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## MARTINGALE THEORY ON REAL FINANCIAL DATA

Risk-neutral valuation method of Cox-Ross and the equivalence of no-arbitrage condition with the existence of an equivalent martingale measure of Harrison-Pliska, Harrison-Kreps are the foundations of continuous-time finance.  For Levy processes, one has incomplete markets but the Esscher transform allows one to find at least one martingale measure.  Let us take a peek at real data to check approximate Hurst exponents, which are crude measures to estimate the power law spread in time of the return series, since the fractional Brownian motion $B^H_t$ is a martingale only for $H=1/2$.

Now consider a randomly selected stock and let’s take a look at the cumulative sum of $r_t^2$, plot a log-log plot versus log of time.  Here is the plot, with least-square slope of 0.735.

One easily checks that in this case we have much slower growth than the case of normal $r_t$ as follows:

>> z = np.random.normal(size=5000)
>>> T = range(1,5001)
>>> plt.plot(np.log(T),np.log(np.cumsum(z)))
[<matplotlib.lines.Line2D object at 0x11ae7b310>]
>>> A1,B1,rv1,pv1,se1=stats.linregress(np.log(T),np.log(np.cumsum(z)))
>>> A1
nan
>>> plt.show()
>>> plt.plot(np.log(T),np.log(np.cumsum(z**2)))
[<matplotlib.lines.Line2D object at 0x11a4abd10>]
>>> A1,B1,rv1,pv1,se1=stats.linregress(np.log(T),np.log(np.cumsum(z**2)))
>>> plt.plot(np.log(T),np.log(np.cumsum(z**2)))
[<matplotlib.lines.Line2D object at 0x11a2f1fd0>]
>>> A1
1.0075133667755944

When the power law deviates from $t$ to $t^\alpha$ where $\alpha \neq 1$ we don’t have a Brownian martingale.  The question of interest is to understand equivalent martingale measures in this context.   This is an old issue, and Mandelbrot had suggested fractional Brownian motions in rhese cases, originailly invented by A. N. Kolmogorov.  Now theory for fractional martingales came out of this problem late 2000s:  fractional-martingales.  The characterization theorem for fractional Brownian motion suggests that we might expect empirical processes to be a fractional martingale in the sense that although the empirical return process $r_t$ is not a martingale, the transformation $r_t^\alpha = \int_0^t (t-s)^\alpha dr_t$ might be a martingale.