114 OAS Book 2007 - A Tutorial on Zero Volatility and...

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Unformatted text preview: A Tutorial on Zero Volatility and Option Adjusted Spreads Robert Jarrow Johnson Graduate School of Management Cornell University Ithaca, NY 14853, USA raj15@cornell.edu Summary. This paper provides a brief tutorial on the notions of a zero volatility (ZV) spread and an option adjusted spread (OAS), as applied to fixed income se- curities. Using the standard definitions, it is shown that the zero volatility spread measures the percentage of a securitys spread due to any embedded options and any mispricings. The mispricings could be due to either market or model error. In contrast, the OAS only measures the percentage of the securitys spread due to mis- pricings. Refinements and alternative measures of a bonds embedded optionality and mispricings are also provided. Key words: Option adjusted spreads; zero volatility spreads; HJM model; arbitrage opportunities. 1 Introduction Zero volatility (ZV) and option adjusted spreads (OAS) apply to bonds with embedded options. The embedded options could be call provisions, prepay- ment provisions, or even credit risk (viewed as the option to default). ZV spreads and OAS were first used in the residential mortgage-backed securities market to adjust for prepayment risk. The purpose of this paper is to provide a brief tutorial on the notions of a ZV spread and OAS, using as a frame of reference the HJM [4] arbitrage- free term structure models. The HJM model, as applied to bonds, provides an objective method for valuing embedded options and determining market mispricings. Using these objective measures, we can more easily define and characterize both ZV spreads and OAS. We show that ZV spreads measure the excess spread on a bond due to both the embedded options and any mispricings. The mispricings could be due to either model or market errors. A market error represents an arbitrage opportunity. A model error represents a misspecified model perhaps due to the selection of an incorrect stochastic process, omitted risk premia, omitted risks (e.g., liquidity risk), or market 96 Robert Jarrow imperfections (e.g., transaction costs). In contrast, we show that OAS is a measure of the residual spread in a bond due to only mispricings ( after the removal of all embedded options and interest rate risk); see [2], [3] and [8] for background material. Refinements and alternative measures of a bonds embedded optionality and mispricings are provided. An outline for this paper is as follows. Section 2 presents the theory, sec- tion 3 discusses ZV spreads, and section 4 studies OAS. Section 5 provides a numerical example to illustrate the previous concepts. Section 6 concludes. 2 The Theory Assume a typical Heath, Jarrow, Morton (HJM [4]) economy. Given is a fil- tered probability space ( , F , P, ( F t ) t [0 , ] ) satisfying the usual conditions (see [7]), where is the state space with generic element , F is a set of events, P the statistical probability measure, and ( F t ) t [0 , ] is the filtration.is the filtration....
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114 OAS Book 2007 - A Tutorial on Zero Volatility and...

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