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ruiqi · 2025年05月01日

这里计算过程涉及的知识点是哪个

  1. FRM II
  2. Market Risk Measurement And Management

NO.PZ2022071105000019

问题如下：

A fund manager owns a portfolio of options on TUV, a non-dividend paying stock. The portfolio is made up of

5,000 deep in-the-money call options on TUV and 20,000 deep out-of-the-money call options on TUV. The

portfolio also contains 10,000 forward contracts on TUV. Currently, TUV is trading at USD 52. Assuming 252

trading days in a year, the volatility of TUV is 12% per year, and that each of the option and forward contracts is

on one share of TUV, which of the following amounts would be closest to the 1-day 99% VaR of the portfolio?

选项：

A.

USD 11,557

B.

USD 12,627

C.

USD 13,715

D.

USD 32,000

解释：

C是正确的。我们需要把资产组合map到股票TUV上面。一个深度实值看涨期权的delta接近于1，深度虚值看涨期权的delta接近于0, 一份远期合约的delta等于1。

所以资产组合的delta（Dp） = 1*5,000 + 0*20,000 + 1*10,000 = 15,000, 并且资产组合是gamma中性的。

现有:

ꭤ = 2.326 (99% 置信度)

S = TUV的股价 = USD 52

Dp = 资产组合的delta = 15,000

σ = TUV股票的波动率 = 0.12

所以99%置信度水平下的1天的VaR等于：

ꭤ *S*Dp*σ*sqrt(1/T) = (2.326)*(52)*(15,000)*(0.12/sqrt(252)) = USD 13,714.67

C is correct. We need to map the portfolio to a position in the underlying stock TUV. A

deep in-the-money call has a delta of approximately 1, a deep out-of-the-money call has a

delta of approximately zero and forwards have a delta of 1.

The net portfolio has a delta (Dp) of about 1*5,000 + 0*20,000 + 1*10,000 = 15,000 and is

approximately gamma neutral.

Let:

ꭤ = 2.326 (99% confidence level)

S = price per share of stock TUV = USD 52

Dp = delta of the position = 15,000

σ = volatility of TUV = 0.12

Therefore, the 1-day VaR estimate at 99% confidence level is computed as follows:

ꭤ *S*Dp*σ*sqrt(1/T) = (2.326)*(52)*(15,000)*(0.12/sqrt(252)) = USD 13,714.67

这里计算过程涉及的知识点是哪个

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1 个答案

李坏_品职助教 · 2025年05月01日

嗨，从没放弃的小努力你好：


题目最后问的是VaR是多少，而这道题的资产组合涉及到多个期权。要求出期权的VaR，我们需要用VaR mapping的公式：

这道题需要先求出δ，就是公式里的△。

另外，题目里面的期权都是deep in the money或者deep out of money，这些期权都是gamma约等于0的，所以公式里的gamma(就是1/2后面那个东西)等于0.


所以这道题的期权组合的VaR = delta * VaR(dS)。 而VaR(dS)就是股票的VaR， VaR(dS) = 2.326 * S *σ * sqrt(1/T) 。

所以期权的VaR = 15,000的delta * 2.326 * S *σ * sqrt(1/T)

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