*The pay-off of an call option = max(ST-X,0)
看涨期权的盈利情况为max(ST-X,0)
#The pay-off of a put option = max(X- ST,0)
看跌期权的盈利情况为max(X-ST,0)
In the above table we can summarize our findings that when stock price is more than the strike price (X), the portfolios are worth the stock or share price (ST) and when the stock price is lower than the strike price, the portfolios are worth the strike price (X). In other words, both the portfolios are worth max(ST, X).
上表总结了一下我们的发现：当股价高于行权价X时，投资组合的总值等于票价ST；当股价低于行权价时，投资组合的总值等于行权价X。换句话说，这两个组合的总值均等于ST和X中最大的那个。
Portfolio A: When, ST > X, it is worth ST,
对于组合A来说，如果ST > X，组合的总值等于ST
Portfolio B: When, ST
对于组合B来说，如果ST
Since, both the portfolios have identical values at time T, they must therefore have similar or identical values today (since the options are European, it cannot be exercised prior to time T). And if this is not true an arbitrageur would exploit this arbitrage opportunity by buying the cheaper portfolio and selling the costlier one and book an arbitrage (risk free) profit.
由于这两个组合在到期日T的当天价值相等，因此这两个投资组合当前的现值必须相似或相等（因为期权是欧式期权，不能在到期日T之前行权）。如果不是这样，那么套利机会就会出现，套利者将买入低估值的投资组合，卖出高估值的投资组合以获取套利利润。
This brings us to a conclusion that today portfolio A should be equal to Portfolio B. or,C0+X*e-r*t = P0+S0
因此我们得出结论，
今天，投资组合A的价值应该等于投资组合B，即C0+X*e-r*t = P0+S0
Arbitrage Opportunity through Put-Call Parity
看涨期权和看跌期权平价关系中的套利机会
Let’s take an example to understand the arbitrage opportunity through put-call parity.
先看一个例子以理解看涨期权和看跌期权平价关系中的套利机会。
Suppose, share price of a company is $80/-, the strike price is $100/-, the premium (price) of a six month call option is $5/- and that of a put option is $3.5/-. The risk free rate in the economy is 8% per annum.
假设，股价
@80美元，期权为6个月的看涨期权的期权费
@5美元，看跌期权的期权费@3.5美元，行权价均为
@100，无风险利率为年化8%。
Now, as per the above equation of put-call parity, value of the combination of call option price and the present value of strike would be,
现在，根据看涨期权和看跌期权平价关系的等式，看涨期权和相当于行权价金额的现值加在一起应等于，
C0+X*e-r*t = 5+100*e-0.08*0.5
= 101.08
And value of the combination of put option and share price is
看跌期权和股价加在一起等于，
P0+S0 = 3.5+80
= 83.5
Here, we can see that first portfolio is overpriced and can be sold (an arbitrageur can create a short position in this portfolio) and second portfolio is relatively cheaper and can be bought (arbitrageur can create a long position) by the investor in order to exploit arbitrage opportunity.
我们可以见到第一个投资组合被高估了，套利者可以卖掉该组合；第二个投资组合相对而言低估了一些，套利者可以借机买入以获取利润。
This arbitrage opportunity involves buying a put option and a share of the company and selling a call option.
套利操作为买入看跌期权和股票，同时卖出看涨期权。
Let’s take this further, by shorting the call option and creating a long position in put option along with share would require below calculated funds to be borrowed by an arbitrageur at risk free rate i.e.
