ATM 看涨期权 Vega 对比

Comparing Vega for ATM Calls

专题: Finance / 金融
难度: L4
来源: QuantQuestion

个人笔记

题目详情

有两份平值（ATM）欧式看涨期权，二者到期时间 $T$ 、波动率 $\sigma$ 、无风险利率 $r$ 完全相同，但一个标的现价 $S_0=100$ ，另一个标的现价 $S_0=200$ 。

问：哪个期权的 Vega 更大？

英文原题

You have two at-the-money (ATM) European call options with identical time to maturity ( $T$ ), volatility ( $\sigma$ ), and risk-free rate ( $r$ ). However, one option is on a stock with a spot price $S_0 = 100$ , and the other is on a stock with $S_0 = 200$ . Which option has the higher Vega?

解析

Black-Scholes 下看涨期权 Vega 为

\nu=\frac{\partial C}{\partial \sigma}=S_0\,\varphi(d_1)\sqrt{T},

其中 $\varphi$ 为标准正态密度。

“ATM”通常指 $K=S_0$ ，此时两份期权的 $d_1$ 相同，所以 $\varphi(d_1)$ 相同， $T$ 也相同。

因此 Vega 与 $S_0$ 成正比， $S_0=200$ 的那份 Vega 更大（约为另一份的 2 倍）。

英文解析

Under the Black-Scholes model, the Vega of a call option is
$\nu=\frac{\partial C}{\partial \sigma}=S_0\,\varphi(d_1)\sqrt{T},$
where $\varphi$ is the standard normal density.

"ATM" typically refers to $K=S_0$ , at which point the $d_1$ of the two options are the same, so $\varphi(d_1)$ is the same, and $T$ is also the same.

Therefore, Vega is proportional to $S_0$ , meaning the Vega of $S_0=200$ is larger (approximately twice that of the other).