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## EQUIVALENT MARTINGALE MEASURE EXISTS FOR THE TIME-FRACTIONAL HESTON MODEL

### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

2:37 PM (39 minutes ago)

 to jfrenkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira

The time-fractional Heston model, or the Zulf model, of option pricing specifies stochastic volatility dynamics via a time-fractional stochastic differential equation:

D^alpha_t V(t) = kappa*(theta-V(t)) + sigma*sqrt(V(t)) dw(t)/dt

generalizing Heston’s 1993 model.  The existence of an equivalent martingale measure for this model is resolved as follows, closely following Heyde-Wong 2006 Lemma 3.1 where the authors resolve this question for the Heston model.  My main contention here is that their beautiful and clever argument can be modified without problems to provide existence of equivalent martingale measure for my model, the time-fractional Heston model.

Heyde-Wong prove that for certain real alpha,beta:

E(exp( -alpha*V(T) – beta*\int_0^T V(u) du)) < infty

is a martingale.  So they consider F(v,t) solving a Cauchy problem:

F_t + (1/2)*sigma^2 v F_{vv} + kappa*(theta-v) F_v = beta*v*F                   (*)

F(v,T) = exp(-alpha*v)

This Cauchy problem has the property that when one defines

M(t) = exp( -beta*\int_0^t V(u) du) F(V,t)

then Ito formula applied to M(t) leads to:​

dM(t) = exp(-beta*\int_0^t V(u) du)[ -beta*v*F + F_t dt + F_v dv + (1/2) F_{vv} d<v>]

and the Cauchy problem condition (*) ensures there are full cancellation for all the terms involving dt

dM(t) = exp(-beta*\int_0^t V(u) du) F_v sigma*sqrt(V(t))*dw(t)

Then they solve the Cauchy problem for F using the same method as Heston showing that M(t) is a martingale.

For the Zulf model, we note that

dV(t) = (T’_{alpha}(t)v(0) + S_{alpha}(t)*kappa*theta – kappa*V(t)) dt + S_{alpha}(t)*sigma*sqrt{V(t)} dw(t)

so we want to replace (*) with the appropriate Cauchy problem for which my initial guess is:

F_t + (1/2)*sigma^2*S_alpha(t)^2*v*F_{vv} + (T’_{alpha}(t)*v(0) + S_{alpha}*kappa*theta – kappa*v) F_v = beta*v*F (**)

F(v,T) = exp(-alpha*v)

where

T_alpha(t) = MittagLeffler(-kappa*t^alpha,alpha,1)

and

S_alpha(t) = t^{alpha-1}*MittagLeffler(-kappa*t^alpha,alpha,alpha)

So long as (**) has reasonable bounded solutions, we will have shown that the Zulf model has equivalent martingale measure.

Please forgive the sloppiness in this attempt.  The substance of the Zulf model is that time-fractional stochastic volatility is the best SCIENTIFIC MODEL for the option pricing problem that empirically explains the natural phenomena of  all features of volatility surface and takes into account my discovery that waiting times of volatility shocks have a Mittag-Leffler distribution rather than merely a GENERALIZATION of the Heston model.  The existence of an equivalent martingale measure is an important technical problem but secondary to the fundamental claim of the Zulf model as the best scientific model for the natural phenomena in volatility surfaces.

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