tradingvolatility.txt

This sensitivity or stock exposure is measured by the slope of the price of
the investment, not the actual level or value of the investment. With volatility trades we
specifically design portfolios in such a \\a\ that the exposure varies constantly with the
underlying share price.

The strict definition of volatility as used by market participants is quite complex
and involves the use of natural logarithms and a knowledge of the statistics
associated with what is known as the lognormal distribution.

on what is known as the implied volatility. The implied
volatility is probably one of the most important measures for the volatility player
and will be dealt with later.

The volatility of a price series is a measure of the deviation of price
changes around the trend.

The strict definition of volatility as used by market participants uses a function
of relative price changes rather than price changes

d some mean trend. One use of the standard volatility measure is to make
probability statements about the approximate likely range of the stock price in the
future. Assuming that the st

1. There is a 66% chance that the stock price is in the range (100 - x)% to (100 + x)% of the price today.
2. There is a 95% chance that the stock price is in the range (100 - 2x)% to (100 + 2x)% of the price today.
3. There is a 99% chance that the stock price is in the range (100 - 3x)% to (100 + 3x)% of the price today.

weeks or 50 weeks? If we
go back as far as 200 weeks we may be including data that are so out of date as to
be increasing, rather than reducing, the error of the estimate. Empirical evidence
suggests that volatility is constantly changing and so it is important not to go too
far back in time. Another issue is that of using inter-day data. In the above we
used one price per day and many practitioners are content with the results obtained
from daily closing prices. Some traders advocate using more than one piece of
information per day such as the open, high, low and close.

There is evidence for what
is known as "reversion to the mean" with volatility. Simply put, this means that if
a particular price series begins to exhibit high volatility this will eventually fall.
Similarly if the series begins to exhibit particularly low volatility then it will
eventually rise. The theory suggests the presence of some long-term average
volatility level. The trick is of course picking the high and low volatility points.

. Most individuals would assign a higher probability to the current stock price
and a lower probability to prices further away. Unless some extraordinary event
occurs one would normally assume that the stock price in three months' time will
be near the present price.

. Because this ratio is so important in constructing hedged portfolios,
the delta is also often referred to as the hedge ratio.

extent of the time decay at each stock price. The
curves are furthest apart with the stock price near the exercise price and most
bunched together at the extremes. It is clear that the near-the-money options suffer
the most time decay. As one moves further away from the exercise price, the
effects of time decay get smaller and smaller. Deep out-of-the-money or deep inthe-money
options have virtually no time decay at all.

mple. The very options that have no time
decay also have very little price curvature. And we need price curvature for the
volatility trade. Options with the most curvature unfortunately suffer the most time
decay. Again we return to this subject later.

The new option
contract value is 100 X (8.95 - 8.19) = $76 lower than the one used in the naive
description of the trade without time value considerations. So what was a profit of
$49 turns out to be a profit ot $49 - $76 = -$27, i.e. a loss. So what sort of price
move in one iliiection do we need to make a profit? It is clear from Figure 4.5 that
it tlu- price had moved as far $106 in the three months then the trade ^vould break
even. If the price had moved higher then a profit would n-siilt. One c

One can repeat this exercise for various different scenarios to get an
idea of what type of price swings one would need to make a profit.

Rehedging more frequently will
certainly capture profits due to small price swings but has the disadvantage of
missing out on the really large profits obtained with large price swings. Rehedging
less frequently gives more scope for the large profits but will mean that the little
profits associated with smaller price swings will be missed.

. With a bigger portfolio, rehedging can be done by adjusting the option or
stock size, and costs such as the bid to offer spread, taxes and commissions will
dictate which side to adjust. The chosen rehedging strategy will be a question of
compromise. If the dealing costs are high (as with options on stocks) one has to
wait for a significant price move to justify the transaction charges. If the dealing
costs are low (as with options on futures) then rehedging can take place relatively
frequently.

experienced. If the price paid is low and the volatility is high, the player will win
overall. If the price paid is high and the actual volatility is low then the player will
lose. If the individual in question executes this trade on many occasions there will
be a long-run average rehedging revenue. If this long-run revenue exactly matches
the price paid, then we can say that the option price was fair in volatility terms.
This, then, gives us another way to see if an option is expensive or not.

