## The binomial option pricing model – part 1

This is post #1 on the binomial option pricing model. Even though this is post #1, there are two previous posts with examples to illustrate how to price options using the one-period binomial pricing model (example of call and example of put). The purpose of post #1:

Post #1: Describe the option pricing formulas in the one-period binomial model.

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The one-period binomial option pricing model

We first consider the pricing of options on stock. The most important characteristic of the binomial option pricing model is that over a period of time, the stock price is assumed to follow a binomial distribution, i.e. the price of the stock can only take on one of two values – an upped value and a downed value. In this post, we describe how to price an option on a stock using this simplifying assumption of stock price movement.

Consider a stock with the following characteristics:

• The current share price is $S$.
• If the stock pays dividends, we assume the dividends are paid at an annual continuous rate at $\delta$.
• At the end of a period of length $h$ (in years), the share price is either $S_h=uS$ or $S_h=dS$, where $u$ is the up factor and $d$ is the down factor. The factor $u$ can be interpreted as one plus the rate of capital gain on the stock if the stock goes up. The factor $d$ can be interpreted as one plus the rate of capital loss if the stock goes down.
• If $\delta>0$, the end of period share price is $S_h=uS e^{\delta h}$ or $S_h=dS e^{\delta h}$. This is to reflect the gains from reinvesting the dividends. Of course if $\delta=0$, the share prices revert back to the previous bullet point.

The end of period stock prices are shown in the following diagram, which is called a binomial tree since it depicts the 2-state stock price at the end of the option period.

$\text{ }$
Figure 1 – binomial tree

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Now consider a European option (either call or put) on the stock described above. When the stock goes up, we use $C_u$ to represent the value of the option. When the stock goes down, we use $C_d$ to represent the value of the option. The following is the binomial tree for the value of the option.

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Figure 2 – option value tree

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Replicating Portfolio
The key idea to price the option is to create a portfolio consisting of $\Delta$ shares of the stock and the amount $B$ in lending. At time 0, the value of this portfolio is $C=\Delta S + B$. At time $h$ (the end of the option period), the value of the portfolio is

$\text{ }$
Time $h$ value of the replicating portfolio

$\displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h}) + B \ e^{r h}&\ \ \ \ \ \ \text{(when stock price goes down)}& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h}) + B \ e^{r h}&\ \ \ \ \ \ \text{(when stock price goes up)} \end{matrix}\right.$

$\text{ }$

This portfolio is supposed to replicate the same payoff as the value of the option. By equating the portfolio payoff with the option payoff, we obtain the following linear equations.

$\text{ }$

$\displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h}) + B \ e^{r h}=C_d&\ \ \ \ \ \ \text{ }& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h}) + B \ e^{r h}=C_u&\ \ \ \ \ \ \text{ } \end{matrix}\right.$

$\text{ }$

There are two unknowns in the above two equations. All the other items – stock price $S$, dividend rate $\delta$, and risk-free interest rate $r$ – are known. Solving for the two unknowns $\Delta$ and $B$, we obtain:

$\text{ }$
$\displaystyle \Delta=e^{-\delta h} \ \frac{C_u-C_d}{S(u-d)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
$\text{ }$

$\displaystyle B=e^{-r h} \ \frac{u \ C_d-d \ C_u}{u-d} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
$\text{ }$

Once the replication portfolio of $\Delta$ shares and $B$ in lending is determined, the price of the option (the value at time 0) is:

$\text{ }$
$C=\Delta S + B \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
$\text{ }$

After plugging in (1) and (2) into (3), the option price formula becomes:

$\text{ }$
$\displaystyle C=\Delta S + B=e^{-r h} \biggl(C_u \ \frac{e^{(r-\delta) h}-d}{u-d} +C_d \ \frac{u-e^{(r-\delta) h}}{u-d} \biggr) \ \ \ \ \ \ \ \ \ (4)$
$\text{ }$

The price of the option described above is $C$, either given by formula (3) or formula (4). One advantage of formula (4) is that it gives the direct calculation of the option price without knowing $\Delta$ and $B$. Of course, if the goal is to create a synthetic option for the purpose of hedging or risk management, it will be necessary to know the make up of the replicating portfolio.

