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8/7/2019 Trading Volatility as an Asset Class Emmanuel Derman
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Page 1 of 37
June 10, 2003
T
RADING
V
OLATILITY
A
S
A
N
A
SSET
CL
ASS
Emanuel Derman
Columbia University
8/7/2019 Trading Volatility as an Asset Class Emmanuel Derman
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June 10, 2003
Summary
Volatility is a useful trading hedge against all kinds of disasters. Howcan you trade it?
Calls and puts dont quite do it. Though calls and puts are sensitive tovolatility, they are not sensitive only to volatility.
How can you do better?
1. W
HY
T
RADE
V
OLATILITY
?
2. V
OLATILITY
T
RADING
WITH
O
PTIONS
: T
HE
V
ALUE
OF
C
URVATURE
3. O
PTIONS
T
RADING
: W
HAT
C
AN
G
O
W
RONG
4. T
HE
V
OLATILITY
S
MILE
V
IOLATES
B
LACK
-S
CHOLES
5. W
HAT
C
AUSES
T
HE
S
MILE
?
6. I
S
THE
S
MILE
F
AIR
?
7. T
RADING
& P
RICING
V
OLATILITY
U
SING
V
OLATILITY
S
WAPS
8/7/2019 Trading Volatility as an Asset Class Emmanuel Derman
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June 10, 2003
W
HY
T
RADE
V
OLATILITY
?
I
I
W
HY
T
RADE
V
OLATILITY
?
Volatility is the simplest measure of a stocks riskiness or uncertainty.Different types: realized volatility, implied volatility, local volatility...
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June 10, 2003
W
HY
T
RADE
V
OLATILITY
?
I
Realized Volatility
The realized daily volatility
d
of an equity index S
i over a period of N
days is the square root of the variance of the daily returns r
i
:
Stock returns are roughly random and
normal; variance grows ~ return time.
Similarly for currencies, commodities, interest rates, volatility itself,
ri
Si
Si
-------- d
2 1
N---- r
i( )
2 1
N---- r
i
i
2
i
=
S S t
annual
16 daily
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June 10, 2003
WHY TRADE
VOLATILITY?
I
Implied Volatility
Black-Scholes: the fair price of a stock option depends on its futurerealized volatility.
Black-Scholes is both a model and a quoting mechanism for optionsprices.
Implied volatility is the value of the volatility you have to insertinto the Black-Scholes formula to make it match the market price ofan option.
You can think of it as the markets expectation of future realizedvolatility, plus a spread.
CBS
C S K t T r , , , , ,( )=
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June 10, 2003
WHY TRADE
VOLATILITY?
I
Bonds and Options /Yields and Volatilities
Bonds Options
Interest rates are the parameterspeople use to quote bondprices.
Volatilities are the parameterspeople use to quote optionsprices
Realized daily interest rates:actual short-term interest rates
Realized daily volatility:the actual volatility of an index
Yield to maturity of a bond:
the average of the future real-ized rates that make that bondprice fair. Its the implied yieldbased on price.
Implied volatility of an option:
the average of future realizedvolatilities that make theoptions price fair, based onBlack-Scholes.
Forward rates:
the future realized rates,moment by moment, that mustcome to pass to make currentyields of all liquid bonds fair.
Local (forward) volatilities:
the future realized index vola-tilities, index level by indexlevel and moment by moment,that must come to pass to makecurrent implied volatilities fair.
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WHY TRADE
VOLATILITY?
I
Why Trade Volatility?
Attractive characteristics:
It grows when uncertainty increases.
It reverts to the mean.
It goes up and tends to stay up when most assets go down.
Types of volatility trading:
Speculative trading of volatility levels in various markets.
Index vs. stock, foreign vs. domestic, short-term vs. long-term, currencies, rates,...
You need views about future uncertainty to trade it. Trading the spread between realized and implied volatility levels.
Hedging implicit volatility exposure.
Hedge funds and risk arbitrageurs are often implicitly shortvolatility. They often take short positions in the spreadbetween stocks of companies planning to merge, assumingthat the spread will narrow if the merger takes place. Ifoverall market volatility increases, these mergers are lesslikely to occur, and the spread may widen.
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VOLATILITY
TRADING WITHOPTIONS
II
II
VOLATILITY TRADINGWITH OPTIONS
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VOLATILITY
TRADING WITHOPTIONS
II
A Linear Security: A stock or index
ASSUMED STOCK EVOLUTION: P&L OFA STOCK POSITION:
If you own a stock or index, you make money if it goes up, lose if itgoes down.
You have a linear position in S over the next instant t.
As an investor, how do you profit no matter whether the index goesup or down?
P&L = S
S0 SS
t
S
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VOLATILITY
TRADING WITHOPTIONS
II
A Curved Security
To make money irrespective of direction, you want a security which
gives you a curved P&L: a quadratic position in (S)2:
To get curvature: delta-hedge away the linear part of a call option.
P&L = (1/2)(S)2
S
curvature
C
S
- =
long call short stock =curvedhedge
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V
OLATILITY
T
RADING
W
ITH
O
PTIONS
II
P&L of Delta-Hedged Options
indexshift S
gain ~ 1/22
S
2
tThe gain due to a move S with
loss ~ (1/2)2S2t
The loss in an instant tif expected vol is
indexshift S
gain - loss ~(1/2)[2 -2]S2 t
A hedged position makesmoney if realized vol exceedsimplied vol, with magnitudedepending on options curvature
realized vol
and stock price S2
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VOLATILITY
TRADING
WITH
OPTIONS
II
P&L Depends on Realized Vol vs. Implied Vol
Long options: you make money if
Short options: you make money if
Net P&L1
2--- S
2
2
2
t=
>