Nassim Taleb, the veteran options trader and the celebrated
author of the book Black Swan was once quite annoyed when
someone on his trading floor suddenly started singing “Volare,
Volare”. It seems most of the traders on the floor of New York
stock exchange have been singing, albeit in a rather hoarse
voice, the same 1950s Italian song for the past one week.
Indeed, the global financial markets have been swinging and
dancing to the tune of the song “Volare” recently.
“Volare” in Italian means “to fly”. It comes from the Latin
word “Volatilis”. Financial markets are indeed flying; or, in the
financial lexicon, markets are crashing and volatility is on the
rise. The VIX index, which is an index of the implied volatility
of the S&P500 stock index was up more than 20% and hit 41.9
at one point in New York trading this week and some traders
cried “Armageddon”. One index trader in Hong Kong
bemoaned to me over the phone last Wednesday that “It’s the
tails again!” And as the Dollar-Swiss Franc volatility hit 23
(read 23%) this week a terrified FX trader in Tokyo shouted
out to his colleague across the floor, “Are we going to see 35
vols in USD/CHF this time?”
Volatility is all that matters in the financial markets these days.
Everybody, from the floor traders to the interbank dealers to
the G7 Finance ministers and financial journalists, talks about
volatility. It has come to dominate our lives.
But, what is volatility? It is both a profound and a silly
question. We all know what “volatility” is and yet all of us will
have difficulty in explaining the concept articulately and in a
concise manner. This is because volatility can mean different
things to different people and in different context. Is it a fear gauge? Is it a statistical quantity? Is
it a parameter in option pricing model? Is it an asset? In fact, it is all of these things and perhaps
much more. Volatility is the most profound, and yet perhaps the most elusive concept both in the
real financial markets and in the theory of quantitative finance. As a veteran options trader once
told me, “Beware of volatility, it has tails and it can fly”.
Rahul Bhattacharya
Inside this Issue August 12, 2011
Volume I, Issue 1
Volatility has tails and it can fly
Understanding Historical Volatility
Product idea for the Equity Market
Derivatives Trading Terminology
The Silent Mafioso – I: The man
who once ruled Deutsche Bank
Book Review: Options on Foreign
Exchange, 3rd
Edition by David
deRosa.
Vanilla times
2
Broadly speaking, there are two definitions, or rather concepts, of
volatility in the financial markets and within the theory of
quantitative finance: historical volatility and implied volatility.
Historical volatility is a statistical concept and it simply means
the standard deviation. The accepted norm in the market – and the
way volatility is understood and quoted in most asset markets – is
that historical volatility is the standard deviation of the asset
returns (and not the asset price). For example, say, over the last
ten days an asset has taken the following (historical) price path:
100, 101, 101.5, 102, 102.5, 104, 105.5, 105, 104.8, 106.
Here’s the algorithm to calculate the historical volatility from the
above time series of prices. First, calculate the log returns of the
above prices by using the formula: log returns is equal to today’s
asset price divided by yesterday’s asset price; then, then calculate
the mean of the new time series of log returns; the mean is
generally denoted by the Greek letter “mu” and is simply the
arithmetic average of the series; finally, calculate the historical
volatility using the statistical formula for standard deviation. If
we implement the above algorithm on an Excel™ spreadsheet and
do all calculations by ourselves then the value of historical
volatility comes out to 0.6378%. If we use Excel’s library
function =STDEV(.) then the value comes to 0.677%
This would be known as the daily volatility, because the asset
price movement is depicted over days. Daily history will produce
daily volatility; weekly history will produce weekly volatility and
so on. Normally, historical volatility, or any other measure of
volatility, is always annualized. To annualize the above figure we
need to multiply it by square root of 252. Thus the annualized
volatility of the above asset price would be 10.13%.
Team Latte
Historical Volatility
Historical Volatility is the
standard deviation of the asset
returns, defined as natural log of
this period’s price divided by
previous period’s price.
Historical Volatility is analogous
to the concept of realized
volatility or actual volatility.
Get access to high value added quantitative finance
content via www.risklatte.com. Mathematical finance, quantitative models of
derivatives valuation, portfolio analysis, financial risk management and
algorithmic and quantitative trading.
3
Given the significantly high volatility in the equity markets
one of the equity linked products that one can consider for
investment purposes is the “Single Asset Best of Option”.
This is typically an institutional product, but can be very
easily tailored to suit small investors, such as high net worth
individuals.
