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1、Chapter 8: How Traders Manage Their Risks1交易组合价值减少10500美元。8.2当波动率变化2%时，交易组合价格增长200X2 = 400美元。8. 3两种情形均为0. 5*30*4=60美元8.4 1000份期权短头寸的Delta等于-700,可以通过买入700份股票的形式使 交易组合到达Delta中性。8. 5 Theta为TOO的含义是指在股价与波动率没有变化的情况下，期权价格每天 下降100美元。假如交易员认为股价及隐含波动率在将来不会改变，交易员可以 卖出期权，并且Theta值越高越好。8. 6当一个期权承约人的Gamma绝对值较大，Gamm
2、a本身为负，并且Delta等于 0,在市场变化率较大的情况下，期权承约人会有较大损失。8. 8看涨及看跌期权的多头头寸都具备正的Gamma,由图6. 9可以看出，当Gamma 为正时，对冲人在股票价格变化较大时会有收益,而在股票价格变化较小时会有 损失，因此对冲人在（b）情形收益更好，当交易组合包含期权的空头头寸时， 对冲人在（a）情形收益会更好。8. 9 Delta的数值说明当欧元汇率增长0. 01时一，银行交易价格会增加 0.01*30000=300美元,Gamma的数值说明,当欧元价格增长0. 01时,银行交易 组合的Delta会下降0. 01*80000=800美元；为了做到Delta
3、中性，我们应该卖出30000欧元；当汇率增长到0 93时,我们期望交易组合的Delta下降为（0. 93-0.9） *80000=24000,组合价值变为27600。为了维持Delta中性,银行应该对2400 数量欧元空头头寸进行平仓，这样可以保证欧元净空头头寸为27600o当一个交易组合的Delta为中性，同时Gamma为负时资产价格有一个较大变动时 会引起损失。因此银行可能会蒙受损失。8.15. The gamma and vega of a delta-neutral portfolio are 50 per $ per $ and 25 per %, respectively. Est
4、imate what happens to the value of the portfolio when there is a shock to the market causing the underlying asset price to decrease by $3 and its volatility to increase by 4%.With the notation of the text, the increase in the value of the portfolio is0.5 x gammer (AS) + vegax AcThis is0.5 x 50 x 32
5、+ 25 x 4 = 325The result should be an increase in the value of the portfolio of $325.8.16.Consider a one-year European call option on a stock when the stock price is $30, the strike price is $30, the risk-free rate is 5%, and the volatility is 25% per annum. Use the DerivaGem software to calculate t
6、he price, delta, gamma, vega, theta, and rho of the option. Verify that delta is correct by changing the stock price to $30.1 and recomputing the option price. Verify that gamma is correct by recomputing the delta for the situation where the stock price is $30.1. Carry out similar calculations to ve
7、rify that vega, theta, and rho are correct.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. Delt
8、a predicts a change in the option price of 0.6274 x 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 x 0.1 = 0.005 which is (to three decimal places) th
9、e 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
10、 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 0.1512.8.17.A financial institution has the following portfolio of over-the-counter options on sterling:TypePositionDelta
11、of OptionGamma of OptionVega of OptionCall-1,0000.502.21.8Call-5000.800.60.2Put2,000-0.401.30.7Call-5000.701.81.4A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.(a) What position in the traded option and in sterling would make the portfolio both gamma neutral and
12、delta neutral?(b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?The delta of the portfolio is-19 000 x 0.50 - 500 x 0.80 - 2,000 x (-0.40) - 500 x 0.70 = -450The gamma of the portfolio is-l5000 x 2.2 - 500 x 0.6 - 2,000 x 1.3 500 x 1.
13、8 = -6,000The vega of the portfolio is1, 000 x 1.8 - 500 x 0.2 - 2,000 x 0.7 - 500 x 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 x 1.5 = +6,000. The delta of the whole portfolio (including traded options) i
14、s then:4, 000 x 0.6-450= 1,950Hence, 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 x 0.8
15、 二 +4,000. The delta of the whole portfolio (including traded options) is then5,000 x 0.6 450 = 2,550Hence, 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.8.18. Consider again the situation in Problem 8.17. Suppose
16、 that a second traded option with a delta of 0.1, a gamma of 0.5f and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega neutral?Let wi be the position in the first traded option and W2 be the position in the second traded option. We require:6, 000 = 1.5vvi + 0.5w24,
17、000 = 0.8wi + 0.6w2The solution to these equations can easily be seen to be w = 3,200, W2 = 2,400. The whole portfolio then has a delta of-450 + 3,200 x 0.6 + 2,400 xO.l = 1,710Therefore the portfolio can be made delta, gamma and vega neutral by taking a long position in 3,200 of the first traded op
18、tion, a long position in 2,400 of the second traded option and a short position in 1,710.8.19. (Spreadsheet Provided)Reproduce Table 82 (In Table 8.2, the stock position is rounded to the nearest 100 shares.) Calculate the gamma and theta of the position each week. Using the DerivaGem Applications B
19、uilders to calculate the change in the value of theportfolio each week (before the rebalancing at the end of the week) and check whether equation (8.2) is approximately satisfied, Note: DerivaGem produces a value of theta “per calendar day. The theta in equation 8.2 is “per year.)Consider the first
20、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 x 49 = $2,557, 800
21、 to 52,200 x 48.12 = $2,511,864 for a loss of $45,936. The net gain is 51,293 - 45,936 = $5,357. The gamma and theta (per year) of the portfolio are -6,554.4 and 430,533 so that equation (8.2) predicts the gain as430,533 xl/52 + 0.5 x 6,554.4 x (48.12 - 49)2 = 5,742The results for all 20 weeks are s
22、hown in the following table.WeekActual Gain ($)Predicted Gain ($)15,3575,74225,6896,093319,742-21,08441,9411,57253,7063,65269,3209,19176,2495,93689,4919,25999618701023,380-18,992111,6432,497122,6451,3561311,36510,92314-2,876-3,3421512,93612,302167,5668,815173,880-2,763186,7646,899194,2955,205204,8044,805