The $\Delta$ calculated in (1) is also called the hedge ratio and is examined in greater details in in this subsequent post.

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Examples

Example 1
Let’s walk through a quick example to demonstrate how to apply the above formulas. Suppose that the future prices for a stock are modeled with a one-period binomial tree with $u=$ 1.3 and $d=$ 0.8 and having a period of 6 months. The current price of the stock is $50. The stock pays no dividends. The annual risk-free interest rate is $r=$ 4%. • Determine the price of a European 55-strike call option on this stock that will expire in 6 months. • Determine the price of a European 45-strike put option on this stock that will expire in 6 months. The two-state stock prices are$65 and $40. The two-state call option values at expiration are$10 and $0. Apply (1) and (2) to obtain the replicating portfolio and then the price of the call option. $\text{ }$ $\displaystyle \Delta=\frac{10-0}{65-40}=\frac{10}{25}=$ 0.4 $\displaystyle B=e^{-0.04(0.5)} \ \frac{1.3(0)-0.8(10)}{1.3-0.8}=-16 e^{-0.02}=$ -$15.68317877

The replicating portfolio consists of holding 0.4 shares and borrowing $15.68317877. Call option price = $50 \Delta+B=$$4.316821227

$\text{ }$

The 2-state put option values at expiration are $0 and$5. Now apply (1) and (2) and obtain:

$\text{ }$
$\displaystyle \Delta=\frac{0-5}{65-40}=\frac{-5}{25}=-0.2$

$\displaystyle B=e^{-0.04(0.5)} \ \frac{1.3(5)-0.8(0)}{1.3-0.8}=13 e^{-0.02}=$ $12.74258275 The replicating portfolio consists of shorting 0.2 shares and lending$12.74258275.

Put option price = $50 \Delta+B=$ $2.742582753 $\text{ }$ Example 1 is examined in greater details in this subsequent post. More Examples Two more examples are in these previous posts: ___________________________________________________________________________________ What to do if options are mispriced What if the observed price of an option is not the same as the theoretical price? In other words, what if the price of a European option is not given by the above formulas? Because we can always hold stock and lend to replicate the payoff of an option, we can participate in arbitrage when an option is mispriced by buying low and selling high. The idea is that if an option is underpriced, then we buy low (the underpriced option) and sell high (the corresponding synthetic option, i.e. the replicating portfolio). On the other hand, if an option is overpriced, then we buy low (the synthetic option) and sell high (the overpriced option). Either case presents risk-free profit. We demonstrate with the options in Example 1. Example 2 • Suppose that the price of the call option in Example 1 is observed to be$4.00. Describe the arbitrage.
• Suppose that the price of the call option in Example 1 is observed to be $4.60. Describe the arbitrage. For the first scenario, we buy low (the option at$4.00) and sell the synthetic option at the theoretical price of $4.316821227. Let’s analyze the cash flows in the following table. $\text{ }$ Table 1 – Arbitrage opportunity when call option is underpriced $\left[\begin{array}{llll} \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Sell synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Short 0.4 shares} & \text{ } & - \ 16 & - \ 26 \\ \ \ \ \ \text{Lend } \ 15.683 & \text{ } & + \ 16 & + \ 16 \\ \text{ } & \text{ } \\ \text{Buy call } & \text{ } & \ \ \ 0 & \ \ \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$ $\text{ }$ The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain:$4.316821227 – $4.00 =$0.316821227.

For the second scenario, we still buy low and sell high. This time, buy low (the synthetic call option at $4.316821227) and sell high (the call option at the observed price of$4.60). Let’s analyze the cash flows in the following table.

