Page 47 - CCFA Journal - Sixth Issue

加中金融 Quant Corner 数量分析

The above conditions ensure that (k  R )  0 for all k  . R 上述条件保证了对所有 ∈ ， (k  R )  0 。请注意映
w
;
w
;
Note further that the function k  w (k ; R ) is strictly 射是全实域严格凸的。以上关于 (k  R )的定义中，
w
,
convex on the whole real line. From the definition of
w (k , ),  增加 会产生所有方差加大，在垂直方向的上升
R
 增加 会产生看涨和看跌两翼的斜率上升，收紧波
 Increasing increases the general level of variance, a 动率的微笑
vertical translation of the smile;  增加 会降低（或升高）左（或右）翼，产生波动
 Increasing increases the slopes of both the put and 率微笑的逆时针旋转
call wings, tightening the smile;  增加 会造成波动率向右移动
 Increasing decreases (increases) the slope of the left
(right) wing, a counter-clockwise rotation of the  增加 会降低波动率微笑上平价期权的曲率
smile;
 Increasing translates the smile to the right; SVI 满足以下无投机条件：1）静态无投机，即不可能今
 Increasing reduces the at-the-money (ATM) 天没有投资而明天预期回报为非零。2）日历无投机，即
curvature of the smile. 日历利差的成本为正。3）垂直无投机，即垂直方向的利
差成本为正。4）水平无投机，即蝴蝶利差也为正。
SVI model stratifies the following arbitrage free conditions. 1)
Static arbitrage free condition: static arbitrage free condition 3. 参数标定
makes it impossible to invest nothing today and receive
positive return tomorrow; 2) Calendar arbitrage free condition: 在此我们介绍 SVI 模型参数的标定，并提供数值范例。
the cost of a calendar spread should be positive; 3) Vertical 在 SVI参数标定时，必须匹配到已知的隐含波动率。当然
(spread) arbitrage free condition: The cost of a vertical spread
should be positive; 4) Horizontal (butterfly) arbitrage free 有很多目标函数。我们选择 Gatheral 和 Jacquier (2014)的
condition: The cost of a butterfly spread should be positive. 方 法 ， 即 误 差 方 差 最 小 化 ， 例 如 使 用 Levenberg-
Marquardt 优化法来取得下述目标函数的最优解
3. Parameter Calibration
n
i
2
We introduce a calibration method for the parameters in the err  min  W i ( i SVI   MarketImpl iedvol )
SVI model and then presents some numerical examples to i 1 model i
demonstrate their accuracy by comparing them with market i i
data. 其中 SVI 是 SVI 参数模型得到的隐含波动率， BS 是观察
到的 Black-Scholes 模型下的隐含波动率。
To calibrate SVI parameters to observed implied volatilities,
there are many ways of defining an objective function. We 最后，我们可以用线性插值法或其它方法获得所有的到
adopt the Gatheral and Jacquier (2014) approach to take a 期时的 SVI 参数。
square-root as an objective function. At each volatility slice the
model fits the implied volatilities using a Levenberg-
Marquardt optimization routine with the following objective
function:

n
i
i
err  min  W i ( SVI   MarketImpl iedvol ) 4. 数值计算范例
2
i 1 model i
where  SVI is the implied volatility using SVI parametrization, 以 2015-08-30 的 SPX 期权报价为例。SPX 的即时价格是
i
1550。表一是平价差的对数值和对应的隐含波动率。
based on the i-th observation;  BS is the implied volatility
i
under Black-Scholes pricing formula, based on the i-th
observation. After we obtain the SVI parameters, we can build 表二是彭博社 SPX 的隐含波动率。分别表示在三个时间
the implied volatility surface. In order to price the option in 点 T=0.136986, T=3.002739 和 T=6.002739 的不同的期权
different time slices, we need to interpolate the SVI parameters. 执行价所对应的隐含波动率。
Usually the linear interpolation is used.
对上述目标函数采用 Levenberg-Marquardt 优化算法可以
4. Numerical Examples
得到下列各图标。
We take SPX option quotes as an example as of 8/30/2015. The
spot price for SPX is 1550, and the log-moneyness are listed in
Table 1.
The market implied volatilities for SPX from Bloomberg are
listed in Table 2. The data for implied volatilities at the time
slice T = 0.136986 are in the first and second rows; and the
third and fourth rows are the data for implied volatilities at time
slice T =3.002739; and fifth and sixth are the data for implied
volatilities at time slice T =6.002739.

Using the objective function above and Levenberg-Marquardt
optimization routine, we obtained the following parameters in
SVI model.

CCFA JOURNAL OF FINANCE February 2022
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