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Tonyzhai · 2022年04月18日

根据公式，不是应该 vega notional ，不应该要除以（2*strike price）吗？

  1. CFA III
  2. Derivative

NO.PZ2018113001000052

问题如下：

Olivia, a fund manager, sells $50,000 vega notional of a one-year variance swap on the S&P 500 at a strike of 20% (quoted as annual volatility).

Now six months have passed, and the S&P 500 has experienced a realized volatility of 16% (annualized). On the same day, the fair strike of a new six-month variance swap on the S&P 500 is 19%.

The the current value of the variance swap sold by Olivia (note that the annual interest rate is 2.5%) is:

选项：

A.

$112,963

B.

$ 998,653

C.

$ 159,228

解释：

A is correct.

Volatility strike on existing swap = 20.

Variance strike on existing swap = 20^2 = 400.

Variance notional = Vega notional/(2*Strike)=50000/(2*20)=1250.

RealizedVol(0,6)^2 = 16^2 = 256.

ImpliedVol(6,12)^2 = 19^2 = 361.

PVt(T) = 1/[1 + (2.5% × 6/12)] = 0.987654

The current value of the swap is

VarSwapt = 1,250 × (0.987654) × [(6/12) × 256 + (6/12) × 361 – 400]

= –$112,962.9263.

Given that Olivia is short the variance swap, the mark-to-market value is positive for her, and it equals $112,963.

中文解析：

本题考察的是对variance swap求value。

直接带入公式：variance  swapt=variance  notional×PVt(T)×{tT×[realized  volatility(0，t)]2+T−tT×[implied  volatility(t,T)]2}−strike2variance\;swap_t=variance\;notional\times PV_t\left(T\right)\times\left\{\frac tT\times\left[realized\;volatility\left(0，t\right)\right]^2+\frac{T-t}T\times\left[implied\;volatility\left(t,T\right)\right]^2\right\}-strike^2计算即可。

需要注意以下两点：

1. 该公式是站在long position的角度，而本题问的是short position，因此注意最后的结果需要加负号

2. strike，implied volatility以及realized volatility在代入计算时，不加百分号，只取百分号前面的数字。

题目给的是 vega notional ，不应该要除以（2*strike price）吗？

为什么答案没有除以，而且题目也没有给strike price？

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Tonyzhai · 2022年04月18日

不用回答了，我看错了题目了，谢谢！

1 个答案

Hertz_品职助教 · 2022年04月18日

嗨，努力学习的PZer你好：


好滴，如果有问题可以再次提问哈，祝同学学习顺利！

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加油吧，让我们一起遇见更好的自己！

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NO.PZ2018113001000052 问题如下 Olivia funmanager, sells $50,000 vega notionof a one-yevarianswon the S P 500 a strike of 20% (quoteannuvolatility).Now six months have passe anthe S P 500 hexperiencea realizevolatility of 16% (annualize. On the same y, the fair strike of a new six-month varianswon the S P 500 is 19%.The the current value of the varianswsolOlivia (note ththe annuinterest rate is 2.5%) is: A.$112,963 B.$ 998,653 C.$ 159,228 A is correct.Volatility strike on existing sw= 20.Varianstrike on existing sw= 20^2 = 400.Variannotion= Vega notional/(2*Strike)=50000/(2*20)=1250.Realizeol(0,6)^2 = 16^2 = 256.Implieol(6,12)^2 = 19^2 = 361.PVt(T) = 1/[1 + (2.5% × 6/12)] = 0.987654The current value of the swisVarSwapt = 1,250 × (0.987654) × [(6/12) × 256 + (6/12) × 361 – 400]= –$112,962.9263.Given thOlivia is short the varianswap, the mark-to-market value is positive for her, anit equals $112,963.中文解析本题考察的是对varianswap求value。直接带入公式计算即可。需要注意以下两点1. 该公式是站在long position的角度，而本题问的是short position，因此注意最后的结果需要加负号2. strike，implievolatility以及realizevolatility在代入计算时，不加百分号，只取百分号前面的数字。 这里给的前半年实现的volatile是16%，这个是年化的。如果算半年的，为什么不用8%呢？谢谢？

2024-08-21 23:46 1 · 回答

NO.PZ2018113001000052 问题如下 Olivia funmanager, sells $50,000 vega notionof a one-yevarianswon the S P 500 a strike of 20% (quoteannuvolatility).Now six months have passe anthe S P 500 hexperiencea realizevolatility of 16% (annualize. On the same y, the fair strike of a new six-month varianswon the S P 500 is 19%.The the current value of the varianswsolOlivia (note ththe annuinterest rate is 2.5%) is: A.$112,963 B.$ 998,653 C.$ 159,228 A is correct.Volatility strike on existing sw= 20.Varianstrike on existing sw= 20^2 = 400.Variannotion= Vega notional/(2*Strike)=50000/(2*20)=1250.Realizeol(0,6)^2 = 16^2 = 256.Implieol(6,12)^2 = 19^2 = 361.PVt(T) = 1/[1 + (2.5% × 6/12)] = 0.987654The current value of the swisVarSwapt = 1,250 × (0.987654) × [(6/12) × 256 + (6/12) × 361 – 400]= –$112,962.9263.Given thOlivia is short the varianswap, the mark-to-market value is positive for her, anit equals $112,963.中文解析本题考察的是对varianswap求value。直接带入公式计算即可。需要注意以下两点1. 该公式是站在long position的角度，而本题问的是short position，因此注意最后的结果需要加负号2. strike，implievolatility以及realizevolatility在代入计算时，不加百分号，只取百分号前面的数字。 请问题目中variance已经是Volatility的平方概念，为何公式里还是需要用平方计算？

2024-04-20 17:50 2 · 回答

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2024-01-29 11:35 1 · 回答

NO.PZ2018113001000052 $ 998,653 $ 159,228 A is correct. Volatility strike on existing sw= 20. Varianstrike on existing sw= 20^2 = 400. Variannotion= Vega notional/(2*Strike)=50000/(2*20)=1250. Realizeol(0,6)^2 = 16^2 = 256. Implieol(6,12)^2 = 19^2 = 361. PVt(T) = 1/[1 + (2.5% × 6/12)] = 0.987654 The current value of the swis VarSwapt = 1,250 × (0.987654) × [(6/12) × 256 + (6/12) × 361 – 400] = –$112,962.9263. Given thOlivia is short the varianswap, the mark-to-market value is positive for her, anit equals $112,963. 中文解析 本题考察的是对varianswap求value。 直接带入公式 variance  swapt=variance  notional×PVt(T)×{tT×[realize volatility(0，t)]2+T−tT×[implie volatility(t,T)]2}−strike2variance\;swap_t=variance\;notional\times PV_t\left(T\right)\times\left\{\frtT\times\left[realize;volatility\left(0，t\right)\right]^2+\frac{T-t}T\times\left[implie;volatility\left(t,T\right)\right]^2\right\}-strike^2varianceswapt​=variancenotional×PVt​(T)×{Tt​×[realizeolatility(0，t)]2+TT−t​×[implieolatility(t,T)]2}−strike2计算即可。 需要注意以下两点 1. 该公式是站在long position的角度，而本题问的是short position，因此注意最后的结果需要加负号 2. strike，implievolatility以及realizevolatility在代入计算时，不加百分号，只取百分号前面的数字。 算的结果是-112963, 怎么理解是盈利呢

2022-02-23 21:20 1 · 回答