$\text{ }$

Table 2 – Arbitrage opportunity when call option is overpriced

$\left[\begin{array}{llll} \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Buy synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Long 0.4 shares} & \text{ } & + \ 16 & + \ 26 \\ \ \ \ \ \text{Borrow } \ 15.683 & \text{ } & - \ 16 & - \ 16 \\ \text{ } & \text{ } \\ \text{Buy call } & \text{ } & \ \ \ 0 & - \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$

$\text{ }$

The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain: $4.60 –$4.316821227 = $0.283178773. These two examples show that if the option price is anything other than the theoretical price, there are arbitrage opportunities and there is risk-free profit to be made. ___________________________________________________________________________________ How to construct a binomial tree In the binomial tree in Figure 1, we assume that the share price at expiration is obtained by multiplying the original share price by the movement factors of $u$ and $d$. The binomial tree in Figure 1 may give the impression that the choice of the movement factors $u$ and $d$ is arbitrary as long as the up factor is greater than 1 and the down factor is below 1. In the next post, we show that $u$ and $d$ have to satisfy the following relation, else there will be arbitrage opportunities. $\displaystyle d < e^{(r-\delta) h} < u \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$ Thus the choice of $u$ and $d$ cannot be entirely arbitrary. In particular the relation (5) shows that the future stock prices have to revolve around the forward price. $\displaystyle dS < Se^{(r-\delta) h} < uS \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)$ The purpose pf the factors $u$ and $d$ in the binomial tree is to incorporate uncertainty of the stock prices. In light of (6), we can set $u$ and $d$ by applying some volatility adjustment to $e^{(r-\delta) h}$. We can use the following choice of $u$ and $d$ to model the stock price evolution. $\displaystyle u = e^{(r-\delta) h \ + \ \sigma \sqrt{h}}$ $\displaystyle d = e^{(r-\delta) h \ - \ \sigma \sqrt{h}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)$ where $\sigma$ is the annualized standard deviation of the continuously compounded stock return, $\sigma \sqrt{h}$ is the standard deviation of the continuously compounded stock return over a period of length $h$. The standard deviation $\sigma$ measures how certain we are that the stock return will be close to the expected return. There will be a greater chance of a return far from the expected return if the stock has a higher $\sigma$. If $\sigma=0$, then there is no uncertainty about the future stock prices. The formula (7) shows that when $\sigma=0$, the future stock price is precisely the forward price on the stock. When the binomial tree is constructed using (7), the tree will be called a forward tree. A note on calculation. If a problem does not specific $u$ and $d$ but assume a standard deviation of stock return $\sigma$, then assume that the binomial tree is the forward tree. We now use a quick example to demonstrate how to price an option using the forward tree. Example 3 Everything is the same as Example 1 except that the up and down stock prices are constructed using the volatility $\sigma=$ 30% (the standard deviation $\sigma$). The following calculates the stock prices at expiration of the option. $\displaystyle uS = 50 \ e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}=$$63.06431255

$\displaystyle dS = 50 \ e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}=$ $41.25989534 $\displaystyle u=\frac{63.06431255}{50}=$ 1.261286251 $\displaystyle d=\frac{41.25989534}{50}=$ 0.825197907 Using formulas (1), (2) and (3), the following shows the replicating portfolio and the call option price. Note that the binomial tree is based on a different assumption than that in Example 1. The option price is thus different than the one in Example 1. $\text{ }$ $\displaystyle \Delta=\frac{8.064312548-0}{63.06431255-41.25989534}=$ 0.369847654 $\displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(0)-0.825197907(8.064312548)}{1.261286251-0.825197907}=$ –$14.95770971

The replicating portfolio consists of holding 0.369847654 shares and borrowing $14.95770971. Call option price = $50 \Delta+B=$$3.534672982

$\text{ }$

The following shows the calculation for the put option.

