Create Synthetic Positions via Put Call Parity Put Call Parity is a theorem that defines a price relationship between a call option, put option and the underlying stock. 看涨 期权和看跌期权之间的平价关系是给看涨期权、看跌期权和标的股票之间的等价关系进行定价的原理。 Understanding the Put Call Parity relationship can help you connect the value between a call option, a put option and the stock. When you see how these building blocks are connected, you will be able to create other synthetic positions using various option and stock combinations. 了解看涨期权和看跌期权之间的平价关系有助于投资者将看涨期权、看跌期权和标的股票之间的关系关联起来。如果搞清楚这些交易组合的内部是如何关联的，就能通过各种期权和股票的组合合成其他种类的交易头寸。 These are the basic components for the put call parity formula: 看涨期权和看跌期权之间最基本的平价关系用等式表示如下： If you are long a call and short a put at the same strike price, in the same expiration month, you are effectively long the underlying shares at the strike price level. 如果买入的看涨期权和卖出的看跌期权的行权价和到期日均相同，那么该组合的损益形态实际上等同于以期权的行权价买入股票。 Using this basic equation, you can add/subtract components from one side to another to create synthetic payoffs of other option strategies. For example, move the put to the other side of the equation by adding it to both sides and subtract the stock leg from both sides, which will give you this: 用这个最基本的平价关系等式，通过将等式某一边的要素加减到等式的另一边，可以构建其他种类期权交易策略的收益形态。举个例子，通过在等式的两边同时加上看跌期权，然后在等式两边同时减去股票，那么就会将看跌期权挪到等式的另一边，新的等式为： Here is how these components work together: The conditions for the "official" theorem to hold true are; The options are of European style Identical strike price for both call and put options No brokerage or exchange fees (called a frictionless market) Interest rates remain constant until the expiration date The stock pays no dividends In practice, however, the Put Call Parity relationship is used for many different asset types as a means of gauging an approximate value of a call or a put relative to its other components. The original formula provides the basis and we'll take a look later in the article how to account for American style stock options that pay dividends. 但是在实际交易中，看涨期权和看跌期权之间的平价关系被用于很多种资产大类作为估算一个看涨期权或看跌期权价值大概水平的手段。最初始的公式提供了一个计算的基础，我们将在后文中探讨如果是美式股票期权在分红的情况下需要对公式做何种调整。 The formula supposes the existence of two portfolios that are of equal value at the expiration date of the options. The premise is that if the two portfolios have identical values at expiration then they must be worth the same value now. If one portfolio was worth more than the other then traders would buy the undervalued asset and sell the overvalued asset until no further opportunity exists - also referred to as the "no arbitrage" principle. 该公式假设存在两个投资组合，在期权到期日当天这两个投资组合的价值相等。前提是如果这两个投资组合在期权的到期日当天价值相等，那么在交易日当天这两个投资组合的价值也要相等。如果其中一个投资组合的价值更高那么投资者就会买入另一个价值被低估的资产并且卖掉价值高估的资产，直到所有的套利空间消失，这也被称为“无套利”原则。 This therefore means that buying a call and put at the same strike price with the same expiration date will have the same value as the stock price minus the strike price. Given this, the payoff profile of each side will also be the same and we can see this with a synthetic long stock profile, which is long call and short put. 因此这就意味着，如果买入的看涨期权和卖出的看跌期权的行权价和到期日均相同，那么该组合的价值等于股价减去期权的行权价。即 Call - Put = Stock - Strike 有鉴于此，等式两边的损益也应该相等，这样做的结果是相当于合成了一个股票多头仓位，而实际上是买入了一个看涨期权+卖出了一个看跌期权。 Let's look at some real world examples of put call parity to understand how prices fit together. 我们看一下真实的案例来了解评价等式关系的机理是如何运行的。 Take a look at the option series below for MSFT. As an example, let's look at the $26 strike and see if the prices in the market prove put call parity. We'll see if we can back out the price of the call option given the prices of the other components. 我们看看图中股价为26美元时的期权报价，看看 市场是否认可看涨期权和看跌期权之间存在平价关系。 If we rearrange the put call parity equation to solve for the call option we have; Call = Stock - Strike + Put 我们重新调整一下看涨期权和看跌期权的关系等式，求： 看涨期权的价值=股价-期权的行权价+看跌期权的价值 Entering in the values from the market; Call = 26.04 - 26.00 + 1.80 其中股价在上图中的右上角，为26.04；期权报价和行权价见红线标记 看涨期权的价值=26.04 - 26.00 + 1.80=1.84 Mmm. The last traded price of the call option in the market, however, is 1.66: a difference of 0.18. Why is this? 