FRM一級估值與風(fēng)險模型例題解析

An analyst is pricing a 2-year European put option on a non-dividend-paying stock busing a binomial tree with two time steps of one year each. The stock price can go up or down by 20% each period, and assuming that the annual risk-free rate will remain constant at 2% over the next two years and the annual stock volatility is 15%. Please calculate the risk-neutral probability of an upward movement.

A. 50%

B. 44.95%

C. 55.05%

D. None of above

答案：C

解析：

關(guān)聯(lián)考點：風(fēng)險中性上漲概率計算

易錯點分析：計算上漲因子u和下跌因子d有兩種方法，第一個方法是：u=上漲幅度，d=下跌幅度；第二個方法：u=e^西格瑪*根號delta；d=e^-西格瑪*根號delta，在題干同時給出上漲幅度和波動率時，理論上兩種方法的計算結(jié)果是一樣的，如果有差異，優(yōu)先使用第一種方法。

Assume we calculate a one-week VaR for a natural gas position by rescaling the daily VaR using the square root rule. Let us now assume that we determine the “true” gas price process to be mean reverting and recalculate the VaR. Which of the following statements is true?

A. The recalculated VaR will be less than the original VaR

B. The recalculated VaR will be equal to the original VaR

C. The recalculated VaR will be greater than the original VaR

D. There is no necessary relation between the recalculated VaR and the original VaR

答案：A--------

解析：The square root rule applies only when the returns are uncorrelated. With mean reversion, natural gas prices are negatively correlated. In that case the VaR computed from the square root rule will overstate the true VaR, and the recalculated VaR that corrects for mean reversion will be less than the original VaR. If the process is mean reverting then we can no longer scale volatility (and associated

VaR) up by the square root of time.

關(guān)聯(lián)考點：VaR計算平方根法則易錯點分析：不滿足每天的收益獨立同分布假設(shè)的時間平方根普通法是乘以√T(1+ρ)，當(dāng)ρ大于0時，說明是帶有趨勢的，平方根法低估VAR；當(dāng)ρ小于0時，說明是均值復(fù)歸的，平方根法則高估VAR。通俗來講，當(dāng)存在均值復(fù)歸時，假如昨天的收益率上升，則今天的收益率下降，如此往復(fù)，在均值附近波動，如果不存在均值復(fù)歸，則收益率曲線要么一直上升要么一直下降，波動率比存在均值復(fù)歸時更大，其VaR值也更大，反之，存在均值復(fù)歸時的VAR值被低估。