$\text{ }$
$\displaystyle \Delta=\frac{0-3.740104659}{63.06431255-41.25989534}=$ -0.171529678

$\displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(3.740104659)-0.825197907(0)}{1.261286251-0.825197907}=$ $10.60320232 The replicating portfolio consists of shorting 0.171529678 shares and lending$10.60320232.

Put option price = $50 \Delta+B=$ $2.026718427 $\text{ }$ ___________________________________________________________________________________ More examples We present two more examples in illustrating the calculation in the one-period binomial option model where the stock prices are modeled by a forward tree. Example 4 The stock price follows a 6-month binomial tree with initial stock price$60 and $\sigma=$ 0.3. The stock is non-dividend paying. The annual risk free interest rate is $r=$ 4%. What is the price of a 6-month 55-strike call option? Determine the replicating portfolio that has the same payoff as this call option.

We will use risk-neutral probabilities to price the option.

$\displaystyle uS = 60 \ e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}=$ $75.67717506 $\displaystyle dS = 60 \ e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}=$$49.51187441

$\displaystyle C_u=$ 75.67717506 – 55 = 20.67717506

$\displaystyle C_d=$ 0

$\displaystyle u=\frac{75.67717506}{60}=$ 1.261286251

$\displaystyle d=\frac{49.51187441}{60}=$ 0.825197907

$\displaystyle p^*=\frac{e^{(0.04-0) 0.5} - 0.825197907}{1.261286251 - 0.825197907}=$ 0.447164974

$\displaystyle 1-p^*=$ 0.552835026

$\displaystyle C=(p^* \times C_u + (1-p^*) \times C_d) e^{-0.02}=$ 9.063023234

$\text{ }$

$\displaystyle \Delta=\frac{20.67717506-0}{75.67717506-49.51187441}=$ 0.790251766

$\displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(0)-0.825197907(20.67717506)}{1.261286251-0.825197907}=$ –$38.35208275 The replicating portfolio consists of holding 0.79025 shares and borrowing$38.352.

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Example 5
The stock price follows a 3-month binomial tree with initial stock price $40 and $\sigma=$ 0.3. The stock is non-dividend paying. The annual risk free interest rate is $r=$ 5%. What is the price of a 3-month 45-strike put option on this stock? Determine the replicating portfolio that has the same payoff as this put option. The calculation is calculated as in Example 3. $\displaystyle uS = 40 \ e^{(0.05-0) 0.25 \ + \ 0.3 \sqrt{0.25}}=$$47.05793274

$\displaystyle dS = 40 \ e^{(0.05-0) 0.25 \ - \ 0.3 \sqrt{0.25}}=$ $34.861374 $\displaystyle C_u=$ 0 $\displaystyle C_d=$ 45 – 34.861374 =$10.138626

$\displaystyle u=\frac{47.05793274}{40}=$ 1.176448318

$\displaystyle d=\frac{34.861374}{40}=$ 0.87153435

$\displaystyle p^*=\frac{e^{(0.05-0) 0.25} - 0.87153435}{1.176448318 - 0.87153435}=$ 0.462570155

$\displaystyle 1-p^*=$ 0.537429845

$\displaystyle C=(p^* \times C_u + (1-p^*) \times C_d) e^{-0.0125}=$ 5.381114117

$\text{ }$
$\displaystyle \Delta=\frac{0-10.138626}{47.05793274-34.861374}=$ -0.831269395

$\displaystyle B=e^{-0.05(0.25)} \ \frac{1.176448318(10.138626)-0.87153435(0)}{1.176448318 - 0.87153435}=$ $38.63188995 The replicating portfolio consists of shorting 0.831269395 shares and lending$38.63188995.

$\text{ }$

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Remarks

The discussion in this post is only the beginning of the binomial pricing model. The concepts and the formulas for the one-period binomial option model are very important. The one-period model may seem overly simplistic (or even unrealistic). One way to make it more realistic is to break up the one-period into multiple smaller periods and thus produce a more accurate option price. The calculation for the multi-period binomial model is still based on the calculation for the one-period model. Before moving to the multi-period model, we discuss the one-period model in greater details to gain more understanding of the one-period model.

