R74 练习: 运用买卖权平价进行期权复制
考纲范围
- explain put-call parity for European options
- explain put-call forward parity for European options
Q1.
According to the put-call parity, which of the following strategies can synthesize a call option?
A. long stock, short put, short bond.
B. short stock, long put, short bond.
C. long stock, long put, short bond.
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答案：C
解析：买卖权平价公式：，变形得 。即 long stock + long put + short bond（借钱）= synthetic call。加号代表long，减号代表short。
选项 判断 解析 A ✗ Short put方向错误，应该是long put B ✗ Short stock方向错误，应该是long stock C ✓ ：long stock + long put + short bond
Q2.
The European options with the underlying firm’s assets are used to explain the firm value under put-call parity. The firm is said to be insolvency when:
A. The strike price of the call option held by the shareholder is below the firm asset value.
B. The strike price of the put option held by the debtholder is above the firm asset value.
C. The strike price of the put option sold by the debtholder is above the firm asset value.
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答案：C
解析：在公司价值的期权分析框架下：股东持有看涨期权（行权价=债务面值），债权人相当于持有无风险债券+卖出看跌期权。当公司资产价值低于债务面值（即put的行权价高于资产价值）时，公司资不抵债（insolvency）。
选项 判断 解析 A ✗ 当行权价（债务面值）低于资产价值时，公司有偿付能力，不是资不抵债 B ✗ 债权人是”卖出”put，不是”持有”put C ✓ 债权人卖出的put行权价（债务面值）> 资产价值 → 公司资不抵债
Q3.
Consider a European call option with one month to expiration and an exercise price of $100 sold for $5. The current price of the underlying stock is $102 and the risk-free interest rate is 4%. If the put-call parity holds, the value of the corresponding put with the same exercise price is closest to:
A. $2.67 B. $3.00 C. $5.00
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答案：A
计算过程：
- 买卖权平价：
- PV(X) = \frac{100}{(1.04)^{1/12}} = \frac{100}{1.00327} = \99.67$
- p = 5 - 102 + 99.67 = \2.67$
选项 判断 解析 A ✓ p = 5 - 102 + 100/(1.04)^{1/12} = 5 - 102 + 99.67 = \2.67$ B ✗ 计算有误 C ✗ 计算有误
Q4.
Which of the following assumptions of put-call-forward parity is correct for a European option?
A. Arbitrage opportunities exist within the spot, forward, and option markets.
B. The value of a European put at expiration is the greater of either zero or the exercise price minus the value of the underlying.
C. The value of a European call at expiration is the greater of either zero or the exercise price minus the value of the underlying.
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答案：B
解析：欧式put到期价值 = ，即行权价减标的价格与零的较大值。买卖权远期平价的假设是无套利（不存在套利机会）。
选项 判断 解析 A ✗ 买卖权平价的假设恰恰是不存在套利机会（no arbitrage） B ✓ 正确。Put到期价值 = C ✗ Call到期价值 = ，不是
Q5.
Under put-call-forward parity, which of the following transaction is equivalent to a long position in a forward contract?
A. Short call, long put, long risk-free bond.
B. Long call, short put, short risk-free bond.
C. Long call, short put, long risk-free bond.
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答案：B
解析：买卖权远期平价：将 替换为 即可。。Long forward = Long call + Short put - PV(X)（即short bond）。
选项 判断 解析 A ✗ 这相当于short forward B ✓ Long forward = Long call + Short put + Short risk-free bond C ✗ Long bond方向错误，应该是short bond
Q6.
Which of the following statements regarding synthetic protective put and put-call forward parity is most inaccurate?
A. In a protective put strategy, a synthetic way to hold underlying at initial is to hold a risk-free bond and enter a forward contract to buy the underlying in the future.
B. Based on put-call forward parity, a long call and a short put have the same effect as a long forward and a short risk-free bond.
C. The risk-free bond held in synthetic protective put at initial has a par value equals to the underlying spot price.
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答案：C
解析：在合成保护性看跌策略中，持有的无风险债券面值等于远期价格的现值加上行权价的现值，而非标的资产的即期价格。
选项 判断 解析 A ✓ 正确。合成持有标的 = 持有无风险债券 + 远期合约买入标的 B ✓ 正确。Long call + Short put = Long forward + Short risk-free bond（基于远期平价） C ✗ 错误。无风险债券面值不等于标的即期价格，而是等于远期价格