嗯，此时看涨期权的最新价为1.66，与计算结果1.84之间差了0.18。为啥是这样？ Well, as mentioned earlier, the basic formula we've used so far assumed European options on stocks that don't pay dividends. But MSFT is a company that does pay dividends to its' stock holders and the options traded are of American style. So how can we account for dividends with put call parity? 正如前面所述，所用的基本公式有一个前提是假设该股票为欧式期权，而且不分红。但是微软公司是向股东分红的，期权类型为美式期权，即期权在最终到期日前的任何一个交易日均可交割。那么我们如何根据分红因素对看涨期权和看跌期权的关系等式进行调整？ Put Call Parity with Dividends 如何根据分红因素对看涨期权和看跌期权的关系等式进行调整 As we know stocks pay dividends and these dividends affect the future valuation of the stock as money is being taken out of the company and paid to its' shareholders. Because options have an expiration date, we need to value the option not against the current price of the stock but against what the expected value will be at the expiration date. This is known as the forward price. 我们知道股票是会分红的，这些分红会影响股价未来的估值，因为分红资金被从公司的账上划走支付给了股东。因为期权是有到期日的，因此我们并不是基于当前的股价而是根据期权到期日当天股价的预期水平对期权进行定价，也就是股票的远期价格。 Let's take the basic put call parity formula Call - Put = Stock - Strike and expand on this to account for the underlying stocks' dividends and interest rates. This makes the stock component Stock + Interest - Dividends. So now we have; 我们在Call - Put = Stock - Strike这个体现看涨期权和看跌期权的平价关系的基本关系等式中加入标的股票的分红和利率等因素，也就是股票部分应为股价+无风险利息-分红，经调整后该公式为： Call - Put = (Stock + Interest - Dividends) - Strike 看涨期权的价值-看跌期权的价值=（股价+无风险利息-分红）-期权的行权价 And finally, as the strike price is the expected exercise price in the future we need to discount that value by the interest rates to a present value by dividing the strike by e ^ rt. 最后，由于行权价是在未来到期日预期将行使的股价，因此我们将用无风险利率将行权价贴现到交易日当天，即用行权价除以e^无风险利率*期限 The complete formula will now read; Call - Put = (Stock + Interest - Dividends) - PV(Strike) Call - Put = FV(Stock) - PV(Strike) Where FV = Future Value and PV = Present Value. 看涨期权的价值-看跌期权的价值=（股价+无风险利息-分红）-期权行权价的现值 看涨期权的价值-看跌期权的价值=股价的现值-期权行权价的现值 Now going back to our MSFT example, let's apply dividends and interest rates and see if the market agrees with put call parity. 现在回到微软股价的案例，把无风险利率和分红因素考虑进去看看市场是否认可在微软股票的看涨期权和看跌期权之间存在平价关系。 First, we need to check if MSFT is paying a dividend prior to the expiration of the options. For US stocks we can find this information easily on Yahoo under "company events". 首先，我们需要看一下在期权到期之前微软是否准备分红。如果是美国的股票，可在雅虎财经网站的“公司事项”栏目中轻松找到分红事项。 MSFT went ex-dividend on the 15th November for a payment of 0.20. Looking back we can see that the data shown on Yahoo confirms that the stock was adjusted on the 15th for a 0.20 dividend; 微软在11月15日每股分红0.20美元，我们回过头来能在雅虎财经的“公司事项”中看到11月15日微软股价已经对0.20美元的分红进行了调整。 Next, we'll add the dividend information into our put call parity equation assuming 0.25% interest rates (World Interest Rates Table) and 94 days until the expiration date. 下面，我们将把分红信息放进看涨期权和看跌期权的平价关系等式中，假设美元的无风险贴现利率为0.25%，到期权的到期日为止还有94天。 Call = FV(Stock) - PV(Strike) + Put FV(Stock) = Stock + Interest - Dividends
FV(Stock) = 26.04 * (1.0025 ^ (94/365)) - 0.20
FV(Stock) = 25.85675
PV(Strike) = 26.00 / (1.0025 ^ (94/365))
PV(Strike) = 25.98329
Call = 25.85675 - 25.98329 + 1.80 看涨期权的价值=股价的现值-期权行权价的现值+看跌期权的价值 股价的现值=26.04*1.0025^(94/365)-0.2=25.85675 期权行权价的现值=26/(1.0025^(94/365))=25.98329 最终，看涨期权的价值=25.85675-25.98329+1.8=1.673463 Versus the actual market price of 1.66. Taking into account brokerage and exchange fees I'd say there is no room to profit here hence proving put call parity exists for American options. 市场上的实际报价为1.66，与计算结果有点偏差，但如果考虑交易点差和手续费等因素，我认为这中间没有套利空间。因此可以证明在美式看涨期权和美式看跌期权之间存在平价关系。 按照原文的方法，我们来验证一下微软当前的看涨期权和看跌期权之间是否存在平价关系。 微软当前的股价为112.14，因此平价期权的行权价也应为112 当前股价为112.14，行权价为112，期权的到期日为2018-9-28，交易日为美国当地时间2018-9-17，期权有效期还剩11天，无风险利率以纽约联邦储备银行网站上公布的隔夜联邦基金利率最新价1.92%为准 2018年8月15日微软每股分红0.42美元，但交易日为2018-9-17，因此到2018-9-28日到期日为止没有分红因素影响期权的定价 看涨期权的价值=未来股价的现值-期权行权价的现值+看跌期权的价值 未来股价的现值=112.14*1.02^(11/365)-0.00=111.786944 期权行权价的现值=112/(1.02^(11/365))=111.933179 最终，看涨期权的价值=111.786944-111.933179+1.41=1.673463 比市场报价1.65略有差异，考虑到交易成本，应不存在大的套利空间 | 
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