更进一步，一方面卖出看涨期权，另一方面持有看跌期权并做多股票，这样的组合操作所需要的资金量将低于套利者为了套利而以无风险利率借入的资金量。如下：
= -5+3.5+80
= 78.5
Hence, an amount of $78.5 would be borrowed by the arbitrageur and after six months this needs to be repaid. Hence, the repayment amount would be
因此，套利者需要借入78.5美元，6个月后偿还，偿还的金额应为，
= 78.5*e0.08*0.5
= 81.70
Also, after six months either the put or call option would be in the money and will be exercised and arbitrageur would get $100/- from this. The short call and long call put option position would therefore leads to the stock being sold for $100/-. Hence, the net profit generated by the arbitrageur is
6个月后，要么是看涨期权要么是看跌期权进入价内期权状态，套利者应会从中拿到100美元。做空看涨期权+持有看跌期权+做多股票的操作最终将股票
@100美元被卖出。因此，套利者的净收益应为，
= 100 – 81.70
= $18.30
The above cash flows are summarized in the Table 2:
以上现金流的发生情况示在下表2：
Table: 2
Steps involved in arbitrage position
套利交易操作步骤
Cost involved
涉及到的成本
Borrow $78.5 for six months and create a position by selling one call option for $5/- and buying one put option for $3.5/- along with a share for $80/-i.e. (80+3.5-5)
借入78.5，期限6个月，卖出看涨期权，期权费@5，买入看跌期权@3.5，股价
@80，即80+3.5-5=78.50
-81.7
After six months, if share price is more than the strike price, call option would be exercised and if it is below strike price then put option would be exercised
6个月后，如股价超过行权价100，看涨期权将被行权；如低于100美元，看跌期权将被行权
100
Net Profit (+) / Net Loss (-)损益净值
18.3
The Other side of Put-Call parity
看涨期权和看跌期权平价关系的另一面
Put-Call parity theorem only holds true for European style options as American style options can be exercised at any time prior to its expiry.
看涨期权和看跌期权平价关系的原理只对欧式期权有效，因为美式期权可以在到期日前的任何一天行权。
The equation which we have studied so far is
到目前为止，我们得出的公式为：
C0+X*e-r*t= P0+S0
This equation is also called as Fiduciary Call is equal to Protective Put.
Here, the left side of the equation is called Fiduciary Callbecause in fiduciary call strategy, an investor limits its cost associated with exercising the call option (as to fee for subsequently selling an underlying which has been physical delivered if the call is exercised).
这个等式意味着Fiduciary看涨期权组合的期权费与保护性看跌期权组合的期权费相等（译者注：Fiduciary Call就是不买入股票，而是改为持有股票的看涨期权，同时用节省下来的相当于股价的资金买入零息债券的交易策略；保护性看跌期权是股票加看跌期权组合，即买入一只股票，同时买入该股票的看跌期权）。等式的左边被称为Fiduciary看涨期权组合，该策略的投资者行使看涨期权的成本有限，也就是在当看涨期权被行使的情况下，卖出标的资产以进行实物交割的费用。
The right side of the equation is called Protective Putbecause in protective put strategy an investor is purchasing put option along with a share (P0+S0). In case, share prices goes up the investor can still minimizes their financial risk by selling shares of the company and protects their portfolio and in case the share prices goes down he can close his position by exercising the put option.
等式的右边被称为保护性看跌期权组合，该策略的投资者在买入股票的同时持有看跌期权，即P0+S0。如果股价上涨，投资者人可以卖出股票以最小化投资组合的风险；如果股价下跌，投资者人可以行使看跌期权以平掉股票仓位。
For example:-
示例
Suppose strike price is $70/-, Stock price is $50/-, Premium for Put Option is $5/- and that of Call Option is $15/-. And suppose that stock price goes up to $77/-.
In this case, the investor will not exercise its put option as the same is out of the money but will sell its share at current market price (CMP) and earn the difference between CMP and initial price of stock i.e. Rs.7/-. Had the investor not been purchased sock along with the put option, he would have been end up incurring loss of his premium towards option purchase.
这种情况下，投资者不会行使看跌期权，因为看跌期权此时已经成为价外期权，但是会以当前的市场价卖出股票，赚取当前股价与初始时股价之间的价差77-70=7美元。如果投资者在买入看跌期权的同时没有买入股票，那么投资将以损失买入期权时支付的期权费而告终。
Determining Call options & Put options premium
如何确定看涨期权和看跌期权的期权费
We can rewrite the above equation in two different ways as mentioned below.
以上期权费定价关系等式可用以下两个公式表达：
P0= C0+X*e-r*t-Sand
C0= P0+S0-X*e-r*t
In this way, we can determine price of a call option and put option.
这样，我们就可以确定看涨期权和看跌期权的期权费了。
For example, let’s assume price of a XYZ company is trading at Rs.750/- six months call option premium is Rs.15/- for strike price of Rs.800/-. What would be the premium for put option assuming risk free rate as 10%.
As per equation mentioned above in point no 1,
根据上面第一个公式，
P0= C0+X*e-r*t-S
= 15+800*e-0.10*0.05-750
= 25.98
原文此处公式有误，e-0.10*0.05应为e-0.10*0.5
= 15+800*e-0.10*0.5-750
= 25.98
Likewise, suppose that in the above example put option premium is given as $50 instead of call option premium and we have to determine call option premium.
