Columbia University CL - Emanuel · PDF fileP age 5 of 37 Ju ne 10, 2003 W HY T RADE V OLATILITY? I Implied Volatility Black-Scholes: the fair price of a stock option depends on its

Page 1 of 37 June 10, 2003

T

RADING

V

OLATILITY

A

S

A

N

A

SSET

CL

ASS

Emanuel Derman

Columbia University

[email protected]

Page 2 of 37 June 10, 2003

Summary

Volatility is a useful trading hedge against all kinds of disasters. How can you trade it?

Calls and puts don’t quite do it. Though calls and puts are sensitive to volatility, 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

Page 3 of 37 June 10, 2003

W

HY

T

RADE

V

OLATILITY

?

I

I

W

HY

T

RADE

V

OLATILITY

?

Volatility is the simplest measure of a stock’s riskiness or uncertainty.Different types: realized volatility, implied volatility, local volatility...

Page 4 of 37 June 10, 2003

W

HY

T

RADE

V

OLATILITY

?

I

Realized Volatility

The realized daily volatility

σ

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

∆SiSi

--------≈ σd2 1

N---- ri( )2 1

N---- ri

i∑

2

–

i∑=

∆S σS ∆t≈

σannual 16 σ× daily∼

Page 5 of 37 June 10, 2003

WHY TRADE VOLATILITY?

I

Implied Volatility

Black-Scholes: the fair price of a stock option depends on its future realized volatility.

Black-Scholes is both a model and a quoting mechanism for options prices.

Implied volatility Σ is the value of the volatility σ you have to insert into the Black-Scholes formula to make it match the market price of an option.

You can think of it as the market’s expectation of future realized volatility, plus a spread.

CBS C S K t T r σ, , , , ,( )=

Page 6 of 37 June 10, 2003


I

Bonds and Options /Yields and Volatilities

Bonds Options

Interest rates are the parameters people use to quote bond prices.

Volatilities are the parameters people use to quote options prices

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 bond price fair. It’s the implied yield based on price.

Implied volatility of an option:

the average of future realized volatilities that make the options price fair, based on Black-Scholes.

Forward rates:

the future realized rates, moment by moment, that must come to pass to make current yields of all liquid bonds fair.

Local (forward) volatilities:

the future realized index vola-tilities, index level by index level and moment by moment, that must come to pass to make current implied volatilities fair.

Page 7 of 37 June 10, 2003


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 short volatility. They often take short positions in the spread between stocks of companies planning to merge, assuming that the spread will narrow if the merger takes place. If overall market volatility increases, these mergers are less likely to occur, and the spread may widen.

Page 8 of 37 June 10, 2003

VOLATILITY TRADING WITH OPTIONS

II

II


Page 9 of 37 June 10, 2003


II

A Linear Security: A stock or index

ASSUMED STOCK EVOLUTION: P&L OF A STOCK POSITION:

• If you own a stock or index, you make money if it goes up, lose if it goes 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 goes up or down?

P&L = ∆S

S0 ∆SS

∆t

∆S

Page 10 of 37 June 10, 2003


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)Γ(∆S)2

∆S

curvature Γ

C

S

- =

long call short stock = curvedhedge

Page 11 of 37 June 10, 2003

V

OLATILITY

T

RADING

W

ITH

O

PTIONS

II

P&L of Delta-Hedged Options

indexshift ∆S

gain ~ 1/2Γσ 2S2∆t

The gain due to a move ∆S with

loss ~ (1/2)Σ2S2∆t

The loss in an instant ∆t if expected vol is Σ

indexshift ∆S

gain - loss ~(1/2)Γ[σ 2 -Σ2]S2 ∆t

A hedged position makes money if realized vol exceeds implied vol, with magnitudedepending on option’s curvature Γ

realized vol σ

and stock price S2

Page 12 of 37 June 10, 2003


II

P&L Depends on Realized Vol vs. Implied Vol

Long options: you make money if

Short options: you make money if

Net P&L 12--- Γ∫ S

2 σ2 Σ2–

∆t=

σ Σ>

σ Σ<

Page 13 of 37 June 10, 2003

OPTIONS TRADING: WHAT CAN GO WRONG

III

III

OPTIONS TRADING:WHAT CAN GO WRONG

Page 14 of 37 June 10, 2003


III

1. It’s Not A Clean Bet on Volatility Alone

is irritating. The Γ (Gamma) (Curvature) of an option as stock price S varies sharply:

If the stock price moves away, you get very little bang for your option buck.

