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很不酷 · 2022年04月05日

Vega Notional的含义

NO.PZ2018113001000077

问题如下:

Marcus, who works for an investment management company, is asked to calculate what the gain would be in 6 months on a purchase of $2,000,000 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%The fair strike of a new 6-month variance swap on the S&P 500 will be 18%.

The annual interest rate is 2.00%

The current value of the variance swap is:

选项:

A.

-$5,445,544.500

B.

-$5,500,000.000

C.

$5,445,544.500

解释:

A is correct

中文解析:

本题考察的是variance swap。

需要求解的是variance swap在6个月时刻的value:

第一步,计算variance notional = $2,000,000/(2×20)=50,000

第二步,计算折现因子PVt (T)=1/[1 + (2.00% × 6/12)] = 0.990099

第三步,直接带入公式计算,VarSwapt= $50,000 × (0.990099) × [(6/12) × 256 + (6/12) × 324 – 400]= -$5,445,544.500.

我记得Vega Notional的含义是volatility变动1%的payoff。那么为什么不可以用Vega notional直接乘以volatility的变动求swap的value?即 PV{2,000,000 x [(16%/2 + 18%/2) - 20%]}

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Hertz_品职助教 · 2022年04月06日

嗨,努力学习的PZer你好:


同学你好

我来重新回答一下这个问题哈

因为大多数的市场参与者习惯于从波动率的角度思考问题,所以对于variance swap就做了两项规定:(1) variance swap的交易规模,以vega notional来表示,注意不是variance notional表示哈; (2)strike (X)表示标的的预期未来方差,以波动率来进行表示。做这种规定仅仅是为了表述起来更加直观,因为vega variance名义上代表了波动率相比于strike volatility变化1%时的平均损益。例如,当vega notional是5万美元时,代表了波动率相比于strike volatility变化1%时的平均损益将接近5万美元。

但是教材上提醒我们这种方式计算的损益只是一个近似值,为了确切的计算损益,所以需要使用variance notional来计算。然后教材就给到了variance notional的计算公式,也就是咱们讲义和解析中用的那个公式哈。

因此咱们在计算variance swap的value的时候,这里的名义本金也是需要使用variance notional的,而不能使用vega notional。

因此:

1.     同学公式中的名义本金不可以是用vega notional的值哈

2.     同学公式中的“(16%/2 + 18%/2) - 20%”这里也是错误的,一是应该使用variance 即方差,不应该直接使用波动率,即应该取平方;二是在教材和咱们的课上也强调到,这里计算的时候只计算百分号前面的数据,即只对16,18,20取平方,因此就有了解析中的“VarSwapt= $50,000 × (0.990099) × [(6/12) × 256 + (6/12) × 324400]”。

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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!

Hertz_品职助教 · 2022年04月06日

嗨,爱思考的PZer你好:


同学你好~

根据基础班讲义P293页讲到的vega notional的概念:The vega notional represents the average profit and loss of the variance swap for a 1% change in volatility from the strike,可以知道vega notional代表了波动率相比于strike volatility变化1%时的平均损益。

所以在求variance swap的value的时候我们需要在vega notional的基础上乘以的是实际的波动率(或者隐含的波动率)相比于strike volatility的变化,只不过在具体计算的时候还需要考虑时间和折现问题。

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虽然现在很辛苦,但努力过的感觉真的很好,加油!

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NO.PZ2018113001000077 问题如下 Marcus, who works for investmentmanagement company, is asketo calculate whthe gain woulin 6 months ona purchase of $2,000,000 vega notionof a one-yevarianswon theS P 500 a strike of 20% (quoteannuvolatility).Now six months have passe anthe S P500 hexperiencea realizevolatility of 16%,Thefair strike of a new 6-month varianswon the S P 500 will 18%. The annuinterest rate is 2.00%The current value of the varianswis: A.-$5,445,544.500 B.-$5,500,000.000 C.$5,445,544.500 A is correct中文解析本题考察的是varianswap。需要求解的是varianswap在6个月时刻的value第一步,计算variannotion= $2,000,000/(2×20)=50,000第二步,计算折现因子PVt (T)=1/[1 + (2.00% × 6/12)] = 0.990099第三步,直接带入公式计算,VarSwapt= $50,000 × (0.990099) ×[(6/12) × 256 + (6/12) × 324 – 400]= -$5,445,544.500. 为什么是除以 (2*20)

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