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无分红 ATM Delta

BS world without dividends

专题
Finance / 金融
难度
L4

题目详情

金融数学题：无分红、r>0r>0：ATM call 的 delta > 0.5 吗。

英文原题

Assume a Black- Scholes world without 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 , is the option's delta greater or less than 0.5? What does it depend on?

解析

无分红下 Δ=N(d1)\Delta=N(d_1)。

ATM 时 ln⁡(S/K)=0\ln(S/K)=0，

d1=(r+12σ2)TσT>0,d_1=\frac{(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}>0,

所以

Δ=N(d1)>0.5.\boxed{\Delta=N(d_1)>0.5}.

英文解析

d1=(r+12σ2)TσT>0,d_1=\frac{(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}>0, Δ=N(d1)>0.5.\boxed{\Delta=N(d_1)>0.5}.