Here we will be sloppy with rigour to get some feel for Ito formula manipulations for stochastic volatility models.

. We want to consider first to be Brownian motion and then fractional Brownian motion. We just assume without justification that when is a fractional Brownian motion then there is an Ito formula that holds in this case with the Wick product definition of stochastic integrals.

So with when is a Brownian motion and when is a fractional Brownian motion, we have the quadratic variation formulae

For any function we have formally

The general idea is to find that should be a martingale for no-arbitrage reasoning and take expectations setting the coefficient of to zero and thereby obtain a partial differential equation that is analogous to the Black-Scholes-Merton partial differential equation.

When has Hurst index then . So purely formally, we can produce using this a partial differential equation for a fractional Brownian motion square-root process stochastic volatility by simply considering the expectations of stochastic integrals to be zero

The delta-hedged portfolio argument as used by Gatheral (gatheral-implied-volatility-surface) would use a portfolio consisting of an option with value , and hedges with the underlying and another instrument with value . Following this argument we would have in this case:

where the right hand side contains the market price of risk (which is assumed to be zero in the Heston paper).

So the Heston 1993 analysis changes here only in the introduction of and which are known powers of when is a fractional Brownian motion of index . There is nothing new here and many people have considered long memory stochastic volatility models since 1998 or so. An example is FractionalConstantElasticityOfVariance-Chan-Ng-2007. The reason it is worth reconsidering this step is that there should exist a partial differential equation that is exact to empirical autocorrelations and arrival distributions that is unclear to me still. Recall that the BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION IS NOT A NATURAL LAW!! It is based on arbitrage considerations and on a theorem that says arbitrage implies that the PRICE PROCESS is a semimartingale approximately. It does not tell us any natural law regarding volatility and leaves this completely open. ARBITRAGE CONSIDERATIONS alone do not produce a SCIENTIFIC LAW!! Scientific laws can be found in the actual observed phenomena of volatility such as:

t

## Leave a Reply