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期权Gamma（Γ）

Gamma

专题
Finance / 金融
难度
L4

题目详情

平值且临近到期时，Gamma 会发生什么？

英文原题

Γ=∂2f∂S2\Gamma = \frac{\partial^2 f}{\partial S^2}

For a European call/put on a stock with dividend yield yy:
>Γ>=>N′(d1) e−yτ S στ ,>>N′(d1)>=>12π >e−d122.>> \Gamma > = > \frac{N'(d_1)\,e^{-y\tau}}{\,S\,\sigma\sqrt{\tau}\,}, >\quad > N'(d_1) > = > \frac{1}{\sqrt{2\pi}}\, > e^{-\tfrac{d_1^2}{2}}. >

Question: At-the-money, near expiry, what happens to gamma?

解析

当 S≈KS\approx K 且 τ→0\tau\to 0 时，

Γ=∂2f∂S2→∞.\Gamma=\frac{\partial^2 f}{\partial S^2} \to \infty.

直觉：到期附近 delta 会在很窄的价格区间里从接近 0 快速跳到接近 1，因此二阶敏感度很大。远离平值时 Γ\Gamma 接近 0。

英文解析

Gamma Γ→∞\Gamma\to\infty as τ→0\tau\to0 if S≈K.S\approx K. Because Δ\Delta transitions rapidly from near 0 to 1. Far from K,K, Γ\Gamma is near 0.