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期权市场的无风险套利
中文版
期权市场中的套利实例
为了清楚地说明，让我们通过一个现实的例子来展示套利。
期权市场中的套利实例
假设市场上有以下价格：
- 标的股票价格：100美元
- 欧式看涨期权（行权价100美元，3个月到期）：8美元
- 欧式看跌期权（行权价100美元，3个月到期）：5美元
- 无风险利率：2%（年化）
我们使用一个经典的套利策略，称为“转换套利”：
转换套利策略
转换套利涉及买入标的股票、买入看跌期权并卖出看涨期权。如果期权与标的股票之间存在定价错误，此策略可以锁定无风险利润。
逐步过程：
-
买入标的股票：
- 以100美元购买1股XYZ公司股票。
-
买入欧式看跌期权：
- 以5美元购买一个行权价为100美元的看跌期权。
-
卖出欧式看涨期权：
- 以8美元卖出一个行权价为100美元的看涨期权。
总初始投资：
- 购买股票：100美元
- 购买看跌期权：5美元
- 卖出看涨期权：-8美元（你收到8美元）
总初始投资 = 100美元（股票） + 5美元（看跌期权） - 8美元（看涨期权） = 97美元
到期时的收益：
无论股票价格在到期时是多少，你的头寸都是对冲的：
-
如果股票价格高于100美元（例如110美元）：
- 看跌期权作废。
- 看涨期权被执行，你以100美元卖出股票。
- 你收到100美元。
-
如果股票价格低于100美元（例如90美元）：
- 看跌期权被执行，你以100美元卖出股票。
- 看涨期权作废。
- 你收到100美元。
在这两种情况下，你到期时都得到100美元。
利润计算：
- 到期时收到的总金额：100美元
- 总初始投资：97美元
利润 = 100美元 - 97美元 = 3美元
这是由于期权相对于标的股票的初始定价错误而获得的无风险利润3美元。
无套利例子
在一个无套利市场中，不会存在这样的差异。看涨期权和看跌期权的价格会与股票价格和无风险利率对齐，以便上述策略不会产生无风险利润。
无套利条件下的期权定价实际例子
在无套利条件下，期权的价格应该符合以下无套利定价公式：
C − P = S − K × e − r t C - P = S - K \times e^{-rt} C−P=S−K×e−rt
其中：
- ( C ) 是看涨期权的价格
- ( P ) 是看跌期权的价格
- ( S ) 是股票价格
- ( K ) 是行权价
- ( r ) 是无风险利率
- ( t ) 是到期时间
实例说明
假设以下市场条件：
- 标的股票价格（S）：100美元
- 行权价（K）：100美元
- 无风险利率（r）：2%（年化）
- 到期时间（t）：3个月（即0.25年）
我们需要验证期权价格是否满足无套利条件。假设当前市场价格：
- 看涨期权价格（C）：8美元
- 看跌期权价格（P）：4.5美元
现在，我们将这些数值代入无套利定价公式来验证：
计算无套利定价公式
首先计算右边的表达式 ( K × e − r t K \times e^{-rt} K×e−rt )：
K × e − r t = 100 × e − 0.02 × 0.25 K \times e^{-rt} = 100 \times e^{-0.02 \times 0.25} K×e−rt=100×e−0.02×0.25
计算 ( e − 0.02 × 0.25 e^{-0.02 \times 0.25} e−0.02×0.25 ):
e − 0.005 ≈ 0.995 e^{-0.005} \approx 0.995 e−0.005≈0.995
因此：
100 × 0.995 = 99.5 100 \times 0.995 = 99.5 100×0.995=99.5
代入公式：
C − P = S − K × e − r t C - P = S - K \times e^{-rt} C−P=S−K×e−rt
左边是：
8 - 4.5 = 3.5
右边是：
100 - 99.5 = 0.5
显然，这里不满足无套利条件。
调整后的无套利定价
为了满足无套利条件，我们需要调整看跌期权的价格，使公式成立：
C − P = S − K × e − r t C - P = S - K \times e^{-rt} C−P=S−K×e−rt
即：
8 − P = 100 − 99.5 8 - P = 100 - 99.5 8−P=100−99.5
8 − P = 0.5 8 - P = 0.5 8−P=0.5
P = 8 − 0.5 = 7.5 P = 8 - 0.5 = 7.5 P=8−0.5=7.5
所以，在无套利条件下，看跌期权的价格应为7.5美元。
总结
- 看涨期权价格（C）：8美元
- 看跌期权价格（P）：7.5美元
在这个调整后的例子中：
8 − 7.5 = 100 − 99.5 8 - 7.5 = 100 - 99.5 8−7.5=100−99.5
0.5 = 0.5 0.5 = 0.5 0.5=0.5
这满足了无套利条件。因此，市场在这种情况下没有套利机会，所有期权价格是合理的。
结合前面的无风险套利实例
前面的套利例子中，通过构建保护性看跌和备兑看涨策略，我们发现期权价格存在偏差，导致无风险利润。现在，我们通过无套利条件调整了看跌期权的价格，使其符合市场有效性，从而消除了套利机会。
这个例子说明了在无套利市场中，期权价格如何通过无套利定价公式保持一致，以防止套利机会。
英文版
Example of Arbitrage in the Options Market
To illustrate more clearly, let’s go through a more realistic example of arbitrage.
