Correlation Brownian motions

Steven Heston applied a correlation approach^[5] to negatively correlate stochastic stock returns ${\displaystyle {\frac {dS(t)}{S(t)}}}$ and stochastic volatility ${\displaystyle \ \sigma (t)}$. The core equations of the original Heston model are the two stochastic differential equations, SDEs

${\displaystyle {\frac {dS(t)}{S(t)}}=\mu \,dt+\sigma (t)\,dz_{1}(t)}$ (1)

and

${\displaystyle d\sigma ^{2}(t)=g[\sigma _{L}^{2}-\sigma ^{2}(t)]\,dt+\xi \sigma (t)\,dz_{2}(t)}$ (2)

where S is the underlying stock, ${\displaystyle \ \mu }$ is the expected growth rate of ${\displaystyle S}$, and ${\displaystyle \ \sigma (t)}$ is the stochastic volatility of ${\displaystyle S}$ at time t. In equation (2), g is the mean reversion rate (gravity), which pulls the variance ${\displaystyle \ \sigma ^{2}(t)}$ to its long term mean ${\displaystyle \sigma _{L}^{2}}$, and ${\displaystyle \ \xi }$ is the volatility of the volatility σ(t). dz(t) is the standard Brownian motion, i.e. ${\displaystyle dz(t)=\varepsilon _{t}{\sqrt {dt}}}$, ${\displaystyle \ \varepsilon _{t}}$ is i.i.d., in particular ${\displaystyle \varepsilon _{t}}$ is a random drawing from a standardized normal distribution n~(0,1). In equation (1), the underlying ${\displaystyle S}$ follows the standard geometric Brownian motion, which is also applied in Black–Scholes–Merton model, which however assumes constant volatility. The correlation between the stochastic processes (1) and (2) is introduced by correlating the two Brownian motions ${\displaystyle dz_{1}}$ and ${\displaystyle dz_{2}}$. The instantaneous correlation ${\displaystyle \ \rho }$ between the Brownian motions is

${\displaystyle \operatorname {Corr} [dz_{1}(t),dz_{2}(t)]=\rho \,dt}$ (3).

The definition (3) can be conveniently modeled with the identity

${\displaystyle dz_{1}(t)={\sqrt {\rho }}dz_{2}(t)+{\sqrt {1-\rho }}\,dz_{3}(t)}$ (4)

where ${\displaystyle dz_{2}(t)}$ and ${\displaystyle dz_{3}(t)}$ are independent, and ${\displaystyle dz(t)}$ and ${\displaystyle dz(t')}$ are independent, t ≠ t’.

The Cointelation SDE^[6] connects the SDE's above to the concept of mean reversion and drift which are usually concepts that are misunderstood^[7] by practitioners.

The binomial correlation coefficient

A further financial correlation measure, mainly applied to default correlation, is the binomial correlation approach of Lucas (1995).^[8] We define the binomial events ${\displaystyle 1_{X}=1_{\{\tau _{X}\leq T\}}}$ and ${\displaystyle 1_{Y}=1_{\{\tau _{Y}\leq T\}}}$ where ${\displaystyle \tau _{X}}$ is the default time of entity ${\displaystyle X}$ and ${\displaystyle \tau _{Y}}$ is the default time of entity ${\displaystyle Y}$. Hence if entity ${\displaystyle X}$ defaults before or at time ${\displaystyle T}$, the random indicator variable ${\displaystyle 1_{X}}$ will take the value in 1, and 0 otherwise. The same applies to ${\displaystyle Y}$. Furthermore, ${\displaystyle P(X)}$ and ${\displaystyle P(Y)}$ is the default probability of ${\displaystyle X}$ and ${\displaystyle Y}$ respectively, and ${\displaystyle P(XY)}$ is the joint probability of default. The standard deviation of a one-trial binomial event is ${\displaystyle {\sqrt {P(X)-(P(X))^{2}}}}$, where P is the probability of outcome X. Hence, we derive the joint default dependence coefficient of the binomial events ${\displaystyle 1_{\{\tau _{X}\leq T\}}}$and ${\displaystyle 1_{\{\tau _{Y}\leq T\}}}$ as

${\displaystyle \rho (1_{\{\tau _{X}\leq T\}},1_{\{\tau _{Y}\leq T\}})={\frac {P(XY)-P(X)P(Y)}{{\sqrt {P(X)-(P(X))^{2}}}{\sqrt {P(Y)-(P(Y))^{2}}}}}}$ (5).

By construction, equation (5) can only model binomial events, for example default and no default. The binomial correlation approach of equation (5) is a limiting case of the Pearson correlation approach discussed in section 1. As a consequence, the significant shortcomings of the Pearson correlation approach for financial modeling apply also to the binomial correlation model.

Copulae and the 2007–08 financial crisis

Numerous non-academic articles have been written demonizing the copula approach and blaming it for the 2007/2008 global financial crisis, see for example Salmon 2009,^[13] Jones 2009,^[14] and Lohr 2009.^[15] There are three main criticisms of the copula approach: (a) tail dependence, (b) calibration, (c) risk management.

(a) Tail dependence

In a crisis, financial correlations typically increase, see studies by Das, Duffie, Kapadia, and Saita (2007)^[16] and Duffie, Eckner, Horel and Saita (2009)^[17] and references therein. Hence it would be desirable to apply a correlation model with high co-movements in the lower tail of the joint distribution. It can be mathematically shown that the Gaussian copula has relative low tail dependence, as seen in the following scatter plots.

