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MoreThanYouEverWantedToKnowAboutVolatilitySwaps.pdf
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Volatility swaps are forward contracts on future realized
stock volatility. Variance swaps are similar contracts on variance,
the square of future volatility. Both of these instruments
provide an easy way for investors to gain exposure to the
future level of volatility.
Unlike a stock option, whose volatility exposure is contaminated
by its stock-price dependence, these swaps provide pure
exposure to volatility alone. You can use these instruments to
speculate on future volatility levels, to trade the spread
between realized and implied volatility, or to hedge the volatility
exposure of other positions or businesses.
In this report we explain the properties and the theory of both
variance and volatility swaps, first from an intuitive point of
view and then more rigorously. The theory of variance swaps
is more straightforward. We show how a variance swap can be
theoretically replicated by a hedged portfolio of standard
options with suitably chosen strikes, as long as stock prices
evolve without jumps. The fair value of the variance swap is
the cost of the replicating portfolio. We derive analytic formulas
for theoretical fair value in the presence of realistic volatility
skews. These formulas can be used to estimate swap
values quickly as the skew changes.
We then examine the modifications to these theoretical
results when reality intrudes, for example when some necessary
strikes are unavailable, or when stock prices undergo
jumps. Finally, we briefly return to volatility swaps, and show
that they can be replicated by dynamically trading the more
straightforward variance swap. As a result, the value of the
volatility swap depends on the volatility of volatility itself.
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