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Brownie · 2022年02月23日

算的结果是-112963, 怎么理解是盈利呢

  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。

算的结果是-112963, 怎么理解是盈利呢

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Hertz_品职助教 · 2022年02月24日

嗨，努力学习的PZer你好：


同学你好~

是这样的哈，咱们在计算variance swap的value或者payoff的时候，讲义上的公式，以及这里解析中的公式都是站在long position的角度来计算的，这一点一定一定要注意哈，何老师在课上也有强调。

然后呢站在long position的角度，利用公式计算出来的结果为正，就说明buyer赚钱。

但是这里注意本题题干第一句话说明了他是sell variance swap，对应原文“Olivia, a fund manager, sells $50,000 vega notional of a one-year variance swap”，因此他是short position，所以当我们仍然用原来的公式计算的时候，结果是long position的结果，对于seller来说，要加负号，所以这也是为什么计算结果为负但是选项为正数的原因哈

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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！

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