If we assume that the relevant distribution is lognormal and that the
dispersion measure used is set equal to the expected volatility, then both fair value
methods give identical results. If the volatility input into both models is high (low)
then both fair values will be high (low). The volatility of the day-to-day price
changes directly affects the distribution of prices on expiry. If the volatility is high
then the distribution of stock prices on the one future expiry day will be very
dispersed. If the volatility is high, then there will be a much higher chance of the
expiring option ending up deep-in-the-money and being worth a considerable sum.
So an option on a highly volatile stock should have a higher fair value even when one considers the static buy and hold
strategy.

The sensitivity of option prices to changes in volatility is similar to the sensitivity
to time. Near-the-money options are most sensitive and deep out-of-the money and deep in-the-money options are less sensitive. The
figures illustrate the importance of getting the volatility right. The standard long
volatility trade was explained using a fixed volatility of 15%. At the start with the
stock price at $99 the option was priced at $5.46. If immediately after putting this
trade on, the market priced all one-year options as if future volatility was only
going to be 10% we see that the option price would immediately drop to $3.49
resulting in a loss without anything really happening. The sensitivity of option
prices to changes in volatility is so important to market participants that it has been
assigned a special symbol. The rate of change of an option price with respect to
volatility is known as vega. Vega usually is defined as the change in option price
caused by a change in volatility of 1%.

3. The vega of an option
not only varies with the underlying stock price but also with time to maturity.
Other things being equal, shorter dated options are less sensitive of volatility
inputs—shorter-dated options have lower vegas. This is

Accurate volatility estimation is very important for very longdated
options and virtually irrelevant for very short-dated options.

value. Options with different times to maturity, different stock prices and
different exercise prices can be compared by simple reference to the implied
volatility.

=========================
an option with a more marked degree of curvature. The long
volatility player is always looking for low-priced curvature. Curvature is so
important in the option market that it also has been assigned a Greek letter—
gamma. This concept is best demonstrated by considering two imaginary option
curves in Figure 4.8.
=============================

price changes. The real use of gamma to the option trader
and the volatility player is in the estimation of future rehedging activity. Gamma is
the change in delta associated with a change in price of the underlying and so the
gamma is also a measure of the rate at which we rehedge a delta neutral position. T

The long volatility trade generates profits from the curvature and these
profits are locked in by the rehedging process. So the gamma of the position is a
direct measure of the potential profit due to volatility and often the term long
gamma is used instead of long volatility. The term long gamma also refers to the
fact that the gamma of the position is positive.

Unfortunately these are the very options that suffer the most theta decay.

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o rise. As one crosses higher and
higher delta contours the portfolio becomes increasingly unbalanced on the long
side. There will be a point, depending on the rehedging strategy, at which
rebalancing must take place. One can in fact use the contours to dictate when to
rehedge. If we make an arbitrary rule that we only adjust the stock hedge each time
the delta changes say by 0.10 then each time the stock price crosses one of the
fixed contours we would rehedge. It is thus possible to see on a traditional stock
price versus time chart when a rehedging trade occurs.

. This particular simulation shows a high profit not only because of the high
volatility but also because the stock price path happened to stay in the high gamma
region right up to expiry.

Understanding the complexities of the effects of the stock price, the time to expiry
and the volatility on the option price, the delta and the gamma is essential in
managing a volatility strategy. The simple long volatility strategy will make a profit
in the right circumstances and will make a loss in the wrong circumstances. The
right circumstances are buying relatively low implied volatility options and
subsequently experiencing relatively high volatility. This can mean buying volatility
at 25% and experiencing 35% or buying at 5% and experiencing 15%. 1 he wrong
circumstances are buying what one thought was relatively low volatility and
actually experiencing even lower volatility. With any trading strategy it is useful to look at what is the most one can make or lose. We
need to examine the worst and best case scenarios.

