解析的公式需要展示下，为啥exogenous spread是这么算的呢

问题如下：

A hedge fund has a 25,000-share position in an undervalued and relatively illiquid stock XYZ that has a current stock price of GBP 48 (expressed as the midpoint of the current bid-ask spread). The daily return for XYZ has a mean of 0%, an estimated volatility of 0.32% and avolatility spread of 0.0016. The average bid-ask spread is GBP 0.22. The risk division of the fund usually assumes that the returns are normally distributed and estimates the liquidity adjusted 1-day 95% VaR of the position using the constant spread approach. Suppose that the CRO asks the risk division to determine the liquidity adjusted 1-day 95% VaRusing the exogenous spread approach instead, assuming the volatility spread multiplier k of 3. What would be the increase in the liquidity adjustment? (Practice Exam)

选项：

A.

43.65%

B.

45.71%

C.

69.61%

D.

89.36%

解释：

Explanation: Before considering liquidity adjustment, the 1-day 95% VaR of the position is obtained as:

VaR = P[1 – exp(µ – σz)] = GBP 48*25,000*[1 – exp(0 – 0.0032*1.645)] = GBP 6,300.20

The liquidity adjusted VaR (LVaR) derived using the constant spread approach adds half of the bid-ask spread (as a percent) to the VaR calculation, using the following formula:

LVaR = VaR + Liquidity Cost (LC) = VaR + ½*(Spread * P)

where Spread is equal to the actual spread divided by the midpoint and P is the value of the position.

Therefore,

Daily 95% VaR = (48*25,000)*[1 – exp(0 – 1.645*0.0032)] = GBP 6,300.20

Liquidity cost = (0.5)*(0.22/48)*(48*25,000) = GBP 2,750

And so,

LVaR = VaR + LC = GBP 9,050.20 and so, the liquidity adjustment = 2,750/6,300.20 = 43.65% of VaR.

Using the exogenous spread approach, the liquidity cost (LC) is derived by

LC = (0.5)(48*25,000)*[(0.22/48) + (3*0.0016)] = 2,750 + 2,880 = GBP 5,630

The LVaR using the exogenous approach will be higher than LVaR obtained using the constant spread approach by GBP 5,630 and so, the liquidity adjustment = 5.630/6,300.20 = 89.36% of VaR.

Therefore, the increase in the liquidity adjustment when using the exogenous approach compared to using the constant spread approach = 89.36 – 43.65 = 45.71%