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## ELEMENTARY ITO FORMULA MANIPULATIONS FOR STOCHASTIC VOLATILITY MODELS

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

$dS = \mu S dt + \sigma \sqrt{v_t} dw^1_t$

$dv_t = a(b-v_t) dt + \gamma \sqrt{v_t} dw^2_t$

$\rho = Cov(w^1_t,w^2_t)$.  We want to consider $dw^2_t$ first to be Brownian motion and then fractional Brownian motion.  We just assume without justification that when $w^2_t$ 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 $g(t)=1$ when $w^2_t$ is a Brownian motion and $g(t) = C t^{2H-1}$ when $w^2_t$ is a fractional Brownian motion, we have the quadratic variation formulae

$d\langle S \rangle = \sigma^2 v_t dt$

$d\langle v \rangle = \gamma^2 v_t g(t) dt$

For any function $F(S,v,t)$ we have formally

$dF = \frac{\partial F}{\partial t} dt + \frac{\partial F}{\partial S} (\mu S dt + \sigma \sqrt{v_t} dw_t) + \frac{1}{2} \frac{\partial^2 F}{\partial S^2} \sigma^2 v_t dt$

$+ \frac{\partial F}{\partial v}(a(b-v_t)dt + \gamma \sqrt{v_t} dw^2_t) +\frac{1}{2} \frac{\partial^2 F}{\partial v^2}\gamma^2 v_t g(t) dt$

$+ \rho \frac{\partial^2 F}{\partial v \partial S} \sigma\gamma v_t g_2(t) dw^1_t dw^2_t$

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

When $w^2_t$ has Hurst index $H>0$ then $g_2(t) = t^{H-1/2}/(H-1/2)$.  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

$F_t + \mu S F_S + \frac{1}{2} \sigma^2 F_{SS} v_t + F_v a(b-v_t) + \frac{1}{2} F_{vv} \gamma^2 v_t g(t)$

$+ \rho F_{vS} \sigma \gamma v_t g_2(t) = 0$

The delta-hedged portfolio argument as used by Gatheral (gatheral-implied-volatility-surface) would use a portfolio $\Pi = V - \Delta S - \Delta_1 V_1$ consisting of an option with value $V$, and hedges with the underlying and another instrument with value $V_1$.  Following this argument we would have in this case:

$V_t + \frac{1}{2} v S^2 V_{SS} + \rho g_2(t) \sigma\gamma v_t V_{vS} + \frac{1}{2} F_{vv} \gamma^2 v_t g(t) + rSV_S - rV = -(\alpha - \phi\beta)V_v$

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 $g(t)$ and $g_2(t)$ which are known powers of $t$ when $w^2_t$ is a fractional Brownian motion of index $H$.  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