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## FUNDAMENTAL THEOREM OF ASSET PRICING FOR NON-MARKOVIAN PROCESSES

The ‘fundamental theorem of asset pricing’ refers to the connection between fair markets in the sense of no arbitrage opportunities existing in them and the existence of a probability measure under with the ‘deflated asset prices’ are martingales.  For a subtle and detailed discussion of the various ways in which there have been rigorous attempts to make this equivalence precise, see schachermayer-fundemantal-theorem-asset-pricing.  A mathematically precise proof of this theorem in a discrete time setting can be found in rogers-no-arbitrage-existence-martingale-measure-1994.   The focus has been on semi-martingales for the price process $S(t)$.  Recall that the Black-Scholes-Merton framework is one where $dS(t)/S(t) = d\log(S(t)) = \sigma(t) dB_t + \mu(t) dt$ which is obviously a semi-martingale.  But of course not many empirical price series are semi-martingales: the vast majority of assets are non-Markovian even for daily data.  Even though the autocorrelations of returns is negligible, the autocorrelations of their powers are not and have very slow decay even for daily data, and the Hurst exponents measuring deviation from Markovian processes is widely distributed far from $H=1/2$.  For a known Hurst exponent $H_0 = \gamma/2$ a class of models that take into account the non-Markovian aspect are those for which the probability density function $p(x,t)$ evolves by a non-Markovian Fokker-Planck equation

$D^\gamma_t p(x,t) + L_x p(x,t)$ (1)

where the operator $L_x$ is the truncated Levy operator, i.e. one whose Fourier symbol is given by $\psi_{\alpha,\chi}(\xi)$ with Levy index $\alpha$ and cutoff parameter $\chi$ (the untruncated version is simply $-c|\xi|^\alpha$ and the truncation is a fine tuning that fits return tails better than the stable distributions originally proposed by Benoit Mandelbrot in 1963 for cotton prices.

Now we proceed to construct an equivalent martingale measure assuming that $X(t) = \log(S(t))$  has a probability density $p(x,t)$ following (1).  We will follow the paper by Gerber and Shiu where they introduce the use of Esscher transform to determine an equivalent martingale measure and make appropriate changes to reflect the central feature missing for their arguments in this situation.  If $M(z,t) = E[ e^{zX(t)}]$ is the moment generating function, then they are in the Markovian infinitely-divisible distribution case where $M(z,t) = M(z,1)^t$.  In our situation, we have instead $M(z,t) = M(z,1)^{t^\gamma}$.  In order to see this, note that one can solve (1) by taking Fourier transforms and the basic fact that fractional version of the exponential function solving $D^\gamma_t f(t) + \lambda f(t) = 0$ is given by the Mittag-Leffler function $f(t) = E_\gamma( - \lambda t^\gamma)$, so the solution of (1) has Fourier transform $\hat{p}(\xi,t) = \exp[ - \psi_\alpha(\xi) t^\gamma ]$.  Now the moment generating function for the stochastic process $X(t)$ with density $p(x,t)$ can be thought of as an imaginary Fourier transform:

$M(z,t) = \int_y e^{zy} p(y,t) dy = \int_y e^{-i\(izy)} p(y,t) dy = \hat{p}(iz,t)$

Now it is clear from the explicit expression for the Fourier transform that $M(z,t) = M(z,1)^{t^\gamma}$.  Now following Gerber and Shiu we introduce the density $p(x,t,h)$ for a real valued parameter $h>0$:

$p(x,t,h) = e^{hx}p(x,t) /\int_{-\infty}^{\infty} e^{hx} p(x,t) dx = \frac{e^{hx}p(x,t)/M(x,t)$

The moment generating function for $p(x,t,h)$ is

$M(x,t,h) = M(x+h,t)/M(h,t) = ( M(x+h,1)/M(h,1))^{t^\gamma}$

We need to solve the equation $e^{rt} = E^*[ e^{X(t)} ]$ which does not seem directly possible with only an Esscher transform independently of $t$.  On the positive side, Norros et. al. have shown that one can consider the transformation $M_t = \int_0^t s^{H-1/2}(t-s)^{H-1/2} dB^H_t$ to produce a martingale.

Norros et al approximate an fBM B^H_t with Hurst exponent H by
(t^{2H-1}/const) G_t
where G_t is the martingalification of B^H_t.  Now the Esscher transform has the property M(1,t;h)=M(1,1;h)^{t^gamma} so we can look at
E[ exp(G(t)) ] ~ E[ exp(const t^{1-2H} X_t) ] = E[ exp( const*t^{(1-gamma)} X_t )]
for the Esscher transform of the process G_t instead of that of X_t (whose pdf is the solution of the fractional diffusion equation D^gamma + L_x = 0) .  Then we look at the Esscher transform with the constant h for X_t to look for a cancellation of t^gamma and get a martingale measure for G(t).  There is a martingale measure for G(t) anyway because it’s a diffusion.  So this could be the solution for an equivalent martingale measure.  Obviously with the explicit arbitrage examples for fBM one has to be careful but this approximation by diffusion is apparently quite good.
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