Financial Correlation Modelling – Bottom-up Approaches

Gunter Meissner

“Fortune sides with him who dares”

– Virgil

In this chapter we address correlation models, which were specifically designed to measure the association between financial variables. We will concentrate on bottom-up correlation models that collect information, quantify it and then aggregate the information to derive an overall correlation result.

CORRELATING BROWNIAN MOTIONS (HESTON 1993)

One of the most widely applied correlation approaches used in finance was generated by Steven Heston in 1993. Heston applied the approach to negatively correlate stochastic stock returns dS(t)/S(t) and stochastic volatility σ(t). The core equations of the original Heston model are the two stochastic differential equations (SDEs)

$$\frac{dS\left(t\right)}{S\left(t\right)}=\mu dt+\sigma (t)d{z}_1(t)$$

(5.1)

and

$$d{\sigma }^2\left(t\right)=a\left[m_{\sigma }^2-{\sigma }^2\left(t\right)\right]dt+\xi \sigma (t)d{z}_2(t)$$

(5.2)

where

S: variable of interest, eg, a stock price

µ: expected growth rate of S

σ: expected volatility of S

dz: standard Brownian motion, ie, $dz\left(t\right)=\epsilon \left(t\right)\sqrt{dt},\epsilon t$ is i.i.d. (independently and identically distributed). In particular ε(t) is a random drawing from a