Deriving Fractional Brownian Motion Increment Variance by Substitution

Summary

The document works through the moving-average representation of fractional Brownian motion and asks how to simplify the variance of an increment between two ordered times. It splits the stochastic integral into regions before the first time and between the two times, then applies Itô's isometry. This yields a sum of a squared-kernel integral and a power term proportional to the time interval raised to twice the Hurst parameter.

The proposed simplification factors out the interval length by changing variables in the first integral, so the remaining expression depends on the Hurst parameter rather than the specific pair of times. The question highlights a likely point requiring care: the substitution must shift and rescale the integration variable consistently, including the kernel arguments and integration limits. The document provides no answer or completed proof, and it concerns a mathematical derivation rather than a trading method or empirical result.

Key ideas

• The moving-average representation expresses fractional Brownian motion through an integral against ordinary Brownian motion.
• Itô's isometry converts the increment's second moment into integrals of squared kernels.
• The increment variance can be reduced to a function of the time difference through a shift and rescaling of the integration variable.
• The derivation is posed as a question and does not supply a completed substitution or proof.

Tags

# Literature on Empirical Option Pricing


# Literature on Empirical Option Pricing












When I started combing through the literature I was astonished about how little the option pricing models are tested against market data and benchmarks are limited. The main barrier is of course finding affordable options market data is not easy. But it is essential to assess the performance of the proposed model with some other models accepted by the academia.

For example, while Duan's famous paper paper on option pricing with GARCH has no empirical section, Heston and Nandi's own GARCH paper has detailed info on the data used and benchmarks their model with other models.

I am looking for scientific papers with a data section on empirical option pricing. They usually either propose a model and benchmark using market data or completely devote themselves to benchmark models.

p.s. I am more interested in exchange traded equity, etf and index options, so American and European option pricing models are priority (others are also welcome). No limit on the model type, the only requirement is it should spit out a price. Benchmark papers are also welcome.

## Answer by Ascorpio (score 2)

https://quant.stackexchange.com/a/14746

I think you are right. Now when I check papers I've used for my thesis I don't see almost any with empirical data section.

Maybe this one will be helpful:

Roswell E. Mathis, III, Gerald O., Bierwag Pricing Eurodollar Futures Options with Ho and Lee and Black, Derman, and Toy Models: An Empirical Comparison

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