推广做黄页网站,为什么用asp做网站,uml电子商务网站建设文档,网站建设平台分析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#xff0c;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#xff1a;
标的股票价格#xff1a;100美元欧式看涨期权#xff08;行权价100美元#xff0c;3个月…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−PS−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S100美元行权价K100美元无风险利率r2%年化到期时间t3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C8美元看跌期权价格P4.5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−rt100×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 99.5 100×0.99599.5
代入公式 C − P S − K × e − r t C - P S - K \times e^{-rt} C−PS−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−PS−K×e−rt
即 8 − P 100 − 99.5 8 - P 100 - 99.5 8−P100−99.5 8 − P 0.5 8 - P 0.5 8−P0.5 P 8 − 0.5 7.5 P 8 - 0.5 7.5 P8−0.57.5
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C8美元看跌期权价格P7.5美元
在这个调整后的例子中 8 − 7.5 100 − 99.5 8 - 7.5 100 - 99.5 8−7.5100−99.5 0.5 0.5 0.5 0.5 0.50.5
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: $100European call option (strike price $100, 3 months to expiry): $8European put option (strike price $100, 3 months to expiry): $5Risk-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: $100Purchase of put option: $5Sale 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: $100Total 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−PS−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): $100Strike price (K): $100Risk-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): $8Put 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−rt100×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 99.5 100×0.99599.5
Substitute into the formula: C − P S − K × e − r t C - P S - K \times e^{-rt} C−PS−K×e−rt
Left side: 8 − 4.5 3.5 8 - 4.5 3.5 8−4.53.5
Right side: 100 − 99.5 0.5 100 - 99.5 0.5 100−99.50.5
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−PS−K×e−rt
So: 8 − P 100 − 99.5 8 - P 100 - 99.5 8−P100−99.5 8 − P 0.5 8 - P 0.5 8−P0.5 P 8 − 0.5 7.5 P 8 - 0.5 7.5 P8−0.57.5
Therefore, under the no-arbitrage condition, the put option price should be $7.5.
Summary
Call option price ( C): $8Put option price ( P): $7.5
In this adjusted example: 8 − 7.5 100 − 99.5 8 - 7.5 100 - 99.5 8−7.5100−99.5 0.5 0.5 0.5 0.5 0.50.5
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。