Example of Arbitrage in the Options Market
Assume the following market prices:
- Underlying stock price: $100
- European call option (strike price $100, 3 months to expiry): $8
- European put option (strike price $100, 3 months to expiry): $5
- Risk-free interest rate: 2% (annualized)
We will use a classic arbitrage strategy known as a “conversion arbitrage.”
Conversion Arbitrage Strategy
Conversion arbitrage involves buying the underlying stock, buying a put option, and selling a call option. If there is a pricing discrepancy between the options and the underlying stock, this strategy can lock in a risk-free profit.
Step-by-Step Process:
-
Buy the underlying stock:
- Purchase 1 share of XYZ company stock at $100.
-
Buy a European put option:
- Purchase a put option with a strike price of $100 for $5.
-
Sell a European call option:
- Sell a call option with a strike price of $100 for $8.
Total Initial Investment:
- Purchase of stock: $100
- Purchase of put option: $5
- Sale of call option: -$8 (you receive $8)
Total initial investment = $100 (stock) + $5 (put option) - $8 (call option) = $97
Payoff at Expiration:
Regardless of the stock price at expiration, your positions are hedged:
-
If the stock price is above $100 (e.g., $110):
- The put option expires worthless.
- The call option is exercised, and you sell the stock at $100.
- You receive $100.
-
If the stock price is below $100 (e.g., $90):
- The put option is exercised, and you sell the stock at $100.
- The call option expires worthless.
- You receive $100.
In both cases, you receive $100 at expiration.
Profit Calculation:
- Total amount received at expiration: $100
- Total initial investment: $97
Profit = $100 - $97 = $3
This $3 risk-free profit is due to the initial mispricing of the options relative to the stock.
Example of No-Arbitrage
In a no-arbitrage market, such discrepancies would not exist. The prices of call and put options would align with the stock price and the risk-free interest rate, preventing such risk-free profits from being made.
Example of No-Arbitrage Pricing in the Options Market
In a no-arbitrage market, option prices should satisfy the following no-arbitrage pricing formula:
C − P = S − K × e − r t C - P = S - K \times e^{-rt} C−P=S−K×e−rt
where:
- ( C ) is the price of the call option
- ( P ) is the price of the put option
- ( S ) is the stock price
- ( K ) is the strike price
- ( r ) is the risk-free interest rate
- ( t ) is the time to expiration
Example Illustration
Assume the following market conditions:
- Stock price (S): $100
- Strike price (K): $100
- Risk-free interest rate ( r): 2% (annualized)
- Time to expiration (t): 3 months (or 0.25 years)
We need to verify if the option prices meet the no-arbitrage condition. Assume the current market prices are:
- Call option price ( C): $8
- Put option price ( P): $4.5
Let’s plug these values into the no-arbitrage pricing formula to verify:
Calculating the No-Arbitrage Pricing Formula
First, calculate the right side of the equation ( K × e − r t K \times e^{-rt} K×e−rt ):
K × e − r t = 100 × e − 0.02 × 0.25 K \times e^{-rt} = 100 \times e^{-0.02 \times 0.25} K×e−rt=100×e−0.02×0.25
Calculate ( e − 0.02 × 0.25 e^{-0.02 \times 0.25} e−0.02×0.25 ):
e − 0.005 ≈ 0.995 e^{-0.005} \approx 0.995 e−0.005≈0.995
Thus:
100 × 0.995 = 99.5 100 \times 0.995 = 99.5 100×0.995=99.5
Substitute into the formula:
C − P = S − K × e − r t C - P = S - K \times e^{-rt} C−P=S−K×e−rt
Left side:
8 − 4.5 = 3.5 8 - 4.5 = 3.5 8−4.5=3.5
Right side:
100 − 99.5 = 0.5 100 - 99.5 = 0.5 100−99.5=0.5
Clearly, this does not satisfy the no-arbitrage condition.
Adjusted No-Arbitrage Pricing
To satisfy the no-arbitrage condition, we need to adjust the put option price so that the formula holds:
C − P = S − K × e − r t C - P = S - K \times e^{-rt} C−P=S−K×e−rt
So:
8 − P = 100 − 99.5 8 - P = 100 - 99.5 8−P=100−99.5
8 − P = 0.5 8 - P = 0.5 8−P=0.5
P = 8 − 0.5 = 7.5 P = 8 - 0.5 = 7.5 P=8−0.5=7.5
Therefore, under the no-arbitrage condition, the put option price should be $7.5.
Summary
- Call option price ( C): $8
- Put option price ( P): $7.5
In this adjusted example:
8 − 7.5 = 100 − 99.5 8 - 7.5 = 100 - 99.5 8−7.5=100−99.5
0.5 = 0.5 0.5 = 0.5 0.5=0.5
This satisfies the no-arbitrage condition. Thus, the market in this case has no arbitrage opportunities, and all option prices are fair.
Relating to the Previous Risk-Free Arbitrage Example
In the previous arbitrage example, we identified a pricing discrepancy through the protective put and covered call strategy, leading to a risk-free profit. Now, by adjusting the put option price to meet the no-arbitrage condition, we ensure market efficiency and eliminate the arbitrage opportunity.
This example illustrates how option prices, in a no-arbitrage market, are aligned by the no-arbitrage pricing formula to prevent arbitrage opportunities.
后记
2024年6月16日于上海。基于GPT4o模型。