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Kathy苏苏 · 2023年02月22日

VEGA的公式

NO.PZ2020021205000042

问题如下:

What is the vega of a European put option on a stock index when the index level is USD 1,500, the strike price is USD 1,400, the risk-free rate is 5%, the dividend yield is 2%, the volatility is 18%, and the time to maturity is three months. How can this be interpreted?

选项:

解释:

The vega is SoT\sqrt T\\N'(d1 )eqTe^{-qT}\\

In this case, S0 = 1,500, K = 1,400, r = 0.05, q = 0.02, = 0.18, and T = 0.25.

d1=ln(1500/1400)  +  (0.05  0.02+  0.182/2)  X  0.250.180.25\frac{\ln(1500/1400)\;+\;(0.05\;-0.02+\;0.18^2/2)\;X\;0.25}{0.18\sqrt{0.25}}\\= 0.8949

and vega is

1 500 X 0.25\sqrt{0.25}\\*12πe0.89492/2×e0.02×0.25\frac1{\sqrt{2\mathrm\pi}}e^{-0.8949^2/2}\times e^{-0.02\times0.25}\\=199

This means that the value of a long position increases by 199 X 0.01 = 1.99 if volatility increases by 1% (= 0.01) from 18% to 19%. Similarly, it decreases by 1. 99 if the volatility decreases from 18% to 17%.

老师,VEGA的计算公式是什么?在基础讲义哪页?答案中公式和后面带的数据好像不一致。考纲要求Vega计算吗?谢谢


3 个答案

李坏_品职助教 · 2023年02月23日

嗨,爱思考的PZer你好:


Vega计算是不会考的,我把公式带入输入给你算一下:

vega = S0 * N'(d1) * 根号T,这里N'(d1) 是标准正态分布的概率密度函数。之前我们学的N(x)叫做累积概率函数,N‘(d1)其实就是N(d1)对d1求一阶偏导。

N‘(d1)的公式我之前给错了,下面给出正确的:

N'(d1) =1/ 根号(2π) * exp(-1/2 * d1^2) = (1/根号下6.283) * exp(-1/2 * 0.8949^2) = 0.2673,

所以vega = S0 * N'(d1) * 根号T = 1500 * 0.2673 * 根号0.25 = 200.48.


标准答案和我的结果略有误差。

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虽然现在很辛苦,但努力过的感觉真的很好,加油!

李坏_品职助教 · 2023年02月22日

嗨,从没放弃的小努力你好:


答案解析里面显示的不太对,以基础版讲义为准。N'(d1) = 根号下(2π)*e^(-d1^2/2),我觉得考到的概率非常低。

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虽然现在很辛苦,但努力过的感觉真的很好,加油!

李坏_品职助教 · 2023年02月22日

嗨,爱思考的PZer你好:


在基础班讲义的Other Greek Letters里面:

vega在考试中不会直接要求计算,但是要知道普通期权的vega都是大于零的,而barrier options的vega可能是负数。另外vega的含义是期权价格对标的资产(股票)波动率的一阶导数。

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就算太阳没有迎着我们而来,我们正在朝着它而去,加油!

Kathy苏苏 · 2023年02月23日

能写下这题VEGA计算代入公式的数据的具体过程吗?根据您给的公式,我计算不出199。另外,N'(d1)是N(d1)对谁求导?谢谢老师,辛苦了。

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NO.PZ2020021205000042问题如下Whis the vega of a Europeput option on a stoinx when the inx level is US1,500, the strike priis US1,400, the risk-free rate is 5%, the vinyielis 2%, the volatility is 18%, anthe time to maturity is three months. How cthis interpretep.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #484247}span.s1 {color: #4e2c3f}span.s2 {color: #6b5547}span.s3 {color: #303b5f}span.s4 {color: #4a5465}p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #484246}The vega is SoT\sqrt T\\T​N'( )e−qTe^{-qT}\\e−qTp.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #4b464c}span.s1 {font: 6.0px Helvetica}In this case, S0 = 1,500, K = 1,400, r = 0.05, q = 0.02, = 0.18, anT = 0.25.=ln⁡(1500/1400)  +  (0.05  −0.02+  0.182/2)  X  0.250.180.25\frac{\ln(1500/1400)\;+\;(0.05\;-0.02+\;0.18^2/2)\;X\;0.25}{0.18\sqrt{0.25}}\\0.180.25​ln(1500/1400)+(0.05−0.02+0.182/2)X0.25​= 0.8949anvega is 1 500 X 0.25\sqrt{0.25}\\0.25​*12πe−0.89492/2×e−0.02×0.25\frac1{\sqrt{2\mathrm\pi}}e^{-0.8949^2/2}\times e^{-0.02\times0.25}\\2π​1​e−0.89492/2×e−0.02×0.25=199This means ththe value of a long position increases 199 X 0.01 = 1.99 if volatility increases 1% (= 0.01) from 18% to 19%. Similarly, it creases 1. 99 if the volatility creases from 18% to 17%.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #3f3a41}span.s1 {color: #6b6b6b}p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #4b464c}span.s1 {font: 6.0px Helvetica}p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #484449}span.s1 {font: 6.0px Helvetica}span.s2 {font: 8.0px Helvetica}span.s3 {font: 10.0px Helvetica}p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #484549}p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #544f}p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helveticcolor: #4532}p.p4 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.0px Helveticcolor: #473e44}span.s1 {font: 7.0px Helvetica}span.s2 {font: 13.0px Helvetica}span.s3 {color: #42333a}span.s4 {color: #484e6f}span.s5 {color: #6b636b}span.s6 {font: 6.0px Helvetica}span.s7 {color: #7f7b7f}span.s8 {color: #786356}span.s9 {font: 8.0px Helveticcolor: #4532}span.s10 {font: 11.0px Helveticcolor: #4532}span.s11 {color: #4b5a6c}span.s12 {color: #675248}span.s13 {color: #303c63}span.s14 {color: #363536}老师,第一个红框是求导?第二个红框是什么意思?

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vega还需要掌握计算吗?N'()也不知道怎么算

2020-03-02 12:22 1 · 回答