P1.T4.805. Linear and non-linear derivative value at risk (VaR) (Allen)

[Nicole Seaman]

Learning objectives: Explain and give examples of linear and non-linear derivatives. Describe and calculate VaR for linear derivatives. Describe the delta-normal approach for calculating VaR for non-linear derivatives.

Questions:

805.1. A fund manager's $1.0 million bond portfolio contains the following two long bond positions:

• 50% invested in a zero-coupon bond with 5.0 years to maturity, plus
• 50% invested in a zero-coupon bond with 8.0 years to maturity

For these risky bonds the yield curve is flat at 8.0% per annum with annual compounding. However, with respect to yield changes at their respective maturities, the correlation between these two bonds is imperfectly at ρ = 0.70. The monthly yield volatility is 100 basis points; i.e., the annual basis-point volatility is 1.0% * sqrt(12) = 3.46%. Yield changes are assumed to be normally distributed. Which of the following is nearest to the bond portfolio's 99.0% one-month value at risk (relative VaR: worst expected loss relative to expected future value)?

a. 95,000
b. 105,750
c. 129,900
d. 178,400


805.2. A fund manager has $200,000 invested equally in two equity call option positions:

• 50% invested in an out-of-the-money (OTM) option position with a (per-option, aka percentage) delta of 0.50, while the underlying stock has a volatility of 20.0% per annum
• 50% invested in an in-the-money (ITM) option position with a (per-option, aka percentage) delta of 0.70 while the underlying stock has a volatility of 32.0% per annum

The correlation between the two underlying stocks is 0.40. If we assume 250 trading days per year and normal i.i.d. returns on the stock, which of the following is nearest to the portfolio's 95.0% ten-day value at risk?

a. $4,255
b. $9,193
c. $10,660
d. $15,720


805.3. Consider a stock with a price of $100.00 and a volatility of 36.0% per annum with normal i.i.d. returns. A call option on this stock currently exhibits the following two Greeks:

• Option (percentage) delta: 0.6040
• Option (percentage) gamma: 0.0110

Assume there are 250 trading days per year. If we use the delta-gamma approach (aka, truncated Taylor Series) to approximate the risk based only on the option's given delta and gamma risk factors, which of the following is nearest to the ten-day 95.0% value at risk (VaR)?

a. $6.38
b. $7.15
c. $9.99
d. $11.07

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