Fundamentals of futures and options markets 9th by john c hull 2016 chapter 19

Volatility Smiles
Chapter 19

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Volatility Smile
A volatility smile shows, for options with a
certain maturity, the variation of the implied
volatility with the strike price
 The volatility smile is the same whether
calculated from European call options or
European put options. (This follows from putcall parity.)
 It is also approximately the same when
calculated from American options


Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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The Volatility Smile for Foreign
Currency Options
(Figure 19.1, page 414)

Implied
Volatility

Strike
Price
Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Implied Distribution for Foreign
Currency Options

Lognormal
Implied

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Properties of Implied Distribution
for Foreign Currency Options
 Both

tails are heavier than the lognormal
distribution
 It is also “more peaked” than the normal
distribution

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Possible Causes of Volatility Smile
for Foreign Currencies
 Exchange

rate exhibits jumps rather
than continuous changes
 Volatility of exchange rate is stochastic

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Historical Analysis of Daily
Exchange Rate Changes (10 exchange
rates, 2005-2015; Table 19.1, page 415)
Real World (%)

>1 SD
>2SD
>3SD
>4SD
>5SD
>6SD

23.32
4.67
1.30

0.49
0.24
0.13

Normal Model
(%)

31.73
4.55
0.27
0.01
0.00
0.00

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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The Volatility Smile for Equity
Options (Figure 19.3, page 417)
Implied
Volatility

Strike
Price
Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Implied Distribution for Equity
Options

Lognormal
Implied

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Properties of Implied Distribution
for Equity Options
 The

left tail is heavier than the lognormal
distribution
 The right tail is less heavy than the
lognormal distribution

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Reasons for Smile in Equity
Options
 Leverage
 Crashophobia

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Other Volatility Smiles?
What is the volatility smile if
 True distribution has a less heavy left tail
and heavier right tail
 True distribution has both a less heavy left
tail and a less heavy right tail

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Ways of Characterizing the
Volatility Smiles


Plot implied volatility against K/S0



Plot implied volatility against K/F0




Tarders may define an option as at-the-money when K
equals the forward price, F0, not when it equals the spot price
S0

Plot implied volatility against delta of the option


Traders may define at-the money as a call with a delta of 0.5
or a put with a delta of −0.5. These are referred to as “50delta options”

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Volatility Term Structure
In addition to calculating a volatility smile,
traders also calculate a volatility term structure
 This shows the variation of implied volatility with
the time to maturity of the option
 The volatility term structure tends to be
downward sloping when volatility is high and
upward sloping when it is low


Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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Example of a Volatility Surface
(Table 19.2, page 419)

Strike Price
0.90

0.95

1.00

1.05

1.10

1 mnth 14.2

13.0

12.0

13.1

14.5

3 mnth 14.0

13.0

12.0

13.1

14.2

6 mnth 14.1

13.3

12.5

13.4

14.3

1 year

14.7

14.0

13.5

14.0

14.8

2 year

15.0

14.4

14.0

14.5

15.1

5 year

14.8

14.6

14.4

14.7

15.0

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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The Impact of a Large Jump (pages
420 to 421)
 At

the money implied volatilities are higher
that in-the-money or out-of-the-money
options (so that the smile is a frown!)

Fundamentals of Futures and Options Markets, 9th Ed, Ch 19, Copyright © John C. Hull 2016

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