Page 30 - CCFA Journal - Fourth Issue

Quant Corner 数量分析
他们开发了一种期权定价模型，该模型遵循与 Bachelier 模型很
相似的路径。模型之间的差异主要在于对过程动态的选择。
Bachelier模型遵循算术布朗运动，而 Black-Scholes模型遵循几何
布朗运动。
像 Bachelier一样，Black-Scholes模型基于相对较少的基本假设：
1. 欧洲期权只能在到期时行使。
2. 存在无风险资产，我们可以借入和投资现金（零息债券）。
3. 在期权的整个存续期内都没有股息/应计利息。
4. 买/卖期权和 Delta 套期保值没有交易成本。
5. 标的资产遵循连续的布朗运动，收益呈正态分布（该模型无
法处理资产价格动态中的跳跃和不连续性）。
6. 对所有期权的行权价和到期日的波动率都是恒定的。
在以上的假设中，尽管第 4 个假设并没有太大的区别，但第 5
个和第 6 个假设却非常重要，因为实践证明它们在经验上是错
误的。
为什么这些假设如此重要？首先让我们了解这些假设的含义。
在金融市场中，假定资产价格遵循随机过程。在该过程中存在
一定的轨迹和漂移，但沿该轨迹的变化是随机的，此过程称为
布朗运动。当 Black-Scholes 在 1973 年发表论文时，他们假设价
格动力学遵循连续的几何布朗运动，这意味着证券不能改变符
号，即价格不能从正变为负，反之亦然；此外，他们假设增量
呈正态分布。他们对连续性的假设也造成了问题，因为他们认
为可以持续消除 delta 风险，而从技术上讲这是不可能的，即使
在某种程度上可能，但其一定会带来不可忽略的交易成本。
Black-Scholes 关于波动率所做的最后一个假设可能是最可疑的
假设，这为我们今天所知的“波动率建模”铺平了道路。
随机波动率时代
随着从业人员开始将 Black-Scholes 公式用作整个行业的标准，
人们对 Black-Scholes 波动率假设的批评逐渐增多，因为人们已
经观察到波动率远非恒定的，它随着时间的推移而变化，并且
波动率也遵循某种随机性。 1976 年，Latane 和 Rendelman 发表
了他们的论文“期权价格所隐含的股票价格比率的标准偏差”，
该文件首次建议隐含波动率应源自市场上的交易期权（I.S.D –
隐含标准偏差）。由于从业人员喜欢 Black-Scholes 定价模型的
简单性和封闭形式公式的解决方案，他们非常不愿意将其抛诸
脑后，而是开始对模型进行调整以适应他们的市场和假设。这
是“随机波动率时代”开始的时候。
首先，让我们了解一下随机波动率的含义。任何随机波动率模
型都假设波动率本身遵循随机过程，与底层资产的收益相平行
且具有一定程度的相关性。基本上来说，
随机波动率的全部或一部分过程受其底层
资产的动力学控制，假设动态过程的每个
衍生产品都将遵循相同的过程。让我们姑
且先保留这个假设。
直到 1987 年 10 月 19 日（黑色星期一），
期权交易者才完全理解，波动率水平持平
的假设在金融市场中并不成立，因为对于
一个投资组合应该有额外的成本来确保自
己免受股票的不利波动影响。如果我们看
一下 Black-Scholes 的“理论”波动率表
面，则该表面应类似于以下表面：
CCFA JOURNAL OF FINANCE
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加中金融
Black & Scholes model, like Bachelier’s, based on relatively few base
assumptions:
European options can only be exercised at expiration.
Risk Free asset — there is a risk free asset that we can borrow and invest
our cash in(zero coupon bond).
Absence of dividend/accrued interest throughout the lifetime of the
option.
There are no transaction costs when buying/selling options, and delta
hedging.
The underlying asset follows a continuous BM, and returns are normally
distributed (the model cannot handle jumps and discontinuity in asset
price dynamic).
Volatility is constant across all strikes and maturities.
While the 4th assumption doesn’t make a huge difference, the 5th &
6th assumptions are quite significant, as they have been proven to be
empirically wrong in practice.
Why would these assumptions be so significant?
Let’s first understand what these assumptions mean…
In financial markets assets prices assumed to follow stochastic (random)
process, where there is certain trajectory (drift) but the variations are
random along that trajectory. This process is known as Brownian
motion. When B&S wrote their 1973 paper they assumed that the
dynamic follows continuous Geometric Brownian motion, which
means that securities cannot change sign (i.e. price cannot change from
positive to negative, and vice versa), Moreover, they assumed that
increments distribute normally (i.e. Gaussian distribution). Their
assumption of continuity is also creating issues, as they assumed that
one can continuously eliminate the delta (underlying price change) risk,
while this is technically impossible (and even if possible to some degree,
it comes with non-negligible transaction costs).
The last assumption Black Scholes made regarding the volatility is
probably the most questionable assumption, which paved the way to
what we know today as “Volatility Modeling”.
The Era of Stochastic Volatility
As practitioners started to use Black-Scholes formula as industry-wide
standard, criticism with regards to B&S volatility assumption grew, as it
was well observed that volatility is far from being constant (it evolves
and changes through time, and follows some kind of random process
itself). In 1976 Latane and Rendelman published their paper “Standard
Deviations of Stock Price Ratios Implied in Option Prices”, which
suggested (for the first time) that implied volatility should be derived
from traded option in the market (I.S.D — implied Standard Deviation).
As practitioners liked B&S pricing model for its simplicity and robust
(closed-form) solution, they were pretty reluctant to throw it out of the
window, and started calibrating the model to fit their markets and
assumption. This is when the “Stochastic Volatility Era” begins…
Let’s first understand what stochastic volatility means. Any stochastic
volatility model assumes that the volatility itself
follows a random process (parallels and with
some degree of correlation to the returns of its
underlying asset), so basically the stochastic
volatility’s process is being (partly) controlled by
the dynamic of the underlying it describes
(hypothetically speaking each derivatives of the
previous dynamic will follow the same process,
but let’s leave it for now…)
It wasn’t until Oct 19th, 1987 (Black Monday)
that option traders fully comprehended that the
assumption of flat volatility surface doesn’t hold
water in financial markets, as there should be an
additional cost to insure ourselves against
adverse move in our stock portfolio. If we look
at the Black-Scholes “theoretical” volatility
surface it should look similar to this surface:
June 2021