5.35. The position is the same as a European call to buy the asset for K on the date.
5.36.
(a) When the CP rate is 6.5% and Treasury rates are 6% with semiannual compounding, the
CMT% is 6% and an Excel spreadsheet can be used to show that the price of a 30-year bond with a 6.25% coupon is about 103.46. The spread is zero and the rate paid by P&G is 5.75%. (b) When the CP rate is 7.5% and Treasury rates are 7% with semiannual compounding, the CMT% is 7% and the price of a 30-year bond with a 6.25% coupon is about 90.65. The spread is therefore
max[0, (98.5 × 7/5.78 ? 90.65)/100] or 28.64%. The rate paid by P&G is 35.39%. .
5.37. The trader has to provide 60% of the price of the stock or $2,400. There is a margin call when the margin account balance as a percent of the value of the shares falls below 30%. When the share price is S the margin account balance is 2400 + 200× (S?20) and the value of the position is 200×S. There is a margin call when 2400 + 200 × (S?20) < 0.3 × 200 × S or
140 S < 1600 or S < 11.43
that is, when the stock price is less than $11.43.
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Chapter 6: How Traders Manage Their Exposures
6.15. With the notation of the text, the increase in the value of the portfolio is
0.5?gamma?(?S)2?vega???
This is
0.5 × 50 × 32 + 25 × 4 = 325
The result should be an increase in the value of the portfolio of $325.
6.16. The price, delta, gamma, vega, theta, and rho of the option are 3.7008, 0.6274, 0.050,
0.1135, ?0.00596, and 0.1512. When the stock price increases to 30.1, the option price increases to 3.7638. The change in the option price is 3.7638 ? 3.7008 = 0.0630. Delta predicts a change in the option price of 0.6274 × 0.1 = 0.0627 which is very close. When the stock price increases to 30.1, delta increases to 0.6324. The size of the increase in delta is 0.6324 ? 0.6274 = 0.005. Gamma predicts an increase of 0.050 × 0.1 = 0.005 which is (to three decimal places) the same. When the volatility increases from 25% to 26%, the option price increases by 0.1136 from
3.7008 to 3.8144. This is consistent with the vega value of 0.1135. When the time to maturity is changed from 1 to 1?1/365 the option price reduces by 0.006 from 3.7008 to 3.6948. This is consistent with a theta of ?0.00596. Finally when the interest rate increases from 5% to 6% the value of the option increases by 0.1527 from 3.7008 to 3.8535. This is consistent with a rho of 0.1512.
6.17. The delta of the portfolio is
?1, 000 × 0.50 ? 500 × 0.80 ? 2,000 × (?0.40) ? 500 × 0.70 = ?450 The gamma of the portfolio is
?1, 000 × 2.2 ? 500 × 0.6 ? 2,000 × 1.3 ? 500 × 1.8 = ?6,000 The vega of the portfolio is
?1, 000 × 1.8 ? 500 × 0.2 ? 2,000 × 0.7 ? 500 × 1.4 = ?4,000
(a) A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a gamma of 4, 000 × 1.5 = +6,000. The delta of the whole portfolio (including traded options) is then:
4, 000 × 0.6 ? 450 = 1, 950
Hence, in addition to the 4,000 traded options, a short position in ￡1,950 is necessary so that the portfolio is both gamma and delta neutral.
(b) A long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a vega of 5, 000 × 0.8 = +4,000. The delta of the whole portfolio (including traded options) is then
5, 000 × 0.6 ? 450 = 2, 550
Hence, in addition to the 5,000 traded options, a short position in ￡2,550 is necessary so that the portfolio is both vega and delta neutral.
6.18. Let w1 be the position in the first traded option and w2 be the position in the second traded option. We require:
6, 000 = 1.5w1 + 0.5w2 4, 000 = 0.8w1 + 0.6w2
The solution to these equations can easily be seen to be w1 = 3,200, w2 = 2,400. The whole portfolio then has a delta of
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?450 + 3,200 × 0.6 + 2,400 × 0.1 = 1,710
Therefore the portfolio can be made delta, gamma and vega neutral by taking a long position in 3,200 of the first traded option, a long position in 2,400 of the second traded option and a short position in ￡1,710.
6.19. (Spreadsheet Provided) Consider the first week. The portfolio consists of a short position in 100,000 options and a long position in 52,200 shares. The value of the option changes from $240,053 at the beginning of the week to $188,760 at the end of the week for a gain of $51,293. The value of the shares change from 52,200 × 49 = $2,557, 800 to 52,200 × 48.12 = $2,511,864 for a loss of $45,936. The net gain is 51,293 ? 45,936 = $5,357. The gamma and theta (per year) of the portfolio are ?6,554.4 and 430,533 so that equation (6.2) predicts the gain as
430,533 ×1/52 + 0.5 × 6,554.4 × (48.12 ? 49)2 = 5,742 The results for all 20 weeks are shown in the following table.
