Fractional Brownian motion Maturity T=1

[pavladoc]

Hello I am new and please excuse me for my mistakes in English

I'm working on a pricing issue with fractional brownian motion with this model :

\(dS_t=S_t({\mu}dt+{\sigma}dZ^H_t)\)
\(S_t=S_0exp({\mu}t-\frac{{\sigma}^2}{2}t^{2H}+{\sigma}Z_t)\)

in this article (page 7): http://lipas.uwasa.fi/~tsottine/talks/Vaasa2001.pdf

where \(Z_t\) is a fractional brownian motion with Hurst H. (H in [0;1])

In the same article(page 8), the exactly formula for an call-option pricing with this model is given by :

\(S_0\Phi(y_1(H))-Ke^{-rT}\Phi(y_2(H))\)

where

\(y_1(H)=\frac{log\frac{S_0}{K}+rT+\frac{\sigma^2}{2}T^{2H}}{{\sigma}T^H}\)
\(y_2(H)=\frac{log\frac{S_0}{K}+rT-\frac{\sigma^2}{2}T^{2H}}{{\sigma}T^H}\)

Where T is the time to maturity, it can take different time units (seconde, hour, day, month ...)
If I take \(T=1\), the price will stay constant what ever the value taken by H.
I want to check it out with a Monte Carlo simulation (with the FFT method, Wood and Chan), and I've found that for T=1, the price depends on the value taken by H. More precisely, for all T, the price depends on H.
So can anyone help me and tell me what's wrong in the formula, and why we can't work with T=1 with it ?
Thank a lot !
Best Regards
Pavladoc

 

[Yike Lu]

Fractional Brownian Motions are long memory/path dependent.

 

[pavladoc]

Thanks Yike for your answer.

does that mean, if I want to work with T = 1 day, than I will need to use a time unit shorter than one day ?
I mean, for instance , estimate r and sigma basing on Yield value every hour and use T=24 to get the price for one day ?

Knowing that Fractional Brownian Motions are long memory/path dependent, is it meaningless working with T<1 ?
Should I always give a value greater than 1 to T, because of the long memory/path dependence ?

 

[Yike Lu]

Standard practice is to use T = 1 as one year (so that volatility numbers seem "normal") and recalculate everything else based on that. The day count convention can vary, for example 252 trading days vs 365, as can the intraday convention (390 minutes in a NYSE trading day, but it could be 24 hours for certain instruments like FX).

There is no point to simulating a FBM if you are not doing a well discretized path. I'm not sure what's wrong with the explicit pricing formula (probably has to do with the path dependence and the arbitrage they mentioned), but I would trust the MC more than the formula.