a transition density function of this SCP in closed form which could be used easily to calibrate SCP models to historical data.
As an illustrating example of our new approach, we compute the price of a quantity adjusting option (Quanto) and discuss concisely the effect of considering stochastic correlation on pricing the Quanto.
Keywords
stochastic correlation quanto option correlation risk OrnsteinUhlenbeck process transition density hyperbolic tangent function1 Introduction
Correlation is a well established concept for quantifying the relationship between financial assets. It plays an essential role in several financial applications, e.g. the arbitrage pricing model [1] is based on correlation as a measure for the dependence of assets. Also in portfolio credit models, the default correlation is one fundamental factor of risk evaluation, see e.g. [2] and [3].

If the random variables \(X_{1}\) and \(X_{2}\) are independent, then it follows \(\rho_{1,2}=0\). However, the converse implication does not hold, since in (1) only the two first moments are included. For example, we compute \(\rho_{1,2}=0\) for \(X_{2}=X_{1}^{2}\). Indeed, \(X_{1}\) and \(X_{2}\) depend even almost perfectly on each other. This illustrates that the correlation coefficient only recognizes linear dependencies between random variables.

Correlation is invariant under strictly increasing linear transformations, but, in contrast to Copula methods, not invariant under nonlinear strictly increasing transformations. For example, in general the correlation of the random variables \(X_{1}\) and \(X_{2}\) does not equal the correlation of the random variables \(\ln X_{1}\) and \(\ln X_{2}\), i.e. after a transformation of the financial data the correlation may change.

Usually, the given marginal distributions and pairwise correlations of a random vector cannot determine its joint distribution.

