Fin 500Q – Quantitative Risk Management Homework #7
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Fin 500Q – Quantitative Risk Management
Homework #7 Solutions
1. Suppose a portfolio is made up of two at-the-money calls (stock price equals exercise price), one on stock 1 and one on stock 2. The maturities of both options is one year, the riskless rate is 6%, and neither stock pays a dividend. Let S1 = 500, σ1 = 0.3, S2 = 250, σ2 = 0.1, and ρ 12 = -0.5.
(a) Find the one-day Delta VaR0 .05 of the portfolio position. You may ignore the average change in the value of the portfolio.
Answer: For the call option on stock 1, we compute d1 = ln(S1 /K1 )T(曰)+0 .5σ1(2))T = 0.35 and
d2 = d1 - σ 1 ^T = 0.05 and find price C1 = S1 Φ(d1 ) - K1 e — R曰 T Φ(d2 ) = 73.5854 and delta ∆C,1 = Φ(d1 ) = 0.6368. Similarly, for the call on stock 2 we get C2 = 18.6483 and ∆C,2 = 0.7422. Therefore, the initial portfolio value is W = C1 + C2 = 92.23, with weights w1 = ∆C,1 . S1 /W = 3.45, w2 = ∆C,2 . S2 /W = 2.01, and wB = 1 - w1 - w2 = -4.46. The annual variance of the portfolio return is
w 1(2)σ1(2) + w2(2)σ2(2) + 2w1 w2 ρ 12 σ 1 σ2 = 0.9048.
Thus, we find that the daily VaR in dollars is VaR0 .05 = 1.645 . 92.23 . ^0.9048/252 = 9.09. (b) Find the contribution of the position in each option to the VaR of the portfolio.
Answer: The beta of the first call on the portfolio return is given by
w1 σ 1(2) + w2 ρ 12 σ 1 σ2
w 1(2)σ 1(2) + w2(2)σ2(2) + 2w1 w2 ρ 12 σ 1 σ2
Similarly, we find that β2 = -0.035. Recall from the “Value at Risk” notes that
∆VaRi = - [µi + zα . σRp . βi],
where σRp is the volatility per dollar of the portfolio. Therefore:
∆VaR1 = 1.645 . 0.31 . ^0.9048/252 = 0.0306 ∆VaR2 = 1.645 . (-0.035) . ^0.9048/252 = -0.0034.
The component VaR of option 1 is |W | . w1 . ∆VaR1 = 9.7313 and the component VaR of option 2 is |W | . w2 . ∆VaR2 = -0.6401.
2. Suppose a speculator writes (i.e., sells) an at-the-money straddle on an underlying stock with current price of 1 000, with the parameters RF = 3%, σ = 0.45, q = 0, T = 5. Find:
(a) The 1-month Delta VaR0 .05 of the short straddle position. You may ignore the average change in the straddle value.
Answer: Using the Black-Scholes formulas, we find that the vale of the call is $431.58 and the value of the put is $292.29. Therefore, the value of the short straddle is -$723.87. The deltas of the call and put are 0.7429 and -0.2571, respectively. Therefore, the net delta of the short position is -0.4857. The monthly VaR0 .05 of the portfolio is 1.645 . ^0.48572 . 10002 . 0.452 /12 = 103.79.
(b) The contribution of the short call and put to the VaR.
Answer: Since the call and the put option are written on the same stock, we can compute the contribution of the short call to the VaR as -0.7429/(-0.4857) . 103.79 = 158.74 and the contribution of the put as 0.2571/(-0.4857) . 103.79 = -54.95.
(c) The 1-month Delta-Gamma VaR0 .05 of the short straddle. You may ignore the average change in the straddle value.
Answer: The monthly variance of the stock is 10002 . 0.452 /12 = 16875. The gammas of the put and the call are both 0.00033066. Therefore, the gamma of the short straddle is -2 . 0.00032052. Using the formula for the Delta-Gamma VaR with zero mean, we have
VaR0 .05 = 1.645 . ╱ |-0.4857| . ^16875 - . (-2 . 0.00032052) . 16875、 = 118.43.
(d) Now suppose the speculator goes long on an additional put option with exercise price of 900 and maturity of 2 years. Then what would be the new 1-month Delta VaR5% of the entire position? You may ignore the average change in the value of the portfolio.
Answer: The delta of the additional put is -0.2816. The net delta of the three options is -0.4857 - 0.2816 = -0.7673. The Delta VaR is then 1.645 . ^0.76732 . 16875 = 163.97.
2023-05-03