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The latest generation of volatility derivatives goes beyond
variance and volatility swaps and probes our ability to price
realized variance and sojourn times along bridges for the
underlying stock price process. In this paper, the authors
give an operator algebraic treatment of this problem, based
on Dyson expansions and momentmethods, and discuss applications
to exotic volatility derivatives. The methods arequite flexible
and allow for a specification of the underlying process,
which is semiparametric or even non-parametric, including
state-dependent local volatility,jumps, stochastic volatility
and regime switching. The authors find that particularlyvolatility
derivatives are well suited to be treated with moment methods,
wherebyone extrapolates the distribution of the relevant
path functionals on the basis of afew moments. The authors
consider a number of exotics such as variance knockouts,conditional
corridor variance swaps, gamma swaps and variance swaptions
andgive valuation formulas in detail.
Volatility derivatives are designed to facilitate the trading
of volatility. A basic contract is thevariance swap which,
upon expiry, pays the difference between a standard historical
estimateof daily return variance and a fixed rate determined
at inception (Dupire, 1992; Carr and Madan,1998; and Derman
et al., 1999). A variant in this is the corridor variance
swap, which differsfrom the standard variance swap only
in that the underlying price must be inside a specifiedcorridor
because its squared return to be included in the floating
part of the variance swap payout (Carr and Lewis, 2004).
A further generalization is the conditional variance swap,
whichpays the realized variance of an asset again within
some corridor, whereby the average is takenonly over the
period when the spot is in the range. The advantage of conditional
corridor variance swap is that it allows one to take a view
on volatility that is contingent upon the pricelevel, gaining
exposure to volatility only where required. Although conditional
variance swaps appear to be traded (Securities, 2006), this
is the first article in the open literature proposinga pricing
methodology.
The authors present a numerical method, which extends
to other exotic volatility contracts such as, variance swaptions
without difficulties (Carr and Lee, 2007). The authors also
take theliberty of inventing exotic volatility derivatives
such as variance knockout options. These do notseem to be
much traded although they should be. A variance knockout
can be regarded as avariation on the theme of barrier knockout
options whereby the knockout condition is not triggered
by crossing a certain level. Instead, a variance knockout
vanishes in case realizedvariance prior to maturity, which
exceeds a certain pre-assigned threshold. The benefit of
avariance knockout over a barrier knockout is that its hedge
ratios are smoother.We use operator methods for pricing
in this paper and the presentation isself-contained for
the specific purpose at hand but see the review article
(Albanese, 2006) fora more extended discussion of operator
methods. One point we should stress is that themathematical
and numerical methods we present would work efficiently
no matter whatunderlying model for the stock price process
is chosen. We select one for the purpose ofdiscussing a
concrete case and generating sample graphs, but the reader
can do better ifhe/she intends to refine it. Our mathematical
and numerical methods are model agnostic as theydo not rely
on closed form solvability and their performance is not
linked to the modeldefinition. Any model for the stock price
dynamics would work well, the only limitation beingthat
market models cannot be accommodated within the formalism
we propose.
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