調查 Black-Scholes 公式正確性的預印本
最近的預印本出現在 arXiv 上。在已經假設標的物現貨價格採用Black-Scholes 模型的情況下，它質疑 Black-Scholes 公式對歐式看漲期權價格的適用性。
摘要摘錄：
如果正確，我們的結果使 Merton (1971) 的連續時間預算方程和 Black 和 Scholes (1973) 的套期保值論點和期權定價公式無效。
據稱癥結在於自籌資金條件（通常在嚴格推導公式時假設）不合適並且具有不切實際的後果，即
因此，他的分析隱含地假設投資組合再平衡是確定性的，不依賴於資產價格的變化。
假設預印本是準確的。根據現代投資銀行推動衍生品定價的（相當標準化的）模型，**哪些類型的衍生品可能會被嚴重錯誤定價（如果有的話）？**請記住，使用模型/結果可能會產生“高階”效應，由於對 Black-Scholes 公式的某種程度的依賴，其有效性也可能受到質疑。
For example, given the way volatility is often quoted (the right number for the Black–Scholes formula to arrive at a certain price), valuation of vanilla European options is basically guaranteed to be unaffected even if the preprint is accurate.
It’s a common critique that the original Black-Scholes derivation is somewhat imperfect with respect to the self-financing property, as fesman mentions.
Self-financing trading strategies are a key concept in maths finance. After all, we can all replicate any payoff using a non-self-financing strategy by simply injecting funds as necessary. Thus, pricing by arbitrage typically requires identifying a self-financing portfolio which replicates the targeted payoff.
In doubt it’s always useful to consult the writing of the great people in the quant finance community. The issue about self-financing in the Black-Scholes derivation has been addressed, amongst others, by Gordon and Peter Carr.
In his stellar answer, Gordon points out that the standard textbook derivation of the Black-Scholes formula is not self-financing. Whilst it gives the correct result (and yes, the final Black-Scholes formula is correct), the standard derivation is wrong.
In Carr’s paper, he discusses in question V how the Black-Scholes derivation can be made rigorous. To this end, he carefully thinks what meaning the differentials need to have and how one can bypass the self-financing property.