Lognormal-mixture dynamics and calibration to market volatility smiles
Volatility Smiles Chapter 16
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Transcript of Volatility Smiles Chapter 16
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Volatility Smiles
Chapter 16
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Put-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
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Implied Volatilities
• When pbs=pmkt, it must be true that cbs=cmkt
• 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
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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
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The Volatility Smile for Foreign Currency Options
(Figure 16.1)
ImpliedVolatility
StrikePrice
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Implied Distribution for Foreign Currency Options
• The implied distribution is as shown in Figure 16.2.
• Both tails are heavier than the lognormal distribution
• It is also “more peaked than the lognormal distribution
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The Volatility Smile for Equity Options (Figure 16.3)
ImpliedVolatility
Strike
Price
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Yahoo股票期权隐含波动率微笑
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Implied Distribution for Equity Options
The implied distribution is as shown in Figure 16.4.The right tail is less heavy and the left tail is heavier than the lognormal distribution
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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
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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)
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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
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Volatility Term Structure
The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low
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Dell股票期权隐含波动率的期限结构
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Example of a Volatility Surface(Table 16.2)
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