This option pays the best of (or the maximum of) the equity
return and a certain fixed percentage. For example, the
product can be structured to pay at maturity, say, one year,
50% of the return of S&P500 index and 4%. In other words,
the product pays either 4% fixed return on the notional capital
or 50% of the return of S&P500 after one year, whichever is
greater. Say, S&P500 index generates a return of 10% after
one year. Then 50% of 10% return is 5%, which is greater
than 4%. Thus, the investor would get 5% on his invested
capital. However, say, if S&P500 after one year generates a
returnof -3% (negative 3%), then 50% of that return would be
-1.5% (negative 1.5%) which is less than 4% and hence the
investor will get 4% return on his invested capital.
This option can be decomposed into a long coupon paying
bond with a coupon of 4% and a leveraged call option. The
strike price of the call option would be 1.08 times the current
(today’s) value of the S&P500 index.
Team Latte
Tails: “Tails” represent the implied volatility of the out of
money (OTM) options with respect to the at the money
(ATM) options in a Black-Scholes world. Looked at another
way, the “tails” represent the fourth moment of a Normal
distribution.
Fat Tails: “Fat Tails” means extremely large Kurtosis, or the
fourth moment, of a Gaussian (Normal) probability
distribution. One would need a wider graph to represent a fat
tailed Normal distribution because the standard deviation, the
second moment of the distribution, blows up and makes tail of
the distribution on both sides of the mean move upwards from
Product Idea Single Asset, Best of Option
The payoff of the option is given by:
Algebraically decomposing the above
product gives:
The above can be further decomposed
as:
This clearly shows that the payoff can
decomposed into a coupon paying
bond (with a coupon of 4%) and a
call option with a strike price of 1.08
times the current price of the asset.
4
the X axis, where returns are plotted. Many statisticians and
mathematicians talk of infinite variance to denote fat tails.
However, experienced option traders consider “fat tails” as
simply the result of rapidly changing volatility captured by
volatility of volatility (vvol). Even if the volatility of an asset is
low, a high volatility of this low volatility can cause fat tails.
VVol: “VVol” stands for volatility of volatility and option
traders believe that the reason the out of the money (OTM)
options have higher implied volatility than the at the money
(ATM) options is because of the existence of volatility of
volatility. In a Black-Scholes world, where options are priced
using a Black-Scholes model, volatility of an asset is assumed to
be constant. However, in the real world, the volatility of assets
such as stocks, FX, bonds, interest rates, etc. display variability.
One way to capture this variability statistically is by estimating
the dispersion or the standard deviation of the volatility, which
in other words, is thought of as volatility of volatility.
Quoted Volatility: In the FX options market, traders quote
option prices in terms of volatility (percentage volatility or
volatility points) rather than in dollars and cents. A 25 delta,
USD call / JPY put option quote could be 15.50 / 16.50, which
would denote a bid volatility of 15.50% and an ask volatility of
16.50%.
Team Latte
Trading Terminology
Tails: Implied volatility of Out of
the money (OTM) options
Fat Tails: Large Kurtosis, or the
fourth moment of a Gaussian
distribution
VVol: Volatility of Volatility, which
is one way to capture variability
of volatility.
Quoted Volatility: Method of
quoting option prices in the FX
options market.
A comprehensive
reference book of formulas, math equations,
algorithms and short explanation for CFE
students and quantitative finance professionals.
5
It feels strange to talk about a
once famous banker ten years
after his death, for ten years is
too short a period for putting
things in “historical perspective”
and too long for any meaningful
analysis in the global financial
markets.
He is the founder and architect of
modern day Deutsche Bank who
died at a very young age in a
tragic accident in December
2000. He was perhaps one of the
most colourful and handsome investment bankers in the City of
London. He was an expert on financial derivatives and a pioneer
in the field of derivatives banking He had a mistress and jet
setting lifestyle. But above all, he was a ruthless manager whose
strategies sometimes were perhaps more in line with that of a
mafia don than a Dartmouth MBA.
His name is Edson Mitchell and he was the head of global
markets at Deutsche Bank and a member of the Bank’s Board
until his tragic death in a plane crash on December 22, 2000;
and his colleagues at Merrill Lynch, the bank where he worked
before joining Deutsche Bank, used to call him The Silent
Mafioso.
