A Financially Motivated Extension of the Heston Model for a Joint ℙ- and ℚ-Dynamics Analysis of Variance

Abstract

The empirical, or “ℙ,” distribution determines profits and losses from market price changes, while the risk- neutral “ℚ” distribution is what is embedded in derivatives prices. Expected returns under the ℙ distribution include expected risk premia, while under the ℚ, those risk premia are embedded in the probabilities. Risk-neutral probabilities, and hence market prices for derivatives, move randomly because expectations about true (ℙ-density) probabilities change and also because risk premia are variable too. While changes in the ℚ density are observable in option prices, the breakdown between true expectations and risk premia is not. Although evidence suggests that risk premia may fluctuate as much as probability beliefs do, it is rare for their dynamics to be formally modeled. That is what Rebonato and Ng do in this article looking at exchange rates. Starting with the assumption of a standard Heston model for dynamics under the P measure, they construct a corresponding volatility process under the ℚ measure by modifying the long-term variance level and the reversion speed. Rather than modeling these as constant parameters, they introduce mean-reverting diffusions for both parameters. The effects of these generalizations are explored by holding one constant while the other is varied. After showing the extra flexibility that the “complex Heston” formulation allows, they devise an analytic approximation formula using the average projected parameter values from the dynamic model as the constant parameters in the standard Heston specification and show that the approach works very well. Finally, they take the model to the data for the EURUSD, JPYUSD, and GBPUSD exchange rates and demonstrate that the new model can fit well over time without the need to refit parameter values.