Hi my name is Patrick Boyle welcome to my YouTube channel today we're going to
learn about variance swaps new derivatives that we'll be looking at
today so a variance swap is an over-the-counter financial derivative
which allows investors to trade future realized variance against current
implied variance. Today we're talking about variance while
yesterday we did talk about volatility swaps so you can watch that video as well
but how variance swaps work is they act like a forward contract on the future
realized variance of a given underlying asset. The reason that variance is used
rather than volatility is because variance which is volatility squared can
be replicated with a static hedge and so essentially these things are more traded
than volatility swaps because they're easier to hedge. At inception of the
trade the strike is usually chosen such that the fair value of the swap is zero
variance swaps allow investors to speculate on or to hedge risks
associated with volatility one leg of the swap will pay an amount based upon
the realized variance of the price changes of the underlying product.
Usually these price changes will be daily log returns based upon daily
closing prices. The other leg of the swap will pay a fixed amount which is the
strike quoted at the deals inception. Thus the net payoff to the
counterparties will be the difference between these two levels, and will be
settled in cash at the expiration of the deal. The long party receives realized
variance and pays the strike variance at maturity and a short party receives the
strike variance and pays realized variance at maturity. The annualized
realized variance is calculated based on a pre specified set of sampling points
over the life of the swap. normally an investment
Bank is the calculation agent for any variance swaps traded. The calculation
agent usually has some discretion over whether a market disruption event occurs
or the treatment of situations such as if a stock is delisted. This can lead to
issues if an investor is long and short identical products versus two different
investment banks. These problems are less of an issue if the counterparties are
joint calculation agents. The profit and loss on a volatility swap relates only
to the realized variance (which is volatility squared) of the underlying and
is unaffected by any directional moves, thus much like volatility swaps it gives
you a simpler and more pure volatility exposure than Delta hedging an option
does. The payoff of a variant swap is convex in volatility as you can see on
the screen right now. This means that an investor who is long a variance swap
will benefit from boosted gains and discounted losses, this bias is reflected
in slightly higher strike prices for variance swaps when compared to
volatility swaps, a phenomenon which is amplified when volatility skew is steep.
The fair value of a variance swap is determined by the cost of the
replicating portfolio of options. This cost especially for index options is
significantly affected by the volatility smile or skew. I've made videos on those
as well which are linked to above. The return calculation for a variance swap
on an index does not adjust for any dividend payments that are paid. This
means that the dividend modeling method can affect pricing. Long-dated dividends
are usually modeled as flat yield and near dated and hence either known or
relatively certain dividends are modeled discretely so they're essentially one by
one modeled. The payoff of a variance swap is variance notional times that
difference between the realized variance and the strike variance. Returns are
computed on a logarithmic basis for variance swaps. Instead of using the
standard calculation for variance the contract terms usually set the mean
return at zero. The reason for this is that its impact on the price is minimal
the expected average daily return is 1/ 252 of the money market rate while
it's omission has the benefit of making the payoff perfectly additive. That means
that 3-month variance plus nine-month variance in three months time would be
equal to 1 year variance. So it's sort of it's just that simplification in the
formula just makes everything a little bit easier. So how do we replicate
variance swaps? Well in order to price variance swaps we should first look at
the replicating portfolio that captures realized variance. The cost of that
portfolio should be the fair value of future realized variance. To achieve such
a portfolio you need to combine options of many different strikes. A portfolio of
options of all strikes, weighted in inverse proportion to the square of the
strike level. Each option weighted by 1 divided by its own strike squared gives
an exposure to variance that is independent of stock price. Intuitively
as the stock price moves up to higher levels, each additional option of higher
strike in the portfolio provides an additional contribution to variance
proportional to that strike. It takes an infinite number of strikes appropriately
weighted to replicate a variance swap. In practice this is neither possible nor
practical, there are only a certain number of options available at any
maturity and if the stock price moves outside the range of available strikes
the reduced Vega of the imperfectly replicated contract will make it less
responsive than a true variance swap. As long as the stock price remains within
the strike range trading the imperfectly replicated contract will allow variance
to accrue at the correct rate. So it's worth noting that when these things
first came out, you were able to trade variance swaps both on indexes and on
single stocks. Now during the credit crunch markets were extremely extremely
volatile and so even indexes were swinging around at movements of 10% a
day. Single stocks were even more volatile than that, and so what was
happening was that banks who had sold or bought but were hedging variance swaps
using this approach were finding that on single stocks, on a daily basis the
market was moving past the available strike prices at least for the static
hedge when it was first implemented and so these hedges weren't working very
well for single stock variance swaps and for that reason these single stocks
variance swaps aren't really available anymore. it's much easier to trade index
variantce but not really to trade these simply because the the banks that were
selling them really just didn't want to to be on the other side of that because
the hedge doesn't work very well in extreme markets. So anyhow how do we
price variance swaps? Variance swaps can theoretically be
hedged as I just described and thus priced as a portfolio of European Call and
put options with weights inversely proportional to the square of the strike.
Care must however be taken with regard to the volatility smile, as the wings (and
by that I mean deep in and out of the money option prices) can have a
disproportionate effect on the price. A volatility smile model is needed to
correctly price variance swaps. If we use the Heston model a closed
form solution can be derived for the fair variance swap rate. To learn more
about this take a look at chapter 12 of my book Trading and Pricing Financial
Derivatives which is linked to in the description below. Okay well you made it
all the way to the end of the video so the rules are, as you know, that you have
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a series on interest rate swaps. See you then, bye