Page 46 - CCFA Journal - Sixth Issue

Quant Corner 数量分析 加中金融

用随机波动率启发模型建造波动率曲面

Construct Volatility Surface Using SVI Model

CCFA

Abstract An implied volatility is the volatility calibrated by the market option price based on the Black-Scholes pricing model. The
purpose of this article is to present the process of constructing volatility surface using the Stochastic Volatility Inspired (SVI)
model. We illustrate the performances by the numerical examples and the results showing that SVI model can capture volatility
smiles.

【提要】隐含波动率是由基于 Black-Scholes 定价模型从市场期权价格校准得到的波动率。本文介绍使用随机波动率启发
(SVI) 模型构建波动率曲面的过程。我们通过数值例子和结果说明其性能，表明 SVI 模型可以捕捉波动率微笑。
Keywords Implied Volatility, SVI, Calibration

1. Introduction 1. 引言

When we value financial contracts such as options, it is a 对期权等金融合约进行估值时，我们通常会使用 Black-
common practice to use the Black-Scholes framework. In Scholes 模型框架，即在 Black-Scholes 模型中，假设波动
Black-Scholes model the volatility is assumed to be constant 率关于期权执行价格是常数。而在现实世界中，期权的
against the strike. However, this stands in contradiction with
the real world, where market prices imply that the volatility 市场价格隐含的波动性一定取决于执行价格。
depends on the strike prices. 市场证据表明，恒定波动率模型在金融产品定价中是不

Market evidences suggest that a constant volatility model is not 合适的，因此产生了波动率微笑模型，我们又称之为波
adequate in financial product pricing. The volatility smile, 动率建模，是从业者和学者的重要研究课题。波动率曲
which we know as volatility modeling, is an important research 面的构建主要方法包括 1）局部波动率模型； 2）随机波
topic for practitioners and academics. The volatility surface 动率模型，如 SABR（Heston and Levy）； 3) SVI、SVI-
construction approaches include 1) Local volatility model; 2) JW、Omega、OmegaQ 等参数或半参数模型；和 4) 直接
Stochastic volatility models such as SABR (Heston and Levy); 模拟隐含波动率动态的市场波动率模型。在本文中，我
3) Parametric or semi-parametric models such as SVI, SVI-JW,
Omega and OmegaQ; and 4) Market volatility model which 们将重点介绍基于最佳市场实践的 SVI 模型和构建波动
directly models the implied volatility dynamics. In this article, 率曲面。
we focus on introducing SVI model and constructing volatility
surface based on the best market practices.

2. The Stochastic Volatility Inspired (SVI) model
2. 随机波动率启发模型（SVI）
The volatility smile arises from the fact that options with
different strikes and maturities have different implied 波动率微笑源于具有不同执行价和到期日的期权具有不
volatilities. The Stochastic Volatility Inspired (SVI) model, 同 的 隐 含 波 动 率 这 一 事 实 。 Gatheral (2004) 引 入的
which is introduced by Gatheral (2004), is a good choice for Stochastic Volatility Inspired (SVI) 模型是捕捉波动率微笑
capturing the volatility smile. It is motivated by the asymptotic 的不错选择。它的动机来自于这是在 Heston 模型 (1993)
extreme strike behaviour of an implied volatility smile, which
is generated in the Heston’s model (1993). Practitioners use 中隐含波动率微笑的渐近极端执行价格行为。从业者使
this to price exotic options consistently with the volatility skew. 用它来为与波动率偏差一致的奇异期权定价。
In this approach, the SVI model is defined at each maturity 在此框架中，对于每一个到期时T 建立一个 5 参数的 SVI
slice T in terms of the five parameters   {a ,b , ,  m , } 模型   {a ,b , ,  m , }使得隐含的总方差 (k  R )是
,
w
R
R
such that the total implied variance (k  ) is
w
,
R
2
w (k ; R )  a  b { (  m )  (  m )   2 }
k
k
2
w (k ; R )  a  b { (  m )  (  m )   2 } 其中 k  ln(S (t / ) K ) ， (t 是 t 时刻的资产价格， K 是
k
k
)
S
where k is log-moneyness with k  ln(S (t / ) K ) . (t is 执行价格。参数  处于以下的定义域中：
S
)
R
the spot price of the underlying at time t and K is the strike. 2



0
a
The parameters  lie in the following definition domain: b  0 ,  ,  [ 1 , 1 ],    1 
R



2
0
b  0 ,  ,  [ 1 , 1 ], and    1 
a

.

CCFA JOURNAL OF FINANCE February 2022
Page 46 第46页