trading volatility as an asset class emmanuel derman

8/7/2019 Trading Volatility as an Asset Class Emmanuel Derman

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June 10, 2003

T

RADING

V

OLATILITY

A

S

A

N

A

SSET

CL

ASS

Emanuel Derman

Columbia University

[email protected]


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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


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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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June 10, 2003

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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June 10, 2003

VOLATILITY

TRADING WITHOPTIONS

II

II

VOLATILITY TRADINGWITH OPTIONS


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June 10, 2003

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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June 10, 2003

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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June 10, 2003

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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June 10, 2003

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=

>

trading volatility as an asset class emmanuel derman

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