Leírás
It is a well-known fact that the correlation between prices of financial products, financial instruments plays an important role in their evaluation and pricing derivatives written on them. Using simply a constant or deterministic correlation may lead to excess risk, since market observations give evidence that the correlation is not a constant quantity. To model the joint behaviour of asset prices correctly is particularly essential when pricing derivatives dependent on those assets. Typical examples are various spread or exchange options, Quanto options or so-called rainbow options. The constant correlation is not sufficient to represent the interdependence of the underlying because from market data we have evidence that interdependence is not linear. In this work, we suppose that the individual asset prices follow one of the usual models of financial mathematics e.g., Geometric Brownian motion or a stochastic volatility model like the Hull--White or Heston models. Eventually, Variance Gamma or other subordinated Brownian motion models may also be considered within our framework. Instead of using a constant correlation we have used so called stochastic correlation i.e. time dependent and random correlation. We suggest creating the stochastic correlation process by using a Jacobi process or a tangent hyperbolic transformation of a diffusion process. Our general approach provides a stochastic correlation which is more realistic to model real world phenomena and could be used in many financial application fields. We illustrate it on an example of two stock price data. Furthermore, using our numerical and simulation methods, we compare our approach of modelling stochastic correlation either by Jacobi or tangent hyperbolic transformation of a diffusion process with the gaussian case and conclude that using constant correlation can lead to underestimated correlation risk, and hence financial loss. The reason is that the Gaussian copula model induced by constant correlation does not allow for market-consistent variability and thus fail to capture the risk. The considered case of stock prices fully justify this statement. In our study we have focused on high frequency stock price data (minute-wise traded) rather than diurnal prices, because of the well known fact that with increasing time-scales prices get closer to the Gaussian model.