The strategy needs volatility to make a profit and the worst case scenario is that in
which there is no volatility at all. There are a number of ways a price series can
have zero volatility.
The simplest case is that of the stock price staying completely constant through
to expiry. As expiry approaches the hedge will be altered but all transactions will
involve stock at the one constant price and on expiry the stock hedge will be
completely unwound at the same constant price. In this situation the only loss to the
strategy will be the time value of the option. Figure 4.16 shows the losses (type 1)
suffered under this type of zero volatility.
A second type of zero volatility (type 2), that in fact can cause higher losses, is
that in which the stock price moves in a gradual and orderly fashion towards the
exercise price, finishing exactly at the exercise price on the expiry date. The option
will thus always expire worthless, resulting in the total loss of the initial option
value. If the option was initially in-the-money then these losses will be slightly offset
by the profits caused by the short stock position. If the option is initially out-ofthe-money
then these losses will be added to by the short stock position

There are many best case scenarios, the most obvious one being that immediately
after implementation of the trade, the stock price swings violently up and down
around the point of maximum gamma (near-the-money). There are, however, two
other interesting extreme situations in which very large profits can be made.

As when using calls, the long volatility strategy using puts produces rehedging
profits because of the presence of price curvature or gamma, and the basic idea
underpinning the strategy is to find cheap options with high gamma.

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d not suit. There are situations, however, when the individual may decide that
if the market rises, volatility will fall and if the market falls, then volatility will
rise. The above combination would perfectly suit such a view. The player would
automatically become short volatility on the way up and long volatility on the way
down.
Another situation that may make such a combination attractive, is one in which
the individual viewed the upper option expensive compared to the lower option.
Here the individual has no view on volatility or the direction of the market price
but wishes to arbitrage out the difference in option values. In order to do this he must short the expensive option
and buy the cheap one. The net combination would be long of the underlying and
in order to remove market risk, the appropriate hedge would be used. The portfolio
would be run either to expiry or until the apparent anomalous price difference
disappeared.


The way in which the implied volatilities vary across strike prices depends on
the market and market conditions. Stock options typically have higher volatilities at
lower strikes and lower volatilities at higher strikes. The standard reasoning behind
this type of volatility profile is that in a falling market everyone needs out-of-themoney
puts for insurance and will pay a higher price for the lower strike options.
Also equity fund managers round the world are long billions of dollars worth of
stock and like writing (selling) out-of-the-money call options against their holdings
as a way of generating extra income. It is believed that this large scale selling of
options has the effect of lowering the implied volatilities of higher strikes. Figure
8.2 shows a graph of such volatilities versus strike prices and this is referred to as a
volatility profile or volatility skew. Practitioners refer to this particular profile as
a volatility smirk.

As with the differences across strike prices, the
differences over expiry cycles are always a function of supply and demand. When
markets are very quiet, the implied volatilities of the near month options are
generally lower than those of the far month and when markets are
very volatile, the reverse is generally true. In quiet markets, option players often
get lulled into believing that the quiet times will persist. In quiet markets no one
wants a portfolio long of near-dated options and so selling is often the best
strategy. Selling in any significant size will drive the price (and the implied
volatility) of the near-dated options down. In very volatile markets, everyone
wants or needs to load up with gamma. Near-dated options provide the most
gamma and the resultant buying pressure will have the effect of pushing prices
(and the implied volatilities) up.
Managing a complex options portfolio thus requires the use of a twodimensional
implied volatility matrix. One dimension is across strike prices and
the other is across time. The individual volatilities used should reflect, as near as
possible, the prices of the options in the market-place. As time passes the structure
of the implied volatility matrix may change and this is yet another source of
uncertainty that has to be considered.

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