I discovered him on a recent trip to London. Over a sumptuous
dinner with an old friend at the famous restaurant La Portes des
Indes at Marble Arch, discussion turned towards financial
derivatives, Bankers Trust, Deutsche Morgan Grenfell, Deutsche
Bank etc., etc. One thing led to the other, as it usually happens at
such dinner table discussions, and along the way my friend
casually mentioned the name of Edson Mitchell. He knew this
man many years ago and had apparently once been interviewed
by him. A half an hour interview with Mitchell had left an
indelible mark on my friend. He thought he had met the
fictitious character, Don Coreleone, from the movie Godfather,
only this real life replica was infinitely more handsome, far
more polite and suave, had an Ivy League MBA and could talk
about financial derivatives and golf in the same dispassionate
Edison Mitchell
The Man who once Ruled
Deutsche Bank
Edson Mitchell can be considered
the founder and the architect of
modern day Deutsche Bank.
He was one of the heads of global
markets at Deutsche Bank until his
tragic death in a plane crash in
December 2000. He had a
multimillion dollar pay package
and a ravishing French mistress.
He was a young and handsome
investment banker who epitomized
the dictum that cheque book can
make any feat possible.
6
tone. Our lives are shaped by characters from novels and
movies; his was shaped by Edson Mitchell.
A half an hour job interview and the impression of a lifetime.
My friend never saw him again, though he ran into him a few
times, in Christmas parties, airports, and financial news
articles. Edson Mitchell not only single handedly transformed
Deutsche Bank into a top tier investment bank from a sleepy
German lender of no particular repute, he also changed the
entire investment banking landscape of the City of London. He
inspired loyalty amongst his subordinates and brought with
him a large army of investment bankers when he defected
from Merrill Lynch to Deutsche Bank. Many of them revered
him and some idolized him. His peers, including Josef
Ackerman, the head of investment banking at Deutsche Bank
at that time, were in awe of him. He was a deity at the altar of
money and power, who epitomized the notion that the cheque
book can make any feat possible. At the turn of the
millennium, he was the future of Deutsche Bank
And, he was in love with Estelle, a thirty something, ravishing
French beauty whom he met in London and who became his
mistress. Today, however, he is the tragic hero, whom no one
wants to remember; least of all, his ex-colleagues and the
senior management team of Deutsche Bank.
After dinner, I realized that someone needs to tell this story
not because Edson Mitchell’s story is a story about Deutsche
Bank, one of the most successful investment banks in the
world today. Someone needs to tell this story because this is a
story about human greed, power, love and lust. It is a story
about a man’s desire to become God.
The story continues in the next issue………….
Rahul Bhattacharya
Who was Edson Mitchell?
He was a pioneer in the financial
derivatives business, especially the
dollar interest rate swaps market.
Before joining Deutsche Bank,
Mitchell worked at Merrill Lynch in
the U.S. for 15 years, turning Merrill
into a global derivatives
powerhouse.
The name of Mitchell’s mistress was
Estelle and it is rumored that he was
living with her in London.
In 1995 Deutsche Bank recruited him
on a two year contract with a pay
package of GBP4.0 million per year.
7
Options on Foreign Exchange, 3rd
Edition by David deRosa: ***
This week’s book is Options on Foreign Exchange, 3rd
Edition by
David deRosa. It is an excellent book and we highly recommend it to
all those who want to get good understanding of FX exotic options
and how they are priced and structured. It is also a book, which has
one whole chapter on volatility where the author clearly and
concisely talks about various concepts surrounding volatility,
especially FX volatility, and some of that explanation is very intuitive
and puts things into historical context. In fact, that is the case,
throughout the book, where many of the advanced mathematical and
quantitative concepts are explained intuitively which enhances a
reader’s understanding of these concepts. In short, this book is a must
have for all those in the business of financial derivatives, especially,
those in the FX options and FX structured products markets. This
book also has a great chapter on Barrier and Binary Currency Option and what is really cool are
the ways the formulas for “One Touch”, “Double No Touch”, “Double Barrier” Binary and other
“out” options are presented and explained. For example, Hui (1996) formula for double barrier
binary option is explained in a lucid manner and so is the formula of Reiner, Rubinstein (1991)
and Wystup (2007) for One Touch binary option. The Chapter on American Exercise Currency
Options is also quite illuminating and easy to follow. Overall, we would rate it Triple Star out of
Five Stars.
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Notes: i. Edson Mitchell’s photographs is courtesy Google images
ii. The cover image of the book Options on Foreign Exchange, 3rd Edition is courtesy of
Google images.
Excel™/VBA based, 100% applications course covering
quantitative modeling for derivatives valuation and risk
analysis, financial risk management, portfolio analysis &
asset allocation models, algorithmic & derivatives trading
and financial mathematics. Write to [email protected]
for more information on CFE.