Volatility Smiles Chapter 18

未来波动率的预测 标的资产未来波动率的估计对于资产定价和风险管理都至关重要，过去人们主要从历史波动率来估计未来波动率，这种用历史来推断未来的做法存在很大的缺陷。而期权价格中则包含着市场对未来波动率的估计，因此我们就可以通过期权价格来提取未来波动率的信息。

隐含波动率 由于其他变量的值在市场上都观察得到，因此实务界就将期权价格代入Black-Scholes期权定价公式（Black and Scholes, 1973），反求出波动率。这样求出来的波动率被称为隐含波动率。

VIX指数（2003年起改为无模型隐含波动率）

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

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

Volatility Smile 由于Black-Scholes期权定价公式是建立在标的资产服从对数正态分布的假定基础上，而这种假定与现实明显不同。于是人们就利用同一期限不同协议价格的期权价格求出对应的隐含波动率，并把它们绘制成曲线，这就是波动率微笑。 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

The Volatility Smile for Foreign Currency Options (Figure 16.1) Implied Volatility Strike Price

Implied Distribution for Foreign Currency Options The implied distribution is as shown in Figure 18.2. Both tails are heavier than the lognormal distribution It is also “more peaked than the lognormal distribution

The Volatility Smile for Equity Options (Figure 16.3) 由于股票收益率存在明显的负偏斜，因此，股票的波动率微笑的斜率一般都为负。 Implied Volatility Strike Price

Yahoo股票期权 隐含波动率微笑

Implied Distribution for Equity Options The implied distribution is as shown in Figure 18.4. The right tail is less heavy and the left tail is heavier than the lognormal distribution

花期银行股票的波动率微笑

花期银行股票的波动率微笑 该股票波动率微笑却呈现两头翘起的典型“微笑”形状，这说明在金融危机最严重之时，市场预测花旗银行未来大涨和大跌的概率都较大，说明市场对花旗银行的未来命运存在较大分歧；此外，看跌期权价格的隐含波动率大于看涨期权的隐含波动率，这说明市场较为悲观。

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

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)

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

Volatility Term Structure The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low

花期银行股票的波动率期限结构

花期银行股票的波动率期限结构 从图上可以看出，期限越长，隐含波动率越低，这说明市场普遍预期，随着时间推移花旗银行股票的波动率会逐渐变小。另外，看跌期权的隐含波动率始终大于看涨期权的隐含波动率，再一次说明此时市场比较悲观。

波动率曲面 把波动率微笑与波动率期限结构结合起来，我们就可以得到波动率曲面（Surface），从而考察市场对资产未来分布的预期。

Example of a Volatility Surface (Table 18.2) 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

无模型（Model-Free）隐含波动率 但无论波动率微笑还是波动率期限结构，它们的计算都是直接利用BS期权定价公式将期权价格翻译成市场预期的波动率，由于BS公式很多假定与现实不符，这种翻译的准确度就令人怀疑，因而我们可以用无模型隐含波动率模型将期权价格准确翻译成波动率曲面。

隐含波动率与未来的波动率

风险溢酬与预期 如果假定波动率风险溢酬是常数，或者能够从其它途径获取波动率的风险溢酬，我们就可以得到现实世界中市场对未来波动率的预期。通过波动率预期与过去波动率对比的时间序列数据，我们可以研究波动率预期的形成机制和特点；通过波动率预期与未来已实现波动率对比的时间序列数据，我们可以研究波动率预期的准确性。而如果能够从其它途径（如调研等）获得现实世界中市场对未来波动率的预期，我们就可以得到波动率风险溢酬，并研究其时间序列特征，特别是在金融危机情形下的特征。