A Tutorial on Zero Volatility and Option
Adjusted Spreads
Robert Jarrow
Johnson Graduate School of Management
Cornell University
Ithaca, NY 14853, USA
[email protected]
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 security’s 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 security’s spread due to mis
pricings. Refinements and alternative measures of a bond’s 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 mortgagebacked 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
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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 bond’s
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 (
Ω, F
τ
, P,
(
F
t
)
t
∈
[0
,τ
]
) satisfying the usual conditions
(see [7]), where
Ω
is the state space with generic element
ω
,
F
τ
is a set of
events,
P
the statistical probability measure, and (
F
t
)
t
∈
[0
,τ
]
is the filtration.
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