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Practice problems

Practice Problems
Practice problems can be found in the companion problem blog via the following links:

basic problem set 1

basic problem set 2

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$\copyright \ \ 2015 \ \text{Dan Ma}$

## Pricing a call option – an example

The example in this post illustrates how to price a call option using the one-period binomial option pricing model. The next post will present an example on pricing a put option. The two posts are designed to facilitate the discussion on the binomial option pricing (given in a series of subsequent posts). Links to practice problems are found at the bottom of the post.

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The example

The following gives the information about the stock:

• The stock of XYZ company is currently selling for $50 per share. The price per share 1 year from now is expected to increase to$65 or to decrease to $40. The stock pays no dividends. Consider a call option with the following specifics: • The underlying asset of the call option is the XYZ stock. • The strike price is$55.
• The option will expire in one year.
• The option is assumed to be a European option, i.e. it can be exercised only at expiration.

The annual risk-free interest rate is 2%. There is a benefit to the buyer of the option described above. If the price of the stock goes up to $65 at the end of the 1-year period, the owner of the option has the right to exercise the option, i.e., buying one share at the strike price of$55 and then selling it at the market price of $65, producing a payoff of$10. If the price of the stock goes down to $40 at the end of the 1-year period, the buyer of the option has the right to not exercise the option. The call option owner buys the stock only when he makes money. What would be the fair price of having this privilege? What is the fair price of this call option? ___________________________________________________________________________________ Pricing the call option In this example, the current stock price is$50 and the stock price can be only one of the two possible values at the end of the option contract period (either $65 or$40). The following diagram shows the future state of the stock prices.

$\text{ }$
Figure 1 – Stock Price

$\text{ }$

The assumption of the 2-state stock prices in 1 year simplifies the analysis of the call option. The value of the call option at the end of 1 year is either $10 (=65-55) or zero. Note that when the share price at the end of the 1-year contract period is less than the strike price of$55, the call option expires worthless. The following diagram shows the value of the call option.

$\text{ }$
Figure 2 – Call Option Payoff

$\text{ }$

In the above diagram, the value of the call option at the end of 1-year is either $10 or$0. The value of the option at time 0 is $C$, which is the premium of the call option in this example. Our job here is to calculate $C$. The key to finding the value of the option is to compare the payoff of the call to that of a portfolio consisting of the following investments:

Portfolio A

• Buy 0.4 shares of XYZ.
• Borrow $15.683 at the risk-free rate. The idea for setting up this portfolio is given below. For the time being, we take the 0.4 shares and the borrowed amount of$15.683 as a given. Note that $15.683 is the present value of$16 at the risk-free rate of 2%. Let’s calculate the value of Portfolio A at time 0 and at time 1 (1 year from now). The following diagram shows the calculation.

