Q:
Describe the three principal derivatives (of first and second order) associated with implied volatility, and what market structures are used to hedge each of them. Give a numerical example of each market structure with its individual components and volatilities.
A:
Vega - ATM(Level) - DN Straddle
Vanna - Skew(slope) - Risk Reversal
Volga - Wings(convexity) - VN Butterfly
The evolution of the volatility surface can be decomposed in three main movements, for each expiry:
• Parallel Shift
• Convexity Increase/Decrease
• Slope Increase/Decrease.
To represent these movements in terms of market instruments, one can consider:
• The ATM straddle volatility as an indicator of the level
• The Vega Weighted Butterfly as an indicator of the convexity
• The Risk Reversal as an indicator of the slope.
Hedging Volatility Risk in a B&S World
In practice, the a trader’s book is frequently updated in terms of the underlying asset price and implied volatility. If the book is re-valued and hedged as in a B&S world, then we know from the previous analysis that
we have to minimize the model risk by minimizing the Vega exposure.
Then the book will be Delta-hedged against the movements of the underlying asset; it will be Vega-hedged against the change in the implied volatility.
Vega-hedging must be considered in a very extended meaning: the portfolio must remain Vega-hedged even after movements in the implied volatility and/or the underlying asset.
So, hedging a book in a B&S world implies setting to zero the following Greeks:
• Delta
• Vega
• Vanna
• Volga
The Delta exposure is (usually) easily set to zero by trading in the underlying asset’s cash market. The volatility-related Greeks are set to zero by trading (combinations of) other options.
Tools to cancel Vega exposures are:
• ATM straddle: this structure has a strong Vega exposure, low Volga exposure and nil Vanna.
• Risk Reversal 25Delta: no Vega and Volga exposures, strong Vanna exposure.
• Vega Weighted Butterfly 25Delta: no Vega and Vanna exposures, strong Volga exposures.
By combining the three structures above, traders make their book Vega-hedged, and the keep this hedging stable to implied volatility and underlying asset movements.
Hedging Volatility Risk in a Stoch Vol World
Motivation:
• Managing the volatility risk on the B&S’s assumption is inconsistent and incomplete.
• All the volatility related Greeks are zeroed, but the model assumes that the impled volatility is constant, so they should not be hedged.
• The book is revalued with one implied volatility (typically the ATM), whereas on the market a whole volatility surface is quoted and it changes over time (the three movements for any expiry have been analyzed before).
• The pricing of exotic options is not consistent with a volatility surface.
Understanding Vanna
Vanna is a greatly under-used, higher-order option tool. Apart from
being useful in its own right by virtue of its plain definition, it is
also a valuable indicator that reveals information about the structure
of an option portfolio,
as well as the dynamic properties of a portfolio
with respect to time. It remains, however, conspicuous by its absence
from many traders’ and risk managers’ typical risk profile matrices.
Vanna is typically defined as the change in option delta for a change
in implied volatility. Usually it assumes a normalized form so as to
show the change in delta for a 1% move in implied volatility. Call
options have positive vanna, and puts have negative vanna. This is
because an increase in implied volatility raises the chance that any
call or put will expire in-the-money and this is synonymous with a
higher, absolute delta.Perhaps a principle reason for the lack of
attention vanna receives is that the risk associated with it might be
thought innocuous. After all, hedging the effects of vanna is simply a
matter of hedging with the spot product. As implied volatility changes,
the portfolio delta changes due to the portfolio vanna and the trader
simply neutralizes the effect by buying or selling the underlying
product in the relevant quantity. This story is however an
over-simplification and also omits vanna’s other potential uses.Firstly,
being a higher-order Greek, its potency is often magnified by compound
events. Let’s take an example of a portfolio that is long calls, short
puts, fully delta-hedged and vega neutral. This position will exhibit
positive vanna, as both the long calls and short puts are vanna
positive. Now, if the spot price is unchanged, it is true to say that
the trader is not greatly concerned by her long vanna exposure. Whether
implied vol rises or falls, he/she can simply re-hedge to maintain
delta-neutrality with little or no profit and loss implications, since
the spot price has not moved far, if at all, from where the original
position was entered into.Now, however, consider two concurrent events;
namely the implied volatility increases and the spot price falls. In
this case, the trader is running into the short put position as implied
volatility is rising, which is unfortunate. But things are compounded
because he/she is also becoming longer delta in a falling market, due to
positive vanna. The trader is long vanna; the implied volatility is
rising, so the delta position increases. This is where you have double
trouble.In short, vanna is a greater risk when multiple events occur. In
this sense, it is a complex risk ideally suited to situations when a
trader’s own outlook is more specific with respect to the covariance of
an underlying’s price and implied volatility movement. When using hedged
options to form a certain strategy, or to structure a desired pay-off
profile with respect to the spot price and implied volatility, vanna
effects should always be included in the calculation.