同样，假设前例中看跌期权的期权费为50美元，那么看涨期权的期权费等于
C0= P0+S0-X*e-r*t
= 50+750-800*e-0.10*0.05
= 39.02
原文此处公式有误，e-0.10*0.05应为e-0.10*0.5
= 50+750-800*e-0.10*0.05
= 39.02
Impact of dividends on put-call parity
股票分红对看涨期权和看跌期权平价关系的影响
So far in our studies, we have assumed that there is no dividend paid on the stock. Therefore, the very next thing which we have to take into consideration is impact of dividend on put-call parity.
到目前为止，我们在研究中假设股票在期权有效期间没有分红。因此，下一个要考虑的因素是股票分红对看涨期权和看跌期权平价关系。
Since interest is a cost to an investor who borrows funds to purchase stock and benefit to investor who shorts the stock or securities by investing the funds.
利息对于借钱买入股票的投资者来说意味着成本，对于做空股票或证券同时将资金用于投资获利的投资者来说意味着利好。
Here we will examine how the Put-Call parity equation would be adjusted if stock pays dividend. Also, we assume that dividend which is paid during the life of the option is known.
下面我们看一下如果股票有分红，那么看涨期权和看跌期权平价关系的等式应如何调整。此外，假设在期权存续期内股票分红金额是已知的。
Here, the equation would be adjusted with the present value of dividend. And along with call option premium, the total amount to be invested by the investor is cash equivalent to present value of zero coupon bond (which is equivalent to strike price) and present value of dividend. Here, we are making adjustment in fiduciary call strategy. The adjusted equation would be
这样，等式将对股票分红金额的现值做出调整。加上看涨期权的期权费，投资金额应等于零息债券的现值和股票分红的现值的总和。下面我们对fiduciary看涨期权策略进行调整，调整后的等式为
C0+(D+X*e-r*t) = P0+ S0 where,
其中，
D = Present value of dividends during the life of
D=期权存续期内股票分红的现值
Let’s adjust the equation for both the scenarios.
我们在两个变形后的公式中均根据分红的现值调整一下。
For example, suppose the stock pays $50/- as dividend then, adjusted put option premium would be
例如，股票分红为50美元，那么调整后的看跌期权的期权费为：
P0= C0+(D+X*e-r*t) – S0
= 15+(50*e-0.10*0.5+800*e-0.10*0.5)-750
= 73.54
We can adjust the dividends in other way also which will yield the same value. The only basic difference between these two ways are while in first one we have added the dividends amount in strike price, in the other one we have adjusted the dividends amount directly from the stock.
我们可以用另一个公式对分红金额进行调整，但结果也一样。这两个公式本质上的唯一区别在于第一个公式是我们将分红金额加到了行权价中，另一个公式是直接在股价中将分红金额剔除。
P0= C0+X*e-r*t– S0-(S0*e-r*t),
In the above formula we have deducted the dividends amount (PV of dividends) directly from the stock price. Let’s look at the calculation through this formula
在这个公式中我们将分红金额的现值直接从股价中剥离出去，结果：
= 15+800*e-0.10*0.5-750-(50*e-0.10*0.5)
= 73.54
Concluding Remarks
结论
Put-Call parity establishes relationship between the prices of a European put options and calls options having same strike prices, expiry and underlying.
看涨期权和看跌期权的平价关系在具有相同的行权价，到期日和标的资产的欧式看涨期权和看跌期权的期权费之间建立了关联关系
Put-Call Parity does not hold true for American option as American option can be exercised at any time prior to its expiry.
看涨期权和看跌期权的平价关系不适用于美式期权，因为美式期权可以在到期日前的任何一天行权
Equation for put-call parity is C0+X*e-r*t = P0+S0.
看涨期权和看跌期权的平价关系用等式表达为C0+X*e-r*t = P0+S0
In put-call parity, Fiduciary Call is equal to Protective Put.
在看涨期权和看跌期权的平价关系中，Fiduciary看涨期权策略中看涨期权的期权费与看跌期权的相等
Put-Call parity equation can be used to determine the price of European call and put options
看涨期权和看跌期权平价关系的等式可被用来给欧式看涨期权和看跌期权定价
Put-Call parity equation is adjusted, if stock pays any dividends.
如果股票有分红，那么看涨期权和看跌期权平价关系的等式可调整
转播
