二元期权（Binary）定价与对冲

在 BS 框架下，二元看涨价格为

$$c_{\mathrm{binary}}=e^{-r\tau}N(d_2).$$

对冲可用 delta：

$$\Delta = e^{-r\tau}N'(d_2)\,\frac{1}{S\sigma\sqrt{\tau}}.$$

局限：当 $S\approx K$ 且 $\tau\to 0$ 时 gamma 极大，需要极高频调仓，实际交易成本/滑点很大；可用很窄的 call spread 近似，但需要多行权价报价。

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Original Explanation

• Price = $c_{\mathrm{binary}} = e^{-r\tau}N(d_2).$
• Hedge by short $\Delta = e^{-r\tau}N'(d_2)\,\frac{1}{S\,\sigma\sqrt{\tau}}$ shares. But near $S=K$ and $\tau\to0,$ gamma is huge => frequent rebalancing is needed => big slippage. Alternatively approximate via a narrow call spread, though we need multiple strikes.