Using Vanna
The uses of vanna however go well beyond those implied by its simple
definition. A secondary use of the vanna metric is as an indicator of
the portfolio’s vega profile with respect to the upside and downside. In
complex inventories containing longs and shorts of varying strike and
quantities, vanna provides a single number that can at least in part
summarize the distribution of option premium across the curve. For
example, if a trader has positions at many strikes both above and below
the current spot price, he/she will typically reduce this for
convenience into vega by bucket or by curve segment. So an array of
longs and shorts can be simplified by taking the weighted average vega
contributed by the inventory in each strike. The outcome may be a
conclusion that, on balance, the portfolio is long calls (upside) and
short puts (downside). Another way to achieve the same goal would be to
view the vega risk up and down on an underlying price-slide risk matrix.
But in many cases, both methods can be approximated by a glance at the
vanna metric. By definition, positive vanna implies a position that is
either net long calls, net short puts or both.The usefulness of this
shortcut can be augmented in two ways. Firstly, by having in mind the
vanna of a certain option or options, the position can be synthetically
converted into a vanna-equivalent position. So, if the trader knows that
a 15% delta call option of the expiry in which he/she is interested has
say a vanna of vx per 1000 lots, and the corresponding -15% delta put
option a vanna of -vy, then the risk reversal has a vanna of plus or
minus (vx + vy), depending on the direction of trade. Having these
numbers memorized or otherwise to hand, can certainly pay dividends in
fast markets or when inventories become highly complex. Knowing how to
neutralize an imbalance in vega spreads between upside and downside
positions can be made less an art, more a science by employing this
synthetic vanna trick.
Applying Vomma
The observant reader may have spotted a subtle flaw in this thinking,
however. The vanna number is unable to provide clarity on whether its
value is being driven by a long or short call position, a long or short
put position or some combination of one and/or the other. This
deficiency is easily overcome by using vanna in combination with vomma.
Vomma is the higher-order Greek giving the change in vega for a change
in implied volatility. By virtue of the fact that at-the-money options
have all but zero vomma, one can attribute a position’s vomma entirely
to its wings (i.e. to the upside and downside premium). And since both
out-of-the-money calls and puts have positive vomma, it provides the
additional information the vanna metric omits.Let’s consider a position
that is positive vanna but vomma-neutral. What can be gleaned from this?
Well, the positive vanna indicates the position is long calls, short
puts or both. But the zero vomma indicates a flat wing position. In
other words, the longs and shorts in the wings must net off. Therefore
this position is both long calls and short puts, in roughly vega neutral
amounts.Also, consider a position that is short vanna and short vomma.
The short vanna suggests a short call and/or long put position. But the
short vomma indicates short calls and/or short puts. One would conclude
that this position is likely to simply be short calls, as the put
position appears to net out to flat.This demonstrates how using vanna
and vomma in tandem, one can generate a useful shorthand for
characterizing upside/downside exposures. To be even more precise, one
can also use the technique applied above of noting the vanna for a
particular wing option, with respect to the vomma. So, if one notes the
vomma of a 15% delta strangle, then one can deduce with reasonable
accuracy the equivalent upside and downside position in terms of 15%
calls or puts as indicated by the vanna/vomma pairing. Indeed, it is a
relatively simple matter to add these benchmark numbers to any
configurable option risk matrix.Another use of vanna is as a proxy for
charm. Given that falls in implied volatility can be considered as
analogous to a reduction in the time to expiry, vanna can be used
indicatively to assess how a portfolio delta will alter over time.
Specifically, positive vanna will be associated with a falling net delta
as time passes. Why not simply use charm in this case? Of course one
could. But as the information is contained indirectly within the vanna
metric and vanna has many other uses besides, it is not uncommon to see
risk matrices omit charm in favour of vanna.
Health Warnings
Having made a case for vanna, a couple of
health warnings are in order. None are specific to vanna, but they still
ought to be declared. As with any vega-related Greek, addition across
durations is not appropriate unless the volatility surface across
durations moves identically. Vanna, as for any option risk metric,
varies with respect to every other variable (spot price, implied
volatility, cost of carry etc.). Perhaps most pertinent as relates to
the above suggestion that vanna has a useful descriptive function, is
the change in vanna with respect to spot price. Vanna is certainly a
risk metric best viewed on a price-slide risk matrix.Vanna is of course a
function of the implied volatility values employed in its calculation.
If these are theoretical rather than the prevailing market’s, then a
discrepancy will exist between the trader’s vanna and that which the
market would assign to its position, should these implied volatilities
differ. Perhaps this is most relevant when the vanna is used in its most
straightforward way; that is when it is used to determine the likely
change in delta for a change in implied volatility. In large implied
volatility movements, the vanna affect is magnified in like proportion.
And so large profits and losses can accrue not simply from having a
large vanna exposure, but also from any discrepancy between the
theoretical vanna and what would be the market’s perception of the
trader’s vanna using prevailing market volatilities.But all in all,
vanna, used correctly, is a most revealing higher-order option Greek
that should undoubtedly be bleeping on any trader or risk manager’s
radar.
