最近刚开始看volatility相关的东西，但是感念有点儿混淆Dupire function不是用公式把implied volatility求出来吗？如果理论上，不是可以完美的重合implied volatility surface吗？他和这种有parameter的volatility model, 比如CEV 和 quadratic的model, 有什么联系吗？这些model算传统意义讲的local volatility model吗？
Local Volatility Model假设Sigma是一个关于t和St的函数，是一个特殊的Stochastic Volatility Model. LV有两种表现形式：其一是Parametric Form, 最流行的就是CEV Model.其二是Non-Parametric Form. 其本质是通过与Option Price（例如backward Kolmogorov equation和Dupire formula）或者Implied Volatility建立关系，然后利用数值方法求解，这里并没有closed form expression。虽然Non-Parametric LV在某一时点上可以perfectly match IV，但是无法reproduce the dynamics of the IV smile. 可以参考Dumas et al.(1998)~
我是这么理解的，如果大神看到觉得不对请指导。Dupire的local volatility model可以说和CEV，stochastic vol没关系。Dupire假设volatility是一个时间和资产价格的函数。它并没有直接假设函数的形式，而是证明在市场给定的option价格的时候，这个时间和资产价格的函数就必须符合某个形式，这样才能match到市场价格。CEV假设让vol是一个特定的函数形式（手机码字我就不写方程了）。还有stochastic vol啊和其他vol models，他们之间各自是独立的model。经 提醒，他们之间就是non-parametric v.s. parametric models 的问题。 Dupire 的就是non-parametric的local model。“local vol”这个表述会被用到这些vol models上，因为他们都符合volatility随着时间价格变化的特性。local（局部的）表达的就是这意思。如果你想叫black vol是global vol，应该也没问题。IMPORTANT! To be able to proceed, you need to solve the following simple math (so we know that you are a human) :-)
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Recent Posts[转载]根据历史股价估算options&implied&volatility
一般估算IV的方法是:
1)根据期权价格, 用B-S model或者逼近公式, 反向推导出
CBOE推出的^VIX指数就是用这个原理来逼近标准普尔500指数的iv.
2)根据历史股价, 计算historical volatility, 比如过去10天daily return的standard
deviation. 另外, 也有一个realized volatility,
它其实就是用一天之内的intraday的价格算出来的historical vol.已经有投行推出了基于realized
vol的options和swap了.
但是方法2计算出来的数值跟方法1算出来的iv差别很大, 比如过去1年半时间,
用SPY的历史股价算出来的iv比^VIX给出的值差一个常数(5%左右),不知道为什么, 不管是用直接的stdev(r[-1],
..., r[-10]), 还是用sd^2=(1-α)*sd[-1]^2+α*r[-1]^2 (JP
Morgan的EMA方法)(这两者的结果基本相等), 而且都试过乘以sqrt(250),sqrt(360)了,
所以也不会是时间的原因.&
我在网上找了下, 给出的答案是iv确实要大于historical vol, 因为variance risk
premium的原因. 这里就不赘述了, 有兴趣的读者自己查一下.&
因此我想到根据average true range的计算方法, 对方法2改进, 看看估算出来的数值是不是更逼近方法1的结果.
这种方法也有类似的研究, 比如Parkinson, M. 1980. The extreme value method for
estimating the variance of&the rate of return.
Journal of Business 53: 61-68. 以及其他人的一些类似的方法. 他们一般都是用平方平均,
所以结果要大于我的简单平均, 因为这个基本不等式:
sqrt((x^2+y^2)/2)&=(x+y)/2
我的historical volatility=过去10天average of Max [
abs(high-low)/previous close, abs(high/pre close-1) , abs(low/pre
close-1) ] *100*sqrt(250)
这里是取10天简单平均SMA,也可以取20或者30天EMA. 这种方法比Parkinson的方法,
多考虑到了股价跳空缺口的因素(gap).
该方法用在SPY,
^VIX, &01Jan2010 - 01Jun2011, 结果相当成功. 但是当波动很大的时候,
比如2009年初, 2010年初, 这种方法估计出来的数值就大于IV很多(图2中volatility
spread超过+10%的部分). 对其他时间段, 或者其他股票, 这种方法的可靠性还需继续检验.
<img src="/blog7style/images/common/sg_trans.gif" real_src ="/2llc12r.jpg" BORDER="0" STYLE="border-top-width: 0 border-right-width: 0 border-bottom-width: 0 border-left-width: 0 border-style: border-color: vertical-align:" NAME="image_operate_56343"
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TITLE="[转载]根据历史股价估算options&implied&volatility" />
这种方法巨大的作用就是仅仅根据历史股价就能计算出任何一个股票的historical
vol了,而且跟IV还很接近. 要知道, 一般股票的历史IV都是没有现成数据的, 除了^VIX.
下图是上证指数从Jan2009-Jun2011的走势图(蓝色),和IV(红色).我们可以看到几个特点:&
1)IV维持在很低的时候,
下面将要暴跌
2)IV暴涨到顶峰的时候,
也就是股票触底反转(不是反弹)的开始
<img src="/blog7style/images/common/sg_trans.gif" real_src ="/2iqeli8.jpg" STYLE="max-width:690" NAME="image_operate_98522"
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我们可以根据这种方法设计交易策略:&当historical
vol跟IV相差很大的时候, 我们可以用经典的delta neutral的方法来trade volatility
如果iv明显高于histo, 则我们short OTM
call, 买正股hedge. 只有通过买卖正股, 才能获得historical vol的exposure.
