Volatility%20Smiles

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Volatility Smiles
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What is a Volatility Smile?

• It is the relationship between implied volatility
  and strike price for options with a certain
  maturity

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Why the volatility smile is the same
for calls and puts

• When Put-call parity p S0e-qT c K er T
  holds for the Black-Scholes model, we must have
• pBS S0e-qT cBS K er T
• it also holds for the market prices
• pmkt S0e-qT cmkt K er T
• subtracting these two equations, we get
• pBS - pmkt cBS - cmkt
• It shows that the implied volatility of a
  European call option is always the same as the
  implied volatility of European put option when
  both have the same strike price and maturity date

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Example

• The value of the Australian dollar 0.6(S0)
• Risk-free interest rate in US(per annum)5
• Risk-free interest rate in Australia(per
  annum)10
• The market price of European call option on
  the Australia dollar with a maturity of 1 year
  and a strike price of 0.59 is 0.0236.Implied
  volatility of the call is 14.5
• The European put option with a strike price
  of 0.59 and
• maturity of 1 year therefore satisfies
• p 0.60e-0.10x1 0.0236 0.59e-0.05x 1
• so that p0.0419 , volatility is also
  14.5

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Foreign currency options
Figure 1
Volatility smile for foreign currency options
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Implied and lognormal distribution for foreign
currency options
Implied
Figure 2
s(???)
Lognormal
K1
K2
S(??)
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Empirical Results
Real word
Lognormal model
Table 1
gt1 SD gt2 SD gt3 SD gt4 SD gt5 SD gt6 SD
25.04 5.27 1.34 0.29 0.08 0.03
31.73 4.55 0.27 0.01 0.00 0.00
Percentage of days when daily exchange rate moves
are greater than one, two, ,six standard
deviations (SDStandard deviation of daily change)
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Reasons for the smile in foreign currency options

• Why are exchange rates not lognormally
  distributed ? Two of the conditions for an asset
  price to have a lognormal distribution are
• The volatility of the asset is constant
• The price of the asset changes smoothly with no
  jump

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Equity options
Implied
Figure 3
volatility
Strike
Volatility smile for equities
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Implied and lognormal distribution for equity
options
Implied
Figure 4
s(???)
Lognormal
s(??)
K1
K2
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The reason for the smile in equity options

• One possible explanation for the smile in equity
  options concerns leverage
• Another explanation is crashophobia

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Alternative ways of characterizing the
volatility smile

• Plot implied volatility against K/S0(The
  volatility smile is then more stable)
• Plot implied volatility against K/F0(Traders
  usually 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 (This approach allows the volatility smile
  to be applied to some non-standard options)

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The volatility term structure

• In addition to a volatility smile, traders use a
  volatility term structure when pricing options
• It means that the volatility used to price an
  at-the-money option depends on the maturity of
  the option

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The volatility surfaces

• Volatility surfaces combine volatility smiles
  with the volatility term structure to tabulate
  the volatilities appropriate for pricing an
  option with any strike price and any maturity

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Table 2 Volatility surface
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The volatility surfaces

• The shape of the volatility smile depends on the
  option maturity .As illustrated in Table 2, the
  smile tends to become less pronounced as the
  option maturity increases

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Greek letters

• The volatility smile complicate the calculation
  of Greek letters
• Assume that the relationship between the implied
  volatility and K/S for an option with a certain
  time to maturity remains the same

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Greek letters

• Delta of a call option is given by

Where cBS is the Black-Scholes price of the
option expressed as a function of the asset price
S and the implied volatility simp
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Greek letters

• Consider the impact of this formula on the delta
  of an equity call option . Volatility is a
  decreasing function of K/S . This means that the
  implied volatility increases as the asset price
  increases , so that
• gt0
• As a result , delta is higher than that given
  by the Black-scholes assumptions

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When a single large jump is
anticipated

• Suppose that a stock price is currently 50 and
  an important news announcement due in a few days
  is expected either to increase the stock price by
  8 or to reduce it by 8 . The probability
  distribution the stock price in 1 month might
  consist of a mixture of two lognormal
  distributions, the first corresponding to
  favorable news, the second to unfavorable news .
  The situation is illustrated in Figure 5.

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When a single large jump is anticipated

• Figure5

Stock price
Effect of a single large jump. The solid line is
the true distribution the dashed line is the
lognormal distribution
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When a single large jump is anticipated

• Suppose further that the risk-free rate is 12
  per annum. The situation is illustrated in Figure
  6. Options can be valued using the binomial model
  from Chapter 11. In this case u1.16, d0.84,
  a1.0101, and p0.5314
• The results from valuing a range of different
  options are shown in Table 3

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When a single large jump is anticipated
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Figure 6
?
50
?
?
42
Change in stock price in 1 month
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Table 3 Implied volatilities in situation where
true distribution is binomial
Strike price ()
Put price ()
Implied volatility ()
Call price ()
42 44 46 48 50
52 54 56 58
8.42 7.37 6.31
5.26 4.21 3.16 2.10 1.05
0.00
0.00 0.93 1.86
2.78 3.71 4.64 5.57 6.50
7.42
0.0 58.8 66.6
69.5 69.2 66.1 60.0 49.0
0.0
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Figure 7 Volatility smile for situation in Table
3
Implied volatility
90
80
70
60
50
40
30
20
Strike price
10
0
44
46
48
50
52
54
56
It is actually a frown with volatilities
declining as we move out of or into the money