Estimation Risk-Neutral Variance of Returns
I am trying to find a method which allows me to estimate [Math Processing Error] $ Var_{\mathbb{Q}}\left(\frac{S_{t_{i+1}}}{S_{t_i}}\right) $ where [Math Processing Error] $ S $ denotes the price process of an underlying stock (which has to be assumed to be stationary) and [Math Processing Error] $ \mathbb{Q} $ should be the risk-neutral (pricing) measure. A first approach would rely on the Breeden-Litzenberger (1978) result which enables to compute marginal distributions of [Math Processing Error] $ \mathbb{Q} $ by observing prices of plain-vanilla options and by using:
[Math Processing Error]$$ Var_{\mathbb{Q}}\left(\frac{S_{t_{i+1}}}{S_{t_i}}\right)= \mathbb{E}\mathbb{Q}\left[\frac{1}{S{t_i}^2}\mathbb{E}\mathbb{Q}[S{t_{i+1}}^2|S_{t_i}]\right]-1. $$ 接下來，我得到[Math Processing Error] $ \mathbb{E}{\mathbb{Q}}[S{t_{i+1}}^2|S_{t_i}] $ 通過觀察當時的期權價格[Math Processing Error] $ t_i $ . 的價值[Math Processing Error] $ \mathbb{E}\mathbb{Q}\left[\frac{1}{S{t_i}^2}\right] $ 可以通過使用平穩性類似地計算。儘管如此，我對這種方法並不滿意。在談論估計時，我將面臨這些估計是在物理測量下進行的問題。有人知道在風險中性度量下估計收益變異數的方法嗎？ 對你的想法感到興奮，泰拉諾
只是嘗試（我不確定我是否很好地理解了這個問題）。
我將假設通常的風險中性動態[Math Processing Error] $ S_t $ :
$$ dS_t = rS_t dt + \sigma S_tdW_t $$ 以便 $ \forall T>t $ 我們有： $$ S_T = S_te^{\left(r-\frac{1}{2}\sigma^2\right)(T-t) + \sigma W_{T-t}} $$ 在這一點上，計算非常簡單直接。
$$ \mathbb{E}\left[ \frac{S_T}{S_t}\right] = e^{r(T-t)} $$ $$ \begin{align} \mathbb{E}\left[ \frac{S^2_T}{S^2_t}\right] &= \mathbb{E}\left[ e^{(2r - \sigma^2)(T-t) + 2\sigma W_{T-t}}\right] \ & = e^{(2r - \sigma^2)(T-t)} \mathbb{E}\left[ e^{2\sigma W_{T-t}}\right] \ & = e^{(2r - \sigma^2)(T-t) + 2\sigma^2(T-t)}\ & = e^{2r(T-t) + \sigma^2(T-t)} \end{align} $$ 這樣我們得到：
$$ Var\left( \frac{S_T}{S_t}\right) = e^{2r(T-t) + \sigma^2(T-t)} - e^{2r(T-t)} = e^{\sigma^2(T-t)} $$ 這是一般的結果。當然，如果我們把 $ T = t_{i+1} $ 和 $ t = t_i $ 我們得到了我們正在尋找的結果。
讓我在你的目標上潑一些水和任何好的證據。對於我關於此的文章，您可以在以下位置找到它：
哈里斯，DE（2017 年）收益分佈。數學金融雜誌, 7, 769-804
讓我們使用比您的假設更弱的假設[數學處理錯誤] $ S_t,\forall{t} $ 是靜止的。讓我們使用更多馬科維茨風格的假設。
Our first assumption is that there are very many buyers and very many sellers. Normally this is to motivate the absence of liquidity costs, but we are going to repurpose it as it has other consequences that no one noticed.
股票以雙重拍賣方式出售。正因為如此，沒有贏家的詛咒。因此，理性的行為是出價於你的期望。隨著許多買家和許多賣家競標他們的預期，limit book 將收斂到正常，因此隨著投標數量變得足夠大，limit book 將呈正態分佈。
我們也可以假設股票價格是從正態分佈中得出的。該假設的弱點在於它不包括諸如佳士得拍賣會受制於贏家詛咒的事物。有關該問題的解決方案，請參見論文。
所以讓 $ R_t=\frac{S_{t+1}}{S_T} $ . 我們將其稱為投資回報。減一使其成為投資回報。我們將忽略 $ -1 $ 因為它沒有任何改變，只是一點點額外的工作。
現在的問題是什麼是分佈[數學處理錯誤] $ R_t $ 作為 $ S_t,S_{t+1} $ 是實際數據，而[數學處理錯誤] $ R_t $ 不是數據，而是統計數據；也就是說，它是數據的函式。
眾所周知，來自柯蒂斯：
Curtiss, JH (1941) 關於兩個機會變數的商的分佈。數理統計年鑑, 12, 409-421,
任意比例的連續隨機變數的解，其中 $ Z=\frac{Y}{X} $ 是
[數學處理錯誤]$$ p(z)=\int_{-\infty}^\infty|x|f(x,zx)\mathrm{d}x $$ 對於處於平衡狀態的正態分佈變數，解是眾所周知的，並且可以以各種形式追溯到費馬和卡爾達諾：$$ \frac{1}{\pi}\frac{\sigma}{\sigma^2+(z-\mu)^2}. $$ 假設允許無限負回報。如果您限制域，則積分常數將從
$$ \pi^{-1} $$到$$ \left[\frac{\pi}{2}+\tan\left(\frac{\mu}{\sigma}\right)\right]^{-1}. $$ 就我們的目的而言，積分常數無關緊要，儘管如果您將其丟到現實世界中，它會在估計中產生嚴重錯誤。
The above distribution is famous for a variety of reasons. When Laplace first sent his proof of what we now call the “central limit theorem” to his former student Poisson, Poisson returned the proof to him with an exception to when the rule holds. It fails to hold when the distribution is as above. From that observation, when that distribution is present, then you can no longer use things such as t-tests, F-tests and so forth, subject to the qualification that as the sample size exceeds 100, the t-test will work if you hold it to one degree of freedom.
