Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.1
Volatility Smiles
Chapter 15
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.2Put-Call Parity Arguments
? Put-call parity p +S0e-qT = c +X e–r T
holds regardless of the assumptions
made about the stock price distribution
? It follows that
pmkt-pbs=cmkt-cbs
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.3
Implied Volatilities
? The implied volatility calculated from a
European call option should be the
same as that calculated from a
European put option when both have
the same strike price and maturity
? The same is approximately true of
American options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.4
Volatility Smile
? A volatility smile shows the variation of
the implied volatility with the strike price
? The volatility smile should be the same
whether calculated from call options or
put options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.5The Volatility Smile for Foreign
Currency Options
(Figure 15.1,page 332)
Implied
Volatility
Strike
Price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.6
Implied Distribution for Foreign
Currency Options
? The implied distribution is as shown in
Figure 15.2,page 332
? Both tails are heavier than the
lognormal distribution
? It is also,more peaked than the
lognormal distribution
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.7The Volatility Smile for Equity
Options (Figure 15.3,page 334)
Implied
Volatility
Strike
Price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.8
Implied Distribution for Equity
Options
? The implied distribution is as shown in
Figure 15.4,page 335
? The right tail is less heavy and the left
tail is heavier than the lognormal
distribution
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.9
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.10
Possible Causes of Volatility
Smile
? Asset price exhibiting jumps rather than
continuous change
? Volatility for asset price being stochastic
(One reason for a stochastic volatility in
the case of equities is the relationship
between volatility and leverage)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.11
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.12
Volatility Term Structure
The volatility term structure tends to be
downward sloping when volatility is high
and upward sloping when it is low
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
15.13
Example of a Volatility Surface
(Table 15.2,page 336)
S tri k e P ri c e
0, 9 0 0, 9 5 1, 0 0 1, 0 5 1, 1 0
1 m n t h 1 4, 2 1 3, 0 1 2, 0 1 3, 1 1 4, 5
3 m n t h 1 4, 0 1 3, 0 1 2, 0 1 3, 1 1 4, 2
6 m n t h 1 4, 1 1 3, 3 1 2, 5 1 3, 4 1 4, 3
1 y e a r 1 4, 7 1 4, 0 1 3, 5 1 4, 0 1 4, 8
2 y e a r 1 5, 0 1 4, 4 1 4, 0 1 4, 5 1 5, 1
5 y e a r 1 4, 8 1 4, 6 1 4, 4 1 4, 7 1 5, 0