Finally, as stated above, the variances of the two random variables \(X_{1}\) and \(X_{2}\) has to be finite. This assumptions is not fulfilled for every standard distribution, e.g. the Student’s tdistribution with \(v\le2\) possess an infinite variance.
As we explained above, the constant correlation coefficient defined by (1) only captures linear relationships between \(X_{1}\) and \(X_{2}\). Therefore, in the model (5) a linear dependence between \(S_{t}\) and \(R_{t}\) is assumed. However, from the market we realize that there is often a nonlinear dependence between \(S_{t}\) and \(R_{t}\). Specifically, a constant correlation means that the two return processes are jointly stationary which is generally not true in the real world. Thus, the dependence can be hardly modelled by a fixed constant, i.e. the constant correlation may not be an appropriate measure of codependence. Using constant (“wrong”) correlation may result some ’correlation risk’. There exist already some works which show that the correlation should not be constant and even changes over a small time interval as the volatility, see e.g. [5]. Several approaches generalize the constant correlation to a timevarying and stochastic concept, like Dynamic Conditional Correlation model in [6], Local correlation models see e.g. [7] and the Wishart autoregressive process proposed by Gourieroux [8] that guarantees the positive definiteness of the variancecovariance matrix.
We observe that the longer a time window (the value of \(n_{T}\)) the less volatile a historical correlation is. In Figure 1, the 15day historical correlation is more variable than the 30day historical correlation which is again more variable than the 60day correlation. With a longer averaging period a longterm correlation is calculated. If we choose \(n_{T}=10\text{ or }15\) days, the estimated correlation for each time t using (6), could be seen as a shortterm correlation of the current market phenomena whose immediate past returns are used for the estimation. It is worthwhile noting that the events, especially, some extreme events in a time window will affect the correlation which would be estimated in the following time windows, even has a delayed effect on the longterm correlation.
If one assumes that the phenomena in the past could have a reflection in the future, one would like to use the historical correlation as a forecast for the future. It could be a better way for correlation forecasting, if one describes the correlation using a meanreverting stochastic process. Besides, modelling correlation as a stochastic process, not only the variation of the shortterm correlation can be reflected, also the attributes of longterm correlation is determined by the longterm parameter values, like longterm mean value and mean reversion speed.
 (i)
only takes values in the interval \((1,1)\),
 (ii)
varies around a mean value,
 (iii)
the probability mass tends to zero at the boundaries −1, +1.
One stochastic correlation process was proposed by van Emmerich [10], including a restriction on the parameter range to ensure that the boundaries −1 and 1 of the correlation process are not attractive and unattainable. A modified Jacobi process is suggested in [11] modelling stochastic correlation. A more general stochastic correlation process was proposed by Teng et al. [12], which relies on the hyperbolic transformation with the hyperbolic tangent function of any meanreverting process with positive and negative values, the properties (i)(iii) above can be thus directly satisfied without facing any additional parameter restrictions. Hence, the subsequent calibration process is much simpler.
In this work, we study the general SCP by Teng et al. [12]. We show that the correlation process by van Emmerich can be obtained by this general method, i.e. the correlation process by van Emmerich turns out to be a special case of the hyperbolic transformation of a stochastic process. Furthermore, we apply this general approach to find a new SCP which has a transition density function in closed form. Finally, as an illustrating example, we compute the price of a Quanto under stochastic correlation by our new SCP and investigate the effect of considering stochastic correlation on pricing the Quanto.
2 A general stochastic correlation model
Here we study the hyperbolic transformation proposed in [12] of a meanreverting process to be a correlation process. We show that the correlation process model of van Emmerich [10] can be obtained by transforming a meanreverting process with the hyperbolic tangent function. We fix a probability space \((\Omega,\mathcal{F},\mathbb{P})\) and an information filtration \((\mathcal{F}_{t})_{t\in\mathbb{R}^{+}}\) satisfying the usual conditions, see e.g. [13].
2.1 The transformed meanreverting process
2.2 Transformation with other functions
2.3 The correlation model of van Emmerich
3 Stochastic correlation with a modified OrnsteinUhlenbeck process
In this section, we specify a SCP by a hyperbolic transformation of the modified OrnsteinUhlenbeck process. The derivation of the transition density function of this SCP is provided in a closed form. Then, we analyse this density function and show how to fit the correlation process to the historical market data.
3.1 The transformed modified OrnsteinUhlenbeck process
Lemma 1
Proof
3.2 Transition density function
For calibration purposes, we first determine the transition density function of (22) with the aid of the FokkerPlanck equation [15]. Then, we obtain the parameters of the correlation process (22) by fitting the density function to the market data.
3.3 Calibration
We assume that the correlation is itself observable. Under this assumption the transition density can be used for calibration purposes. One uses usually maximumlikelihood estimation (MLE) when the density function is known. Considering the density function (39), it will be tedious to determine its likelihoodfunction.
4 Stochastically correlated Brownian motions
The remaining problem is how to incorporate the stochastic correlation process in the financial model, e.g. how to use the stochastic correlation in the option pricing model. In Section 1, we mentioned that a widely used approach for dependence is to consider the (constant) correlated Brownian motions. In order to consider a stochastic correlation, we need the concept of stochastically correlated Brownian motions. In the following, we study the stochastically correlated Brownian motions following the work of van Emmerich [10].
Lemma 2
 (1)
\(W_{1,0}=0\),
 (2)
\(\mathbb {E}[({W_{1,t}})^{2} ]=t\),
 (3)
\(\mathbb {E}[W_{1,t}\vert \mathcal{F}_{s}]=W_{1,s}\), for \(s \le t\).
Proof
5 Pricing quantos with stochastic correlation
In contrast, fixing a value for κ, the price differences between using constant and stochastic correlation become bigger by increasing the value of the diffusion σ (and thus randomness in the SCP process), as shown in Figure 10b.
6 Conclusion
In this work we have revised concisely some stochastic correlation models. Market observations give strong evidence that financial quantities are correlated in a strongly nonlinear, nondeterministic way. Instead of assuming a constant correlation, correlation has to be modelled as a stochastic process. We discussed first the general stochastic correlation model proposed in [12] and proved that the stochastic correlation process in [10] can be obtained by applying this general approach.
We generalized our approach [12] to derive a stochastic correlation model from a hyperbolic transformation of the modified OrnsteinUhlenbeck process allowing for a transition density function in a closed form and an easytohandle calibration to historical data. As an example, we computed the fair price of a Quanto Putoption with stochastic correlation. The numerical results showed that the correlation risk caused by using a wrong (constant) correlation model cannot be neglected.
Declarations
Acknowledgements
The work of the authors was partially supported by the European Union in the FP7PEOPLE2012ITN Programme under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project MultiITN STRIKE  Novel Methods in Computational Finance). Further the authors acknowledge partial support from the bilateral GermanSpanish Project HiPeCa  High Performance Calibration and Computation in Finance, financed by DAAD.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Campbell JW, Lo AW, MacKinlay AC. The econometrics of financial markets. Princeton: Princeton University Press; 1997. MATHGoogle Scholar
 Brigo D, Chourdakis K. Counterparty risk for credit default swaps: impact of spread volatility and default correlation. Int J Theor Appl Finance. 2009;12:100726. MathSciNetView ArticleMATHGoogle Scholar
 Teng L, Ehrhardt M, Günther M. Bilateral counterparty risk valuation of CDS contracts with simultaneous defaults. Int J Theor Appl Finance. 2013;16(7):1350040. MathSciNetView ArticleMATHGoogle Scholar
 McNeil AJ, Frey R, Embrechts P. Quantitative risk management: concepts, techniques, and tools. Princeton: Princeton University Press; 2005. MATHGoogle Scholar
 Schöbel R, Zhu J. Stochastic volatility with an OrnsteinUhlenbeck process: an extension. Eur Finance Rev. 1999;3:2346. View ArticleMATHGoogle Scholar
 Engle RF. Dynamic conditional correlation: a simple class of multivariate GARCH. J Bus Econ Stat. 2002;20(3):33950. MathSciNetView ArticleGoogle Scholar
 Langnau A. Introduction into “local correlation” modelling. arXiv:0909.3441 (2009)
 Gourieroux C, Jasiak J, Sufana R. The Wishart autoregressive process of multivariate stochastic volatility. J Econom. 2009;150:16781. MathSciNetView ArticleGoogle Scholar
 Bowman AW, Azzalini A. Applied smoothing techniques for data analysis. New York: Oxford University Press; 1997. MATHGoogle Scholar
 van Emmerich C. Modelling correlation as a stochastic process. Preprint 06/03, University of Wuppertal (June 2006)
 Ma J. Pricing foreign equity options with stochastic correlation and volatility. Ann Econ Financ. 2009;10(2):30327. Google Scholar
 Teng L, van Emmerich C, Ehrhardt M, Günther M. A versatile approach for stochastic correlation using hyperbolic functions. Int J Comput Math. 2016;93(3):52439. MathSciNetView ArticleGoogle Scholar
 Oksendal B. Stochastic differential equations. Berlin: Springer; 2000. Google Scholar
 Uhlenbeck GE, Ornstein LS. On the theory of Brownian motion. Phys Rev. 1930;36:82341. View ArticleMATHGoogle Scholar
 Risken H. The FokkerPlanck equation. Berlin: Springer; 1989. View ArticleMATHGoogle Scholar
 Wilmott P. Paul Wilmott on quantitative finance. 2nd ed. West Sussex: Wiley; 2006. MATHGoogle Scholar
 Kloeden PE, Platen E. Numerical solution of stochastic differential equations. Berlin: Springer; 1992. View ArticleMATHGoogle Scholar