Week Actual Gain ($) Predicted Gain ($) 1 5,357 5,742 2 5,689 6,093 3 ?19,742 ?21,084 4 1,941 1,572 5 3,706 3,652 6 9,320 9,191 7 6,249 5,936 8 9,491 9,259 9 961 870 10 ?23,380 ?18,992 11 1,643 2,497 12 2,645 1,356 13 11,365 10,923 14 ?2,876 ?3,342 15 12,936 12,302 16 7,566 8,815 17 ?3,880 ?2,763 18 6,764 6,899 19 4,295 5,205 20 4,804 4,805
Chapter 7: Interest Rate Risk
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7.15. The bank has an asset-liability mismatch of $25 billion. The profit after tax is currently 12% of $2 billion or $0.24 billion. If interest rates rise by X% the bank's before-tax loss (in billions of dollars) is 25 × 0.01 × X = 0.25X. After taxes this loss becomes $0.7 × 0.25X = 0.175X. The bank's return on equity would be reduced to zero when 0.175X = 0.24 or X = 1.37. A 1.37% rise in rates would therefore reduce the return on equity to zero. 7.17.
(a) The duration of Portfolio A is
1?2000e?0.1?1?10?6000e?0.1?10?5.95
2000e?0.1?1?6000e?0.1?10Since this is also the duration of Portfolio B, the two portfolios do have the same duration. (b) The value of Portfolio A is
2000e?0.1×1 + 6000e?0.1×10 = 4,016.95
When yields increase by 10 basis points its value becomes
2000e?0.101×1 + 6000e?0.101×10 = 3,993.18 The percentage decrease in value is
23.77?100= 0.59
4,016.95The value of Portfolio B is
5000e?0.1×5.95 = 2,757.81
When yields increase by 10 basis points its value becomes
5000 e?0.101×5.95 = 2,741.45 The percentage decrease in value is
16.36?100?0.59
2,757.81The percentage changes in the values of the two portfolios for a 10 basis point increase in yields are therefore the same.
(c) When yields increase by 5% the value of Portfolio A becomes
2000e?0.15×1 + 6000e?0.15×10 = 3,060.20 and the value of Portfolio B becomes
5000e-0.15×5.95 = 2,048.15
The percentage reductions in the values of the two portfolios are: Portfolio A:
956.75?100 = 23.82
4,016.95Portfolio B:
709.66?100 = 25.73
2,757.81
7.18. For Portfolio A the convexity is
12?2000e?0.1?1?102?6000e?0.1?10?55.40
2000e?0.1?1?6000e?0.1?1014 / 40
For portfolio B the convexity is 5.952 or 35.4025 The percentage change in the two portfolios predicted by the duration measure is the same and equal to ?5.95×0.05 = ?0.2975 or –29.75%. However, the convexity measure predicts that the percentage change in the first portfolio will be
?5.95 × 0.05 + 0.5 × 55.40 × 0.052 = ?0.228 and that for the second portfolio it will be
?5.95 × 0.05 = 0.5 × 35.4025 × 0.052 = ?0.253
Duration does not explain the difference between the percentage changes. Convexity explains part of the difference. 5% is such a big shift in the yield curve that even the use of the convexity relationship does not give accurate results. Better results would be obtained if a measure involving the third partial derivative with respect to a parallel shift, as well as the first and second, was considered.
7.18. The proportional change in the value of the portfolio resulting from the specified shift is
?(2.0 × 9e + 1.6 × 8e + 0.6 × 7e + 0.2 × 6e ? 0.5 × 5e ? 1.8 × 3e) = ?28.3e
The shift is the same as a parallel shift of 6e and a rotation of ?e. (The rotation is of the same magnitude as that considered in the text but in the opposite direction). The total duration of the portfolio is 0.2 and so the percentage change in the portfolio arising from the parallel shift is ?0.2×6e = ?1.2e. The percentage change in the portfolio value arising from the rotation is ?27.1e. (This is the same as the number calculated at the end of Section 7.6 but with the opposite sign.) The total percentage change is therefore ?28.3e, as calculated from the partial durations.
7.19. The delta with respect to the first factor is
0.21×5+0.26×(?3)+0.32×(?1)+0.35×2+0.36×5+0.36×7+0.36×8 = 7.85
Similarly, the deltas with respect to the second and third factors are 1.18 and –1.24, respectively. The relative importance of the factors can be seen by multiplying the factor exposure by the factor standard deviation. The second factor is about (1.18×6.05)/(7.85×17.49) = 5.2% as important as the first factor. The third factor is about (1.24×3.10)/(1.18×6.05) = 53.8% as important as the second factor.
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风险管理与金融机构-约翰-第二版-答案