Net P&L 12--- Γ∫ S

2 σ2 Σ2–

∆t=

ΓS2

100 call

Page 15 of 37 June 10, 2003


III

2. Delta-Hedging In Practice Is Risky

Delta-hedging assumes smooth stock movements, continuous hedging, no transactions costs, a known future volatility.

Real markets violate Black-Scholes assumptions because of:• Jumps.

• Transaction costs.

• Stochastic volatility: the future realized volatility which determines your hedge ratio is not known.

• You cannot really hedge continuously; you must hedge discretely.

All of these imperfections may overwhelm any theoretical gain.

Let’s look at just one example -- discrete hedging.

Page 16 of 37 June 10, 2003


III

Example: Error When Hedging is Discrete

Perfect Black-Scholes world; hedge N times for 1-month ATM put with 20% realized vol. The Black-Scholes value is 2.512 dollars.

Monte Carlo simulation of the P&L:

Your P&L isn’t guaranteed unless you hedge continuously. But that introduces large transactions costs and shifts the expected return.

It’s not easy to make money this way.

-1.5 -1 -0.5 0 0.5 1 1.5Final profit/loss

4

8

12

16

20

24

28

Fre

quen

cy (

out o

f 100

%)

N=21 (roughly daily) Mean=0Standard deviation=0.41

N=84Mean=0Standard deviation=0.20

-1.5 -1 -0.5 0 0.5 1 1.5Final profit/loss

4

8

12

16

20

24

28

Fre

quen

cy (

out o

f 10

0%)

About 16% error, 4 vol pts About 8% error, 2 vol pts

Page 17 of 37 June 10, 2003

THE VOLATILITY SMILE VIOLATES BLACK-SCHOLES

IV

IV

THE VOLATILITY SMILEVIOLATES

BLACK-SCHOLES

The “smile” is the characteristic variation of implied volatility with strike and expiration. It is inconsistent with the Black-Scholes model everyone uses.

Page 18 of 37 June 10, 2003


IV

Implied Volatility Σ Violates Black-Scholes

• Implied volatility is the single value of the volatility you have to insert into the Black-Scholes model to match the market price of an option.

• It is the future volatility the index must have to make the Black-Scholes price fair.

• If the Black-Scholes model is correct, all options would have the same implied volatility.

• That isn’t the case.

Black-Scholes is wrong in principle, not just in practice, and therefore hedging is even more difficult.

Page 19 of 37 June 10, 2003


IV

Pre- and Post-Crash Implied Volatilities:

Since the ‘87 crash there has been a persistent skewed structure in Black-Scholes implied volatilities in most world equity option markets.

Representative implied volatility skews of S&P 500 options. (a) Pre-crash.(b) Post-crash. Data taken from M. Rubinstein, “Implied Binomial Trees” J. ofFinance, 69 (1994) pp 771-818.

0.95 0.975 1 1.025 1.05

Strike/Index

14

16

18

20

Vo

latil

ity

0.95 0.975 1 1.025 1.05

Strike/Index

14

16

18

20

Vo

lati

lity

(b)

Page 20 of 37 June 10, 2003


IV

A Persistent Negative Global Skew/Smile

A persistent large skew,almost linear, and inconsistent with Black-Scholes.

Global Three-Month Volatility Skews

Nikkei 225

S&P 500

Hang Sendg

FTSE 100

DAX

CAC 40

MIB 30

SMI

AEX25D Put Atm 25D Call

20

25

30

35

40

45

50

Mar 99

g

Σ K( ) Σatm b K S0–( )–=

Page 21 of 37 June 10, 2003


IV

The Implied Volatility Surface of Indexes

• Out-of-the-money puts always have higher implied volatilities than out-of-the-money calls.

• Short-term volatilities are usually more volatile.

• Implied volatility is usually greater than recent historical volatility.

The implied volatility surface for S&P 500 index options as a function ofstrike level and term to expiration on September 27, 1995.

