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Title: Fractal activity time risky asset models with dependence
Author: Petherick, Stuart Gary
ISNI:       0000 0004 2752 0380
Awarding Body: Cardiff University
Current Institution: Cardiff University
Date of Award: 2011
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The paradigm Black-Scholes model for risky asset prices has occupied a central place in asset-liability management since its discovery in 1973. While the underlying geometric Brownian motion surely captured the essence of option pricing (helping spawn a multi-billion pound derivatives industry), three decades of statistical study has shown that the model departs significantly from the realities of returns (increments in the logarithm of risky asset price) data. To remedy the shortcomings of the Black-Scholes model, we present the fractal activity time geometric Brownian motion model proposed by Chris Heyde in 1999. This model supports the desired empirical features of returns including no correlation but dependence, and distributions with heavier tails and higher peaks than Gaussian. In particular, the model generalises geometric Brownian motion whereby the standard Brownian motion is evaluated at random activity time instead of calendar time. There are also strong suggestions from literature that the activity time process here is approximately self-similar. Thus we require a way to accommodate both the desired distributional and dependence features as well as the property of asymptotic self-similarity. In this thesis, we describe the construction of this fractal activity time based on chi-square type processes, through Ornstein-Uhlenbeck processes driven by Levy noise, and via diffusion-type processes. Once we validate the model by fitting real data, we endeavour to state a new explicit formula for the price of a European option. This is made possible as Heyde's model remains within the Black-Scholes framework of option pricing, which allows us to use their engendered arbitrage-free methodology. Finally, we introduce an alternative to the previously considered approach. The motivation for which comes from the understanding that activity time cannot be exactly self-similar. We provide evidence that multi-scaling occurs in financial data and outline another construction for the activity time process.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available