You can find a discussion of this at:
Fama, E. F. and Roll, R. (1968). Some properties of symmetric stable distributions. Journal of the American Statistical Association, 63(323): pp. 817–836.
However, the Fama and Roll discussion does not apply to the case of limiting liability to $ -100% $ . 我正在為另一篇論文中的那些人建立一個單獨的討論。
這種分佈的下一次出現是奧古斯丁·柯西和艾琳妮-朱爾斯·比奈梅之間的戰鬥。Augustin Cauchy 剛剛在一篇期刊文章中提出了一種回歸方法。Bienaymé 撰寫了一篇文章，表明普通最小二乘法是進行回歸的“最佳”方法。Cauchy 將此視為人身攻擊，然後著手確定 OLS 何時總是以機率 1 失敗。
只要存在上述分佈，OLS 就會產生純粹的虛假結果。原因是上述被稱為“柯西分佈”的分佈沒有均值，因此不可能有變異數。
而柯西主值是[數學處理錯誤] $ \mu $ , higher moments do not exist, even about the Cauchy principal value. The second raw moment is infinity or does not exist depending on how you define the integral.
As to estimating the scale parameter of returns, you cannot use a non-Bayesian method. There does not exist an unbiased admissible Frequentist estimator for real data. I have estimated the scale parameter for all disaggregated equity securities in another paper, but for one security, what you should do is solve:
[數學處理錯誤]$$ \Pr(\sigma|\mathbf{R},\mu)=\int_{-\infty}^\infty\frac{\prod_{i=1}^n{\left[\frac{\pi}{2}+\tan\left(\frac{\mu}{\sigma}\right)\right]^{-1}}\frac{\sigma}{\sigma^2+(R_i-\mu)^2}\Pr(\mu;\sigma)}{\int_0^\infty\int_{-\infty}^\infty{\prod_{i=1}^n{\left[\frac{\pi}{2}+\tan\left(\frac{\mu}{\sigma}\right)\right]^{-1}}\frac{\sigma}{\sigma^2+(R_i-\mu)^2}\Pr(\mu;\sigma)}\mathrm{d}\sigma\mathrm{d}\mu}\mathrm{d}\mu. $$ 後驗密度[數學處理錯誤] [σ數學處理錯誤] [St+1數學處理錯誤] $ \sigma $ 表現良好。如果您需要點估計，您可以最小化密度上的成本函式。您甚至應該能夠使用二次損失，因為對於足夠平坦的先驗，後驗密度應該收斂到兩個標準偏差分佈的比率分佈。然而，我還沒有花時間證明這一點。它可能不是真的，但它應該是 $ \sigma $ 是標準差的比率 $ S_{t+1} $ 和標準差 $ S_t $ .
你會想要限制你的先驗機率， $ \Pr(\mu,\sigma) $ 到適當的先驗，因為我沒有在文獻中找到針對截斷情況的廣義貝氏規則，並且沒有理由相信後驗在聯合分佈下表現良好 $ (\mu,\sigma) $ 與製服或其他不當的事前。
由此，足以證明均值變異數金融不存在。因此，任何 $ \beta $ 原始數據中的樣式模型無效。在對數轉換的數據中，概似函式是雙曲正割分佈並且它不允許任何類似於共變異數矩陣的東西，因此這是值得懷疑的。既然沒有什麼可以改變，那麼你在測量什麼？
This is not to say they cannot co-move. When looking at multiple firms returns of the firms cannot be independent, though asymptotically none of them can covary. This is part of what makes this distribution famous. The variables are not independent, but they do not covary as the sample size goes to infinity.
Finally, risk-neutral behavior cannot exist at the margin. I know I am making your day.
There are two arguments for this. The first isn’t a true argument, but should warrant a pause. If you assume risk-aversion, then from deFinetti coherence principle and assumption of a willingness to accept all finite bets at stated prices, then Kolmogorov’s axioms fall out as theorems. If you do not assume risk-aversion, then this does not happen. You then have to add the assumptions that:
$$ \Pr(A)\ge{0}, $$ $$ \Pr(\Omega)=1, $$對於任意不相交集的可數序列$$ \Pr(\cup_{i=1}^\infty{A_i})=\sum_{i=1}^\infty\Pr(A_i). $$ 當大自然提供了一種解決方案，既能最大限度地減少假設，又能經常與現實相匹配時，人們應該暫停一下。 第二個論點來自理性。如果邊際參與者喜歡冒險，那麼他們會支付溢價來承擔風險。這和說的一樣 $ K_{t+1}=RK_t+\epsilon_{t+1},R<1,\forall{t} $ . Given sufficient time the capital stock of the planet would go to zero and all humans would die.
This does not mean that risk-loving actors do not exist, nor does it mean that they are never the marginal actor. It implies that they can be the marginal actor only a minority of the time.
The assumption of risk-neutrality was only ever a mathematical convenience created by using the normal distribution. The probability of risk-neutrality must be zero from this second argument.
It goes like this, risk-neutrality exists at exactly one point. A single point over a continuum of possible points has measure zero and hence a probability of zero. Even if it were true, it could never be measured and risk-loving behavior is impossible. Hence, by being required to use Bayesian statistics, risk-neutral behavior is functionally excluded as a possibility.
For an extended discussion of the Cauchy distribution see: Why the Cauchy Distribution Has No Mean