Page 22 of 37 June 10, 2003


IV

More recent S&P 500 smiles

Single stock smile

SPX Imp Vol (12/26/01)

0%

10%

20%

30%40%

50%

60%

70%

0.5 0.75 1 1.25 1.5

Strike/Spot

Imp

Vo

l 1 Mon

3 Mon

6 Mon

12 Mon

slope ~ 1 vol pt. per 1% change in strike

VOD Imp Vol (12/27/01)

30%

40%

50%

60%

70%

0.5 0.75 1 1.25 1.5

Strike/Spot

Imp

Vo

l 1 Mon

3 Mon

6 Mon

12 Mon

Page 23 of 37 June 10, 2003


IV

Some currency smiles....

I

ATM Strike = 0.90 9.85 123.67

MXN/USDUSD/EUR JPY/USD

Page 24 of 37 June 10, 2003

T

HE

V

OLATILITY

S

MILE

V

IOLATES

B LACK -S CHOLES

IV

Negative Correlation Between Implied Vols and Index Levels

But be careful - ATM vol isn’t something you own. You own a specific strike vol.

Three-Month Implied Volatilities of SPX Options

INDEX

ATM

09-0

1-97

10

-01-

97

11-0

3-97

12

-01-

97

01-0

2-98

02

-02-

98

03-0

2-98

04

-01-

98

05-0

1-98

06

-01-

98

07-0

1-98

08

-03-

98

09-0

1-98

10

-01-

98

11-0

2-98

15

20

25

30

35

40

45

50

55

60

65

65070075080085090095010001050110011501200

Page 25 of 37 June 10, 2003

T

HE VOLATILITY SMILE VIOLATES BLACK-SCHOLES

IV

Patterns of Change in the Volatility Surface

Volatility surfaces fluctuate in modes:

:

Modes, or factors, are movements of the entire surface.

As with interest rates, there are parallel shifts, “steepening”, and “twists.”

Modes are useful when:• They can be understood intuitively.

• Few modes explain most of the variation.

• Modes are historically stable.

One finds about 85% of moves are parallel, 10% changes in slope with respect to time, and 5% related to out-of-the-money short term puts.

∆Σ β1 ( volatility level mode ) β2 ( term structure mode ) β3 ( skew mode )

other modes

+ +

+

=

Page 26 of 37 June 10, 2003

WHAT CAUSES THE SMILE?

V

V


Page 27 of 37 June 10, 2003


V

Causes of The Smile

Behavioral causes:• Supply and demand - purchases of collars

• Expectation of changes in volatility

• Fear of crashes (down for equities, up for goldAfter a crash, realized and implied volatilities will be highercorrelation of individual stocks in a jump down increases

• Level- and time-dependent effectsresistance levels in stockssupport levels in currencieschange of volatility behavior at low interest rates

Also:

• Fat tails in distributions

• Leverage effects

• Transactions costs?

• Uncertain future volatility

All of these effects make Black-Scholes wrong: the underlier doesn’t carry out simple Brownian motion.

Page 28 of 37 June 10, 2003


V

Models for the Smile

Classic Black-Scholes: constant volatility

Local volatility models: correlated volatility

Stochastic volatility models: random volatility

Jump-diffusion models:

small probability of a large jump

S

jump

diffuse

Page 29 of 37 June 10, 2003


V

Which model you use depends on what market you deal with.

• Currencies tend to have stochastic volatilities.

• Interest rates have volatilities that depend on interest levels.

• Stock markets tend to jump in the short run, diffuse over longer times.

•

There is no single correct replacement for Black-Scholes. It’s hard to test options models under the best of circumstances.

Page 30 of 37 June 10, 2003

IS THE SMILE FAIR?

VI

VI

IS THE SMILE FAIR?

Page 31 of 37 June 10, 2003

IS THE SMILE FAIR?

VI

Pre- and Post-Crash SPX Return Distributions

Index Return (%)

Prob

abilit

y (%

)

-20 0 20

01

23

4

Index Return (%)

Prob

abilit

y (%

)

-20 0 20

01

23

45

6

mean =1.8%std. div. = 7.3%

mean = 3.3%std. div. = 7.8%

Three-month, S&P 500 index, observed return distributions. (a) pre-1987 crash (1/70 to 1/87); (b) post-1987 crash (6/87 to 6/99)

Strike Level

Volat

ility (

%)

90 95 100 105 110

1012

1416

1820

2224

post-crashpre-crash

Fair estimated volatility skew.

