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有连续分红：ATM call 的 delta 能 < 0.5 吗

BS world with continuous dividends

专题
Finance / 金融
难度
L4

题目详情

Assume a Black- Scholes world with continuous dividends. Consider a standard European call struck at- the- money (i.e., strike equals current spot) with one year to maturity. If the interest rate is r=0.06r = 0.06 , and dividends are at rate ρ=0.03\rho = 0.03 , can you tell whether the option's delta is greater or less than 0.5? What does it depend on?

解析

连续分红率 qq 时，欧式 call 的 delta 为

Δ=e−qTN(d1),d1=ln⁡(S/K)+(r−q+12σ2)TσT.\Delta=e^{-qT}N(d_1),\quad d_1=\frac{\ln(S/K)+(r-q+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}.

即使 N(d1)>0.5N(d_1)>0.5，乘上 e−qT<1e^{-qT}<1 后也可能小于 0.5。

因此

Δ<0.5 是可能的（取决于 q,σ,T）.\boxed{\Delta<0.5\text{ 是可能的（取决于 }q,\sigma,T\text{）}}.