Rate of convergence of CSYIP-based discrete paths to Brownian motion

Determine the rate of convergence in law of the discrete process X_{k/n}^{(M)} = \sum_{i=1}^{k} \sqrt{\Delta} \, \xi_{n,i}^{(M)} to a standard Brownian motion as \Delta \downarrow 0 in the Cherny–Shiryaev–Yor invariance principle setting where the Bernoulli variables \xi_{n,k}^{(M)} are independent with mean 0 and variance 1 and are constructed from market index data. Quantify the speed of convergence to guide empirical model fitting and approximation accuracy.

Background

The empirical implementation relies on a single historical path of the market index to construct the Bernoulli sequence \xi_{n,k}^{(M)} and its cumulative path X_{k/n}^{(M)}, which approximates Brownian motion under the CSYIP. The quality of regression fits and option pricing accuracy depends on how quickly this discrete approximation converges in law to Brownian motion as the time step decreases.

While weak convergence is guaranteed by the CSYIP under iid assumptions, the authors note that the speed of convergence is unknown, limiting the ability to assess approximation errors and to design improved empirical procedures.