From Stochastic Variance to Stochastic Volatility - Another look at the Heston's Process
March 2, 2011
In a Heston’s process (Heston’s model) variance is assumed to be stochastic and follow a mean reverting Cox-Ingersoll-Ross (CIR) process, which is a generalized Ornstein Uhlenbeck (OU) process. If is the asset price at time and is the corresponding variance of the asset returns at time then the Heston’s model is described by these two equations:
In the above model the variance, of the asset price is stochastic (random) and is modeled as mean reverting CIR process with a constant speed of mean reversion as and a constant long term variance as . The asset, which is also stochastic, follows a geometric Brownian motion with the underlying variance of the asset following a mean reverting CIR process. The stochastic processes for variance and the asset, i.e. the relevant Weiner processes, and are correlated via a correlation factor,.
As one of our CFE students in Hong Kong, so eloquently put it, it is like a process within a process (reminds us of Chris Nolan’s “dream within a dream” in the movie Inception).
The above model, however, gives the process for variance. Of course, a full valuation of the above can be done in a Monte Carlo simulation routine and derivatives payoffs priced using this model. But such a numerical scheme would entail converting the stochastic variances into stochastic volatilities at each point of the path and then feeding that to the asset price process.
However, does an explicit process for the stochastic volatility – instead of stochastic variance – exist? The answer is yes. And even though the underlying math is almost trivial (using Taylor series expansion and Ito’s calculus), it seems most practitioners did not bother with the exact process (equation) of stochastic volatility until as late as 2008.
Zhu (2008) has shown that a stochastic process for volatility can be derived from the stochastic variance process.
Note that, and therefore, rules of ordinary differential calculus give us the first and the second derivative as:
Using a Taylor series expansion, we can write:
For the values of we use Ito’s calculus because the expression for is stochastic.
Therefore, the expression for will become:
With some more algebra the above will reduce to:
Therefore, we see that a Heston’s model for stochastic variance is equivalent to an Ornstein-Uhlenbeck process (mean reverting stochastic process) for volatility but with a very fundamental difference. The mean reversion level, i.e. the long term mean, is now non-stationary (as opposed to stationary mean reversion level for stochastic variance) and dependent on the level of stochastic volatility,.
Therefore, within a Heston’s framework of stochastic variance, the equivalent stochastic volatility process will be described as:
Of course, the stochastic volatility process (equation) will suffer from the same flaw as the stochastic variance, namely, the problem of negative volatility and the breakdown of the Feller condition at various points on the map.
A Simple and Exact Simulation Approach to Heston Model, Zhu, J (2008)
Foreign Exchange Option Pricing, Iain J. Clark (John Wiley & Sons)
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