Page 32 of 37 June 10, 2003

TRADING & PRICING VOLATILITY USING VOLATILITY SWAPS

VII

VII

TRADING & PRICING VOLATILITYUSING

VOLATILITY SWAPS

The cleanest and easiest way to trade volatility is through a volatility swap, which is a forward contract on realized volatility.

Swaps have become very popular because dealers have access to a theory for pricing them relative to the options market.

The theory provides a basis for defining volatility indexes like the VIX in terms of the price of a basket of options that can be traded, and for creating your own volatility swaps.

Page 33 of 37 June 10, 2003


VII

Analogy: Credit Default Swap Market

Why isn’t there a market for volatility alone?

Corporate Bonds Hedged Options

Bond prices are influenced by interest rates and credit spreads

Options prices are influenced by stock prices and volatility

Credit default swaps allow you to simply bet on credit spreads

Volatility swaps let you trade pure volatility.

Page 34 of 37 June 10, 2003


VII

Volatility Contracts : Volatility and Variance Swaps

A Volatility Swap is a forward contract on realized volatility:

Payoff: where is the notional amount.

: realized volatility realized by the index until expiration;

: previously agreed upon “delivery” volatility.

A variance swap is a forward contract on realized variance. It pays

Must specify the precise method for calculating realized volatility, the source and observation frequency of prices, and the annualization factor.

Variance swaps are easier to price and hedge using options.

A dealer will sell you a variance swap. You can trade volatility in onemarket or sector against another easily, without the hassle and errorsof hedging options. But you want to understand how to price it!

σR Kvol–( ) N× N

σRKvol

σR2

Kvar– N×

Page 35 of 37 June 10, 2003


VII

Dealers create a variance swap by hedging a Log Contract

Trading a single option:

An option whose would exactly earn the realized volatility

over the life of the contract, with a delivery price equal to .

• What kind of option has a

?

• An option whose payoff is the natural logarithm of S: – ln(S)

Net P&L 12--- Γ∫ S

2 σ2 Σ2–

∆t=

ΓS2

1=

Σ2

Γ

S

call1/S2

Γ 1 S2⁄∼

ln(S)

S

payoff call

Page 36 of 37 June 10, 2003


VII

Dealers Can Create a Log Contract Out of A Basket of Options with Many Differ ent Strikes

A portfolio of vanilla options with weights inversely proportional to their strike squared can replicate the payoff of a Log Contract. It will be equally sensitive to volatility at all spot level S.

20 60 100 140 180 20 60 100 140 180

20 60 100 140 180 20 60 100 140 180

20 60 100 140 180 20 60 100 140 180

20 60 100 140 180 20 60 100 140 180

strikes: 80,100,120equallyweighted

strikes 60 to 140

strikes 20 to 180spaced 1 apart

spaced 20 apart

strikes 60 to 140spaced 10 apart

(a)

(c)

(e)

(g)

(b)

(d)

(f)

(h)

weightedinverselyproportionalto squareof strike

A 1/K2 densityof puts and callswill give the same payoff as aLog Contract.

It will be equallysensitive to volatilityat all spot levels

Page 37 of 37 June 10, 2003


VII

Pricing of Variance Swaps

• Dealers replicate the log contract by buying and hedging a portfolio of puts and calls (and a forward contract).

• The fair price of the variance swap is given by the market price of the basket of puts and calls, a price which depends on the smile.

You can create your own variance swaps like this to do vol arbitrage.

Difficulties to be aware of• You need a continuum of puts and calls of all strikes to replicate it

exactly.

• In practice you cannot buy very out-of-the-money strikes. So, if the stock price moves too far, your replication will fail.

• If the stock jumps rather than moves smoothly, there are additional replication failures

Volatility swaps are more complex.

Volatility is the square root of variance, and is a more complex derivative of variance. Its value depends not just on volatility, but on the volatility of volatility, and you have to dynamically hedge it by trading variance swaps as the underlier. This is possible too, but needs a model for the volatility of volatility

Documents

Columbia University CL - Emanuel · PDF fileP age 5 of 37 Ju ne 10, 2003 W HY T RADE V OLATILITY? I Implied Volatility Black-Scholes: the fair price of a stock option depends on its