由于正股的historical volatility相对较小, 所以我们可以把正股上的止损放的很紧, 一旦正股跌破止损, 就卖正股,
留着naked call, 或者买个更deep OTM的call对冲. 风险在于股价下跌,但是因为我们卖call,
可以让正股跌破止损的时候我们才开始冒亏损的风险.&
以上网友发言只代表其个人观点，不代表新浪网的观点或立场。Implied Volatility
Implied Volatility ... and Historical Volatility
motivated by a discussion on the
Historical Volatility (HV) is calculated by looking at historical returns and calculating some kind of average deviation from their mean value using the
magic formula for Standard Deviation ... also called Volatility.
>But aren't their several magic formulas for Standard Deviation?
Well ... yes. See .
However, I'm not interested in the particular formula one uses for Standard Deviation, I'm interested in the relationship between
HV (calculated via one of them thar formulas),
and Implied Volatility
Remember the Black-Scholes formula for calculating the "fair" price of a call option?
That guy labelled &#963; is the number one inserts to represent the stock's volatility.
Alas, if one uses HV, it rarely gives the actual price at which the option is selling.
>How come?
Well, I guess that HV is a value generated from past returns and perhaps people who buy options are interested
in future returns, future volatility, future option prices, and the value they place on an option reflects their opinion of future values and ...
>So what volatility should one use, in that Black-Scholes formula?
Well, the value of &#963; which gives the actual option price is called ...
>Don't tell me! It's Implied Volatility, right?
Yes, it's IV.
>So how do they compare ... HV and IV?
That's what I asked myself ... hence this tutorial.
>So what if using HV in Black-Scholes gives a larger option premium?
Larger than it's currently selling for? That'd means that the IV, which gives the actual selling price,
is smaller ... so I guess investors think the option will go down. Anyway, let's play with that ...
Calculating Implied Volatility
Our problem is to calculate IV
(from the Black-Scholes formula) in terms of the other parameters,
namely C, S, K, r,
>Good luck!
I don't think I can do that, but let's give it a try. First some notation:
Define A = C/S, B = K/S, R = rT
and let y = &#963; SQRT(T) =
IV SQRT(T).
The Black-Scholes formula then looks like:
[1] & & & A = N(d1) - B e-R N(d2)
& & & d1 = [ -log(B) + R + (1/2)y2 ] / y
& & & d2 = d1 - y
Our problem is to find y in terms of A, B and R.
We rewrite [1] like so:
[2] & & & Error = N(d1) - B e-R N(d2) - A
If, for a given set of numbers A, B and R, a y-value makes Error = 0, then IV = y / SQRT(T).
>Yeah, so?
Suppose the relevant values are:
Stock Price: S = $10.00
Strike Price: K = $9.80
Time to expiry: T = 0.25 years
Risk-free rate: r = 4%
Actual call option premium: C = $1.00
Then, plotting the Error versus y gives Figure 1.
Then Error = 0 when y = 0.215 or 21.5%
>So, what's your point?
I thought it'd be interesting (now that we have a method for calculating both)
to compare IV and HV and,
in particular, to determine which of the various definitions of Historical Volatility is closest to Implied Volatility.
Implied Volatility versus Historical Volatility
Okay, we'll use a spreadsheet where do the following:
Type in some stock symbol and click a button to download a year's worth of daily stock prices.
Calculate the Standard Deviations (hence Historical Volatiliies) from the downloaded prices.
& & using Annualized Volatility = (Daily Volatility)*SQRT(250)
Look up a call option for that stock ... one that expires in a few months.
Identify C, S,
K and T ... and assume some Risk-free Rate r.
Calculate the parameters A, B and R as indicated above.
Find the Implied Volatility and compare to the Historical Volatility.
Then look at ...
>Why not show the spreadsheet?
Okay, here's an example for a MSFT call option:
Click on the picture to download the spreadsheet
>Not too close, I'd say.
Well, of course, we might calculate "Annualized Volatility" differently, or use three month's worth of daily prices
(instead of a year's worth) or ...
>So, do it.
... we might take another option, maybe one that has a strike price just above the stock price
(rather than just below) and ...
>So, do it.
Okay, here's a picture:
Note that the IV just went up. We could (and many do) take some average of several
IV-values for strike prices near the stock price.
I'd like to point out another interesting thing:
Consider an "in-the-money" option
(meaning K < S). Look again at the B-S equation, namely:
[3] & & & C = S N(d1) - K e-rT N(d2)
If we let y
0 (meaning volatility
(for S / K > 0) and
N(d1) and N(d2) both
>So C = S - K e-rT, right?
Yes, in the limit as the volatility becomes 0.
The interesting point here is that, if S - K e-rT > 0, then the Error > 0
and the plot of Error vs Volatility curve looks like this:
>No implied volatility?
Well, it's clear that the option premium C will be larger than S - K, but will it be larger than S - K e-rT?
If not, then there is NO Implied Volatility that'll give the current option price.
>Does that ever happen?
Yeah, but it depends upon what Risk-free Rate you use. If we change that rate from 5% to 3%, we'd get this guy:
>So ya gotta be careful about your assumed risk-free, right?
Yes, in this example you'd need r less than 4% to get an implied volatility.
>Is that an invention or a real stock?
Today is Aug 9, 2005. The call expires Jan 20, 2006. The stock is GE.Implied Volatility Smirk：隐含波动率微笑Implie
扫扫二维码，随身浏览文档
手机或平板扫扫即可继续访问
Implied Volatility Smirk：隐含波动率微笑
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