Term Structure Models with Unspanned Factors and Unspanned Stochastic Volatility
134 Pages Posted: 9 Dec 2018
Date Written: November 15, 2018
Certain models of the term structure of interest rates exhibit unspanned stochastic volatility (USV). A model has this property if it involves a source of stochastic variation — called an unspanned factor — that does not affect the model’s interest rates directly, but does affect the extent to which future interests are liable to change (that is, interest-rate volatility). This thesis is concerned with these models, from a variety of perspectives.
Firstly, the theoretical foundation of the USV property is addressed. Formal definitions of unspanned factors and USV are developed, generalising ones tentatively proposed in the literature. Several results from these definitions and the accompanying framework are derived. Particularly, the ability to hedge general claims (i.e., the completeness or lack thereof) of these models is examined in detail. Examples are given to illustrate the features of the proposed framework and the necessity of the generalised definitions.
Secondly, the empirical issue of whether USV models are necessary to plausibly represent ob- served interest-rate markets is interrogated. An empirical derivative-hedging approach is adopted, the results of which are contextualised by also treating data simulated from models with USV and non-USV versions. It is shown that hedging effectiveness is relatively robust to the presence of USV, which resolves the apparent conflict between the two studies that have taken a hedging approach to this question. Despite the cross-sectional hedging effects being surprisingly minor, further regression results show that USV models are needed to model the time series of market interest rates.
Finally, the thesis addresses a certain class of models that exhibit USV: those with one spanned factor (driving interest-rate variation) and one unspanned, volatility-related factor. Being the simplest non-trivial USV models, these bivariate USV models are fundamental, and — like one- factor models in general settings — are helpful in introducing and comparing higher-factor models when simple ones are insufficient. These models are shown to exist (contradicting a claim in the literature); to share a particular affine form for their bond pricing functions; and to necessarily exhibit a short-term interest rate with dynamics of a certain type. A specific bivariate USV model is then proposed, which is analysed and compared to others in the literature.
Keywords: Unspanned stochastic volatility (USV), term structure models, interest-rate volatility
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