A young girl demonstrated her ability to freely switch her eyelids among three modes: single, double, and one single with one double. It's so cool! 小孩姐展示任意切换眼皮三种模式 可单可双还可一单一双 太酷了 网友：教教我 阿姨真的很想学这个！神奇绝活 #单双眼皮切换# #搞笑日常#

Not sure how to handle this,Commander? Let Taihou give you a... personal demonstration 指挥官，这根“球杆”的用法，还需要大凤来教你嘛？ #bunnygirl# #AzurLane# #大鳳#

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“要保护好…家人…” “I must... protect... my family..” . photo by @ImMuze character： #祢豆子# #nezuko# #ねずこ# from #demonslayer# #鬼灭之刃# #鬼灭の刃#

请问需要女仆吗？只需要提供十天饭量的那种哦～ Hi~ do you need a maid? You only need to provide 10 days meal~ character：#甘露寺蜜璃# #かんろじみつり# #kanrojimitsuri# 女仆同人. Maid ver. from #demonslayer# #鬼滅の刃# #鬼灭之刃# #kimetsunoyaiba#

和Quant Alex @StochAlex07 讨论： SABR Theta与Spot Theta+Vol Theta+Cross Theta的异同与应用，以及SABR模型自洽性分析。 **English Summary of the Chat** **SABR Theta vs Spot Theta + Vol Theta + Cross Theta** The conversation between **Alex Wu** (white bubbles) and **Jeff Liang** (green bubbles) is a technical discussion focused on **SABR Theta versus Total Theta** (i.e., Spot Theta + Cross Theta + Vol Theta), model self-consistency, PDE residual, and the correct definition of SABR Greeks. ### Key Points Discussed: 1. **SABR Gamma = Spot Gamma** (first major question, raised by Jeff) Jeff asked whether SABR Gamma (\(\partial^2 P / \partial F^2\)) is identical to Spot Gamma and whether it includes the dependence of \(\sigma_B\) on \(F\). He also provided the full chain-rule expansion of SABR Gamma in terms of Black-76 Greeks. Alex confirmed the understanding and **later affirmed in code** that this is exactly how SABR Gamma is implemented in their system. 2. **SABR Theta vs Total Theta and Model Self-Consistency** (main topic, led by Jeff) Jeff shared a clear 3-point understanding: - SABR Theta is computed directly via the SABR approximation formula to obtain \(\sigma_B\), then applying the Black-76 chain rule: \(\partial P/\partial t =\) BS_Theta(\(\sigma_B\)) + BS_Vega \(\cdot \partial\sigma_B/\partial t\). - Total Theta is the exact decomposition from the SABR PDE (Spot Theta + Cross Theta + Vol Theta). - When the model is **fully self-consistent** (Residual = \(\partial P/\partial t + \mathcal{L}P = 0\)), SABR Theta = Total Theta; otherwise the difference is the unexplained PnL caused by the approximation error in the Hagan formula (especially pronounced in long-dated, high vol-of-vol, or high-skew options). 3. **Practical Implication – Theta Decomposition Decision** (comment by Alex) Alex noted that whether to perform Theta decomposition depends on the risk-management approach: - Without decomposition → use SABR Gamma vs. dP/dt. - With decomposition → SABR Gamma maps to Spot Theta, Vanna to Cross Theta, and Volga to Vol Theta. **Overall Tone**: The discussion is highly technical and collaborative. Jeff drives the conversation by asking clarifying questions and presenting a well-structured 3-point summary of his recent study. Alex provides confirmations, practical insights, and code-level validation. Both participants demonstrate a strong command of SABR model nuances, particularly the relationship between approximation error, PDE residual, and real-world risk management.

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