Vega & Volatility

Introduction
Vega is a ratio that measures how much the bet price will change due to
a change in implied volatility.
All option pricing needs to make assumptions about the behaviour of
certain variables. Some are clearly inappropriate, e.g. two of the main
assumptions behind the Black-Scholes option-pricing model are that
volatility remains a constant and that continuous hedging is feasible, both
of which are plainly not true. But financial theorem in general often needs
to make these assumptions in order to draw comparisons between one
instrument and another for evaluation purposes. The gross redemption
yield of a bond is a case in point. In calculating the g.r.y. there is an
implicit assumption that all dividends will be reinvested in the same bond
at the same gross redemption yield through to maturity. Totally unfeasible
of course, but nevertheless the g.r.y. has provided one of the more
enduring measures in the relative evaluation of bonds. By the same token,
some of the underlying assumptions with respect to the pricing of stock,
currency and bond options can also be considered a bit flaky and this
chapter covers, possibly, the most tenuous of them all, volatility.
Volatility is a measure of the movement in the underlying. There are two
measures of volatility, historic and implied. Historic volatility is a measure
of past price movements, while implied is a number thrown out of the
pricing algorithm if the option price is an input instead of volatility.
There will be no inclusion of Wiener processes or Brownian motions in
this chapter so non-mathematicians need not bolt for the door just yet. An
understanding of the normal and lognormal distribution would be an
asset though.

3.1 Normal Distribution
Fig 3.1.1 illustrates two bell-shaped normal distributions. They are
symmetrical about the mean, zero in this case, while the standard
deviation, or volatility in options parlance, is ‘low’ or ‘high’.
If the share price at any time is assumed to be the mean of the
distribution, then the volatility measures the number of times the price has
traded, or maybe settled, away from the mean. If readings were taken
over a year measuring the difference in the price of a stock from one day
to the next, then graphically the readings may be represented as a normal
bell-shaped distribution. If the stock has low volatility then it would be
expected that for a high number of incidences the stock will close at
around the same price the following day. The frequency of the stock
having a big move in one day will be rare and this distribution is reflected
in the ‘Low Volatility’ graph. On the other hand if the stock is volatile
then the incidence of a big move will be greater and the distribution will
look more like the ‘High Volatility’ graph.
In Fig 3.1.2 the area under the graph and above the horizontal axis at
zero frequency is 1, reflecting the total probability of an event taking
place. In this illustration the arithmetic mean has been replaced by the
current underlying price of $100 which is directly below the highest point
of the distribution. If a vertical line was drawn from the peak to the
horizontal axis then 50% of the total area would be to the left of this line
and 50% to the right. This reflects the probability that there is a 50:50
chance of the underlying going up or down. To the right of $100 there is
a vertical line at the strike price of $101. If the area B to the right of the
$101 strike contains 40% of the total area A+B, then the probability of the
underlying ending above the strike is 40% and it would be reasonable to
expect the market price of the $101 upbet to be trading at 40.
If this normal distribution is an accurate fit of the movement of the
underlying, then since the graph is symmetrical at about $100, the
probability of the $99 downbet ending in-the-money, i.e. the underlying
below $99, is also 40%.
3.2 Lognormal Distribution
There is a flaw with using the normal distribution for modelling the price
movements of financial and commodity instruments and that is that
absolute price movements are assumed. For example, if a healthy
company has a share price trading at £1, should the probability of the
share going down £1 be the same as the share going up £1? In effect, are
the odds of the healthy company going bust with absolutely no assets
producing a zero share price the same as the odds of the healthy
company doubling the share price to £2? The answer is no, and in order
to circumvent this problem, the lognormal distribution is used in financial
engineering.
If the underlying moves from $100 to $80 then it has fallen 20%. If the
underlying falls $20, from $120 to $100, then it has only fallen 16.667%.
If the underlying falls from $125 to $100, then that is a 20% fall. In effect
a distribution is required that gives a move from $100 down to $80 the
same probability as a move from $100 up to $125. The lognormal
distribution fits the bill since –log($80/$100)=log($125/$100).
The lognormal distribution is shown in Fig 3.2.1 where there is a fatter
skew to the right of the mean, so that when a lognormal distribution is
used the $80 strike downbet is now worth less than the $120 strike upbet,
reflecting a higher probability of an absolute move of $20 upwards rather
than downwards.

