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Black-Scholes:為何關註波動性?

我們知道Black-Scholes是期權定價的不完美模型。為什麽這麽多的缺陷分析都集中在隱含波動率上?同一時間IV對同一種股票的變化這一事實證明BS存在缺陷(並且不能通過使波動率與時間相關來修復)。它還證明$ \ sigma $實際上不是股票價格的標準差,而只是用於調整結果的一些軟糖因素。

為什麽所有關註這個軟糖因素?一旦你知道模型是有缺陷的,我認為你應該回到繪圖板,看看如何修改BS pde的推導以獲得更復雜的模型。但相反,文獻中的大部分註意力都集中在如何控制軟糖因子上( Derman在The Smile上有12個講座,好像BS方程式從天而降,我們作為凡人的工作只是為了提供midrashim。

My own opinion is that people are too bemused by the famous cancellation of $\mu$ in the BS pde. After all, that's what impressed the Nobel Prize committee, so it must be right. But both Black & Schole's original derivation using asset pricing theory and Merton's later self-financing portfolio argument are enormous over-simplifications of the real world, and contra the Sveriges Rijksbank, Black, Scholes, & Merton did not show that "... it is in fact not necessary to use any risk premium when valuing an option".

默頓的觀點是在數學上更復雜,但也不那麽健壯和靈活。實際上永遠不會接觸連續自籌資金的投資組合:即使IT使持續方面可行,自籌資金方面也將始終用於交易費用。但資產定價參數可以很容易地修改,以允許更大的柔韌性。

解釋波動率微笑的邏輯第一步應該是回歸,而不是做出導致在資產定價論證中取消$ \ mu $的簡單假設。這會立即在模型中引入一個額外的參數,這個參數比IV軟糖因子更有意義,實際上可以解釋很多微笑。例如,您可能會獲得修改後的BS pde $$ rV = \ mu S {\ partial V \ over \ partial S} + {\ partial V \ over \ partial t} + \ frac {1} {2} \ sigma ^ 2S ^ 2 {\ partial ^ 2 V \ over \部分S ^ 2}, $$ 這導致定價公式出現類似的適度變化。

如你所知,現在我不是專家。這些想法是否已經被探索過並且被發現需要?

最佳答案

首先,我可以指出你可能遇到的兩個重大誤解:

  • Implied Volatility (IV) is the input to any vanilla option pricing model (not just Black Scholes (BS) that impacts the pricing the most. You can verify this by flipping through the different risk exposures (greeks and higher order sensitivities) and study mean volatilities in such risk factors and their impact on the pricing of such options.

  • Traders who price and buy/sell options in effect trade future realized volatility/expected variation in the underlying asset returns. Hence, option traders express views on such future asset price variation and thus buy and sell volatility. The term "implied" volatility is in my opinion a bit of a misnomer because the trade starts with an agreed level of volatility and not an option price.

    (In fact, you hardly ever hear any professional traders agreeing on an option price, they most often agree on the exact implied vols they trade at, and often times also trade the delta alongside the option (at least in equity space) in order to have the option delta-hedged at initiation.)

Option pricing models are used to translate the expressed IV -> Price. When you see option prices on your trading screen then those are the outputs of automated pricing applications which as input take among couple other mostly statics, IV.

因此,你認為靜脈註射是一種“軟糖因素”是非常簡單的。事實上,大多數在一個期權交易的東西都是IV。 (當然,您還有其他期權輸入,但您可以交易特定的股息互換或利率衍生品,例如,如果您想表達對此類投入的看法)。期權價格只是一種轉換,以支付交易的隱含波動率。

並且:僅僅因為期權定價模型不完美並不會使它變得毫無價值。事實上,我挑戰你提出一個同樣簡單(計算上和直覺上)並且比B-S更準確的替代模型,我相信市場會接受它並感謝你的努力。

EDIT:

I highly recommend to go through the following short paper: Option Pricing Q&A

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