Page 31 - CCFA Journal - Fourth Issue

加中金融
现实中，惯常的波动率曲面看上去与此大不相同。这将是我们
典型的 S＆P500 波动率曲面：
所有现代随机波动率模型的根源都可以追溯到 Heston 在 1993年
发表的论文，该论文提供了在随机波动率动态条件下为债券期
权和外汇期权定价的一种全新的封闭形式的模型和方法。
每个资产类别都有自己的细微差别和标准。因此，不同的资产
类别采用不同的模型来反映资产的动态：
利率类
利率以收益率曲线的形式建模。评估债券和利率期权的适当模
型，例如利率掉期或掉期期权，应该包含收益率曲线期限结构
的动态变化。该类模型自引入以来，其标准模型一直是 Vasicek
（1977）和 Hull-White（1990），直到引入 SABR 模型（Hagan
等，2002）后开始变化。
SABR 模型是两个因子的随机模型，其中每个参数控制波动率
（alpha）动态的不同方面：
Beta：Beta 控制波动率分布（或 vol-of-vol）的方差和形状。
Rho：rho 是在底层资产当前动态和波动率动态之间关联的参数
（即控制波动率分布的斜率/偏斜）。
由于利率市场在某种意义上是独特的，即现在许多资产都以负
利率交易（或在正利率与负利率之间交替），因此 Black-
Scholes 不能用于这一类衍生品的定价（因为它假设底层价格为
对数正态分布） 。为了适应这种独特的特性，从业者倾向于使
用 Bachelier 模型对这些资产定价。今年早些时候，在 WTI 即月
合约跌至-40 美元之后，CME 将其定价模型更改为 Bachelier 模
型，以应对负资产的冲击。
外汇类
外汇期权市场主要是场外交易，它采用了可以处理第一代，第
二代奇异期权的模型，因为它们本质上是定制的。大多数外汇
从业者使用的模型称为 Vanna-Volga 定价（Malz，1997； Lipton
等，McGhee，2002； Wystup，2003）。该定价模型在外汇市场
中广泛用于对特殊期权，如界限期权和数字支付期权（单/非
触摸，欧洲数字式期权等）进行定价。
Vanna-Volga 定价模型的本质是：假设各类期权市场报价，比如
Vanna 的风险逆转/领口期权，Volga 的蝴蝶差价，反映了特定
资产的隐含风险中性概率分布的情况下，该模型能够量化相对
于 vega 的二阶导数的对冲成本（即 Vanna：dVega / dSpot 和
Volga：dVega / dVol）。
CCFA JOURNAL OF FINANCE
Quant Corner 数量分析
While in reality we know that our usual volatility surface looks very
different from that. This would be our typical S&P500 volatility surface:
We can trace the root of all modern stochastic volatility models to
Heston’s 1993 paper, which offered a new, closed-form, approach for
pricing bond options and foreign-exchange options under stochastic
volatility dynamic.
As not all asset classes were created equal, each asset class has its own
nuances and standards, and therefore, different asset classes adopted
different models to reflect assets’ dynamic:
Interest Rates
Interest rates are modeled in a form of yield curve. An appropriate
model to evaluate bonds , and options on interest (like options on
interest rate swap, or swaptions) should incorporate the dynamic of the
yield curve term structure. Since introduced, the standard models have
been Vasicek (1977) and Hull-White (1990), until the introduction of
SABR model (Hagan et al., 2002)
The SABR model is a two-factor stochastic model, where each
parameter controls a different aspect of the volatility (alpha) dynamic:
Beta- The beta controls the variance and shape of the volatility
distribution(or vol-of-vol)
Rho- The rho is the parameter that correlates between the underlying
spot dynamic and the volatility dynamic (i.e. controls the slope/skew of
volatility distribution).
As the Interest Rates market is unique in a sense that many assets are
trading nowadays with negative rates (or alternate between positive rates
to negative rates), Black-Scholes cannot be used for derivatives pricing
(as it assumes log-normal distribution of underlying price). To
accommodate that unique dynamic practitioners tend to use Bachelier
model for pricing these assets. Earlier this year, in the aftermath WTI
front-month contract’s collapse to -40$ CME changed its pricing model
to Bachelier model to accommodate negative strikes.
Foreign Exchange
Foreign Exchange option market is mainly traded OTC (over the
counter), and as such it adopted a model that can handle 1st, 2nd
generation exotic options (as these are more bespoke in their nature).
The model that most FX practitioners use is known as Vanna-Volga
pricing (Malz 1997, Lipton et. McGhee 2002, Wystup 2003). This
pricing model is widely used in FX market to price exotic options, like
barrier options, and digital payout options (one/no touch, European
Digital, etc.).
The essence of the Vanna-Volga pricing model, is the ability to quantify
the hedging cost of the 2nd order derivatives with respect to Vega (i.e.
Vanna- dVega/dSpot , and Volga- dVega/dVol), under the assumption
that the quoted options strategies (Risk Reversal/Collar for Vanna, and
Butterfly spread for Volga) reflect the implied risk neutral probability
for the particular asset.
June 2021
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