3.3 Historic Volatility & Implied Volatility

Historic volatility is a measurement derived from studying the past price
behaviour of the underlying, while inputting the option price into the
pricing model (instead of volatility) will generate the implied volatility.
Historic Volatility
The numbers required to calculate the ‘fair value’ of an option are the
underlying price, the strike price, the time to expiry, volatility plus interest
rates and yield (which are both assumed to be zero for the present in
order to keep things simple). Since time to expiry, the current underlying
price and the strike price are given, plus zero interest rates and yield are
assumed, the volatility number in evaluating ‘fair’ value is the all important
input. There are many statistical methods of calculating the historic
volatility ranging from simple weighted averages through to more complicated
stochastic methodologies. Many books have been written on this
subject alone and this book is not going to compete in that space, but
one observation can be made.
When assessing an accurate historic volatility for use in an options
pricing, model the algorithm employed will generally be of less
importance than the length of time used over each reading. For example,
assume a trader wishes to trade a twenty-day option, should a 1-day, 20-
day or 1-year moving average with respectively a 5%, 6% and 7% ‘vol’
be used? The twenty-day ‘vol’ would be the most appropriate because if
the one-day 5% volatility is used then there is the ever-present danger of
getting caught short when the market moves. Alternatively, the 7% oneyear
‘vol’ could be too expensive since it is perfectly feasible that the
market may be stagnant for 20 days and then become volatile for the
remaining 232 trading days in the year.
‘Scalpers’ will take an ultra short-term view on ‘vol’ while funds look at
the bigger picture adopting a range of various measures of stochastic
volatilities. There is no such thing as the historic volatility but a myriad of
techniques and timescales all deriving their own interpretation of historic
volatility.
Implied Volatility
Implied volatility is the measure of standard deviation that is an output
from Black-Scholes and other mathematical models available for evaluating
options.
By entering the underlying, strike, time to expiry and option price one
can generate the implied volatility. This volatility can then be used as the
input to create a whole range of option prices for different strikes and
underlying. With conventional options the preferable strike to use for
calculating the implied volatility is the at-the-money option. This is
because at-the-monies are generally more liquid so a keener price is
available, but more importantly they have the most time value which is
the factor on which volatility works. But with binary options the at-themoney
is always 50 so it is necessary to generate two implied volatilities
either side of the at-the-money and, bearing this in mind, the greatest
liquidity will be found in an out-of-the-money upbet and out-of-themoney
downbet.
Historic v Implied Volatility
An options marketmaker will always be bidding and offering around a
volatility level where he feels the weight of options buying equates to the
weight of options selling. To that end historical volatility is of little concern
to him, just a constant assessment as to where the ‘implied’ is at any one
time. Alternatively an options hedge fund may specialise in studying
historic volatility in order to go ‘long’ or ‘short’ premium. The relative
importance of historical and implied volatility is very much dependent on
the trader’s view on random walk theory and Chartism. If the reader is by
nature a chartist then it is more likely that the historic volatility concept
will weigh more heavily than fundamentals. Alternatively the
fundamentalist is more likely to study the schedule of upcoming
announcements on economic data and events and attempt to gauge what
kind of reaction they are likely to precipitate in the marketplace.
A brief anecdote might better describe the gulf that can occur between the
two separate views of volatility. In 1992 the author was trading
conventional options on 90-day Sterling LIBOR futures. The future was
trading at 89.25 with the 89.00 Puts on offer at 1 tick a thousand contracts.
The options had two and a half days to go to expiry. The implied volatility
of the 89.00 Puts was about 14.8% with the at-the-money 89.25 straddle
trading at around 12%. On the (arguable) assumption that the straddle was
a more accurate reflection of historic volatility than the 89.00 Put, even at
1 tick the 89.00 Puts looked expensive with an implied volatility of 14.8%.
An associate bought the thousand on offer. Two days later sterling came
under such pressure that the UK government was forced to raise interest
rates by 2% to 12% leading to the future expiring that day at 88.00 with the
89.00 Puts expiring worth 100. The trader who bought the 89.00 Puts
grossed £1.25m on a £12,500 outlay. Basically the implied ‘vol’ was too
cheap and was not accurately reflecting the tumult that was going to occur
in the next two days. Implied volatility is important, so too is historic
volatility, but what is most important is forecasting where historic volatility
is going to be because that is where the real money is.