The Black-Scholes model was an intellectual breakthrough because for the first time the idea was established that rational prices exist for options independent of investor preferences but it was never a scientific model in the sense of quantitative mathematical models fitting actual data. Since 1973 there have been amazing improvements of the model by extensions of which the most promising seemed to be the stochastic volatility type models of which the most prominent for their reasonable fit to the implied volatility surfaces were the Heston (1993) model and the Bates (1997) and there are others as well. What we do here is turn the issues on their head and ask for an exact model for stochastic volatility and then return to the Black-Scholes argument for a partial differential equation. Heston had chosen to model stochastic volatility by a Feller (1951)/Cox-Ingersoll-Ross (1975) square root process and Bates had followed Merton (1976) by adding normally distributed jumps at Poisson random times. We show that an almost exact fit to volatility proxied by result from the model of a square root process with normally distributed jumps subordinated to inverse stable waiting times. Regardless of option pticing theory, this can be considered then as an exact scientific model for stochastic volatility and serve as the basis for a partial differential equation that must be satisfied for option prices. In Ito form time-change by inverse stable subordinator can be considered by change of density of the terms of the Ito equation for a square root process with jumps:
where and is the density of the inverse stable subordinator.
The diffusion component Ito formula can be used in a straightforward manner. The jump component is more complicated. There is an Ito formula with jumps for example Theorem 4.31 of menaldi-book-06-2013, but jumps were already a nontrivial problem when Robert Merton considered them in Merton1976 one way to deal with option pricing with jumps he considered is summing over jumps for log-normal jump distribution:
We used the jump term in the characteristic probability distribution of David Bates modified to account for the time-change. Bates’ jump term is described here (Bates_Scott):
The Bates PDE (non time-changed case) is standard and here is a useful version of how to deal with the jump term in the PDE.
Recall that a stable subordinator is an increasing Levy process whose density with respect to Lebesgue measure has Laplace transform ; the inverse stable subordinator and more details can be found in Meerschaert et. al. We fit a number of reasonable models of volatility that consider jumps and time changes from the basic square root diffusion and find overwhelming evidence that the combination of normally distributed jumps and inverse stable time change produce tight fits to empirical volatility of stocks. Next, the Ito formula above allows us to repeat the Black-Scholes-Heston standard argument for the partial differential equation that must be solved by assets: let be the asset price, consider the portfolio which we assume is riskless so . Now use Ito formula on and eliminate the stochasticity of by setting and and we obtain
The form of this partial differential equation is such that one can use the Heston method exactly. Recall that Heston solves for the characteristic functions of probabilities such that the plain vanilla call has price and finds these in the form where . The partial differential equation we obtain can be solved by replacing Heston’s solution with . In particular, for our claimed exact models for stochastic volatility, the closed form solution of Heston extends simply and thus we do not lose tractability. Now for the empirical results that provide evidence for our model:
The characteristic function of the square root process as well as the density is available in closed form and can be found for example in Mendoza-Arriega and Lintesky (2014). We derive the jump version using Fourier tranforms conditioned by number of jumps. We use Carr-Wu (2004) CarrWu2004-TCLP Theorem 1 to evaluate the characteristic function of time-change by evaluation of the characteristic function at the Laplace transform of the density of inverse stable subordinator. We fit these theoretical characteristic functions to empirical characteristic functions of using the weight , which is a standard technique (see Jiang-Knight-ECF and references for history of the method) The following are errors of fits to the models using 1000 days of daily return data per stock and shows that the square root with jumps and time-change fit the data extremely well.
The full code to generate these fits is the following: