Abstract
This thesis consists of 4 chapters each of which discusses the properties of various models from Mathematical Finance in particular Rough Volatility models.In Chapter 1 we discuss the origins of Rough Volatility and cover various theoretical topics needed for the later chapters in the thesis. Other competing approaches are briefly discussed.
Chapter 2 (whose contents can also be found in the paper [FGS21]) introduces the Rough Heston model and demonstrates it’s affine structure. In the absence of he semimartingale property of the variance, the affine structure is what we shall use to determine both smalltime and large-time asymptotics for this model as well as the H ↓ 0 limit.
Chapter 3 (based on the preprint titled "Small-time VIX smile and the stationary distribution for the Rough Heston model" found at https://nms.kcl.ac.uk/martin.forde/) stays with the Rough Heston Model but here we examine the small-time asymptotic behaviour of the VIX. We see that the model produces skewness and convexity features similar to those seen in the market.
In Chapter 4 we introduce the Gaussian Multiplicative Chaos (GMC) of the (re-scaled) Riemann-Liouville (RL) process and prove various forms of convergence as H tends to zero by comparing with the Multifractal Random Walk. We show the GMC emerges in the limit from the Rough Bergomi model as H tends to zero. We derive an approximation for the skew. In addition to the original paper on which this chapter is based [FFGS20] we prove convergence in L 1 using the abstract Shamov framework and derive a Karhunen-Loeve type expansion for the H = 0 field.
| Date of Award | 1 Apr 2023 |
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| Original language | English |
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| Supervisor | Martin Forde (Supervisor) & Blanka Horvath (Supervisor) |