Figure 1: Scatter plots of different copula models

As seen in Figure 1b, the student-t copula exhibits higher tail dependence and might be better suited to model financial correlations. Also, as seen in Figure 1(c), the Gumbel copula exhibits high tail dependence especially for negative co-movements. Assuming that correlations increase when asset prices decrease, the Gumbel copula might also be a good correlation approach for financial modeling.

(b) Calibration

A further criticism of the Gaussian copula is the difficulty to calibrate it to market prices. In practice, typically a single correlation parameter (not a correlation matrix) is used to model the default correlation between any two entities in a collateralized debt obligation, CDO. Conceptually this correlation parameter should be the same for the entire CDO portfolio. However, traders randomly alter the correlation parameter for different tranches, in order to derive desired tranche spreads. Traders increase the correlation for ‘extreme’ tranches as the equity tranche or senior tranches, referred to as the correlation smile. This is similar to the often cited implied volatility smile in the Black–Scholes–Merton model. Here traders increase the implied volatility especially for out-of-the money puts, but also for out-of-the money calls to increase the option price..

In a mean-variance optimization framework, accurate estimation of the variance-covariance matrix is paramount. Thus, forecasting with Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective.^[18] Allowing the modeling process to allow for empirical characteristics in stock returns such as auto-regression, asymmetric volatility, skewness, and kurtosis is important. Not accounting for these attributes lead to severe estimation error in the correlations and variances that have negative biases (as much as 70% of the true values).^[19]

(c) Risk management

A further criticism of the Copula approach is that the copula model is static and consequently allows only limited risk management, see Finger (2009)^[20] or Donnelly and Embrechts (2010).^[21] The original copulas models of Vasicek (1987) and Li (2000) and several extensions of the model as Hull and White (2004)^[22] or Gregory and Laurent (2004)^[23] do have a one period time horizon, i.e. are static. In particular, there is no stochastic process for the critical underlying variables default intensity and default correlation. However, even in these early copula formulations, back testing and stress testing the variables for different time horizons can give valuable sensitivities, see Whetten and Adelson (2004)^[24] and Meissner, Hector, and. Rasmussen (2008).^[25] In addition, the copula variables can be made a function of time as in Hull, Predescu, and White (2005).^[26] This still does not create a fully dynamic stochastic process with drift and noise, which allows flexible hedging and risk management. The best solutions are truly dynamic copula frameworks, see section ‘Dynamic Copulas’ below.

Conditionally independent default (CID) correlation modeling

In order to avoid specifying the default correlation between each entity pair in a portfolio a factorization is often applied. This leads to conditionally independent default (CID) modeling. The most widely applied CID model is the one-factor Gaussian copula (OFGC) model. It was the de facto market model for pricing CDOs before the 2007/2008 global financial crisis. The core equation of the OFGC model

${\displaystyle x_{i}={\sqrt {\rho }}M+{\sqrt {1-\rho }}Z_{i}}$ (10)

where ${\displaystyle M}$ and ${\displaystyle Z_{i}}$ are random drawings from ${\displaystyle N(0,1)}$ and ${\displaystyle 0\leq \rho \leq 1}$. As a result, the latent variable ${\displaystyle x_{i}}$, sometimes interpreted as the asset value of i, see Turc, Very, Benhamou and Alvarez et al. (2005),^[30] is also n~(0,1). The common factor ${\displaystyle M}$ can be interpreted as the economic environment, possibly represented by the return of the S&P 500. ${\displaystyle Z_{i}}$ is the idiosyncratic component, the ‘strength’ of entity i, possibly measured by entity i's stock price return. From equation (10) we see, that the correlation between entities i is modeled indirectly by conditioning the latent variable ${\displaystyle x_{i}}$ on the common factor ${\displaystyle M}$. For example, for p =1, the latent variables of all entities ${\displaystyle i=1,...,n,\ x_{i}=M}$, so the ${\displaystyle x_{i}}$ are identical in each simulation. For p = 0, all latent variable for all entities ${\displaystyle i=1,\ldots ,n,\ x_{i}=Z_{i}}$, hence the ${\displaystyle x_{i}}$ are independent. Importantly, once we fix the value of M, the defaults of the n entities are (conditionally on M) mutually independent.

As of 2010, the OFGC is the basis for credit risk management in Basel II. The benefits of the model are simplicity and intuition. One of the main shortcomings of the model is that traders when pricing CDOs randomly alter the correlation parameter for different CDO tranches to achieve desired tranche spreads. However conceptually, the correlation parameter should be identical for the whole portfolio.

Contagion default modeling

Contagion default modeling can be viewed as a variation of CID modeling. As discussed in section 2.3, in the CID framework, correlation is modeled by conditioning on a common market factor M, which impacts all entities to the same degree. The lower the random drawing for M, the higher is the default intensity of all entities (unless ρ = 0). Hence CID modeling can elucidate default clustering. In contrast, contagion approaches model the default intensity of an entity as a function of the default of another entity. Hence contagion default modeling incorporates counterparty risk, i.e. the direct impact of a defaulting entity on the default intensity of another entity. In particular, after a default of a particular entity, the default intensity of all assets in the portfolio increases. This default contagion then typically fades exponentially to non-contagious default intensity levels. See the papers of Davis and Lo (2001)^[31] and Jarrow and Yu (2001),^[32] who pioneered contagion default modeling.