$\text{ }$
Figure 3 – Portfolio A Payoff

$\text{ }$

Note that the payoff of the call option is identical to the payoff of Portfolio A. Thus the call option in this example and Portfolio A must have the same cost. Since Portfolio A costs $4.317, the price of the option must be$4.317. The Portfolio A of 0.4 shares of stock and $15.683 in borrowing is a synthetic call since it mimics the call option described in the example. Portfolio A is called a replicating portfolio because it replicates the payoff of the call option in question. ___________________________________________________________________________________ Arbitrage opportunities In deriving the cost of the call option of$4.137, we rely on the idea that if two investments have the same payoff, they must have the same cost. This idea is called the law of one price, which is a commonsensical idea and is also an important principle in derivative pricing. If the law of one price is violated, in particular if the price of the call option discussed in this example is not $4.317, there would be arbitrage opportunities that can be exploited to gain risk-free profit. What if the law of one price is violated? For example, what if the option were selling for a higher price (say$4.50)? If the price of the replicating portfolio is less than the price of the option, then we can “buy low and sell high” (i.e. buy the replicating portfolio and sell call option) and obtain a risk-free profit of $0.183. The arbitrage is to buy the synthetic call (Portfolio A) at$4.317 and sell the call option at $4.50. The following table shows the Year 1 cash flows of this arbitrage opportunity. $\text{ }$ Table 1 – Arbitrage opportunity when call option is overpriced $\left[\begin{array}{llll} \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Long synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Hold 0.4 shares} & \text{ } & + \ 16 & + \ 26 \\ \ \ \ \ \text{Repay borrowed amount of } \ 15.683 & \text{ } & - \ 16 & - \ 16 \\ \text{ } & \text{ } \\ \text{Short call } & \text{ } & \ \ \ 0 & - \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$ $\text{ }$ The above table shows that buying a synthetic call (holding 0.4 shares and borrow$15.683) and selling a call will have no loss at the end of 1 year. Yet, the time 0 cash flow is $0.183 (=4.50 – 4.317), and is thus a risk-less profit. If the option is underpriced, then we can still buy low and sell high (in this case, buy call option and sell the replicating portfolio) and obtain risk-free arbitrage profit. For example, let’s say you observe a call option price of$4.00. Then the arbitrage opportunity is to buy the call option at $4.00 and sell a synthetic call (Portfolio A) at$4.317. The time 0 payoff is $0.317, which is a risk-less arbitrage profit. The following table shows the Year 1 cash flows. $\text{ }$ Table 2 – Arbitrage opportunity when call option is underpriced $\left[\begin{array}{llll} \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Short synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Short 0.4 shares} & \text{ } & - \ 16 & - \ 26 \\ \ \ \ \ \text{Receive the amount of } \ 15.683 & \text{ } & + \ 16 & + \ 16 \\ \text{ } & \text{ } \\ \text{Long call } & \text{ } & \ \ \ 0 & + \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$ $\text{ }$ ___________________________________________________________________________________ To complete the picture The call option price of$4.317 is derived by showing that the replicating portfolio has the same payoff as the call option. How do we know that the replicating portfolio consists of holding 0.4 shares and the borrowing of $15.683? In general, the replicating portfolio of a European call option consists of $\Delta$ shares of the stock and the amount $B$ in lending at time 0 (borrowing if negative). By equating the payoff of the replicating portfolio and the payoff of the call option in this example, we have the following equations: $\text{ }$ $\displaystyle \begin{array}{ccc} \displaystyle 40 \ \Delta + B \ e^{0.02} & = & 0 \\ \displaystyle 65 \ \Delta + B \ e^{0.02} & = & 40 \end{array}$ $\text{ }$ Solving these two equations, we obtain $\Delta=\frac{10}{25}=0.4$ and $B=-16 \ e^{-0.02}=15.683$. Therefore, the replicating portfolio for the call option in this example consists of 0.4 shares of the stock and$15.683 in borrowing. The net investment for the replicating portfolio is $4.317 (=0.4(50)-15.683). Because there are only two data points in the future stock prices, the option premium is a linear function of $\Delta$ and $B$. The following is the premium of the call (or put) option using the one-period binomial tree $C=\Delta \ S+B$ where $S$ is the stock price at expiration. The above formula gives the cost of the portfolio replicating the payoff of a given option. It works for call option as well as for put option. We will see that for put options, $\Delta$ is negative and $B$ is positive (i.e. shorting stock and lending replicate the payoff of a put). The number $\Delta$ has a special interpretation that will be important in subsequent discussion of option pricing. It can be interpreted as the sensitivity of the option to a change in the stock price. For example, if the stock price changes by$1, then the option price, $\Delta \ S + B$, changes by the amount $\Delta$. In other words, $\Delta$ is the change in the option price per unit change in the stock price.

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Put-call parity

The put-call parity relates the price of a European call with a European put that has the same strike price and the same time to expiration. The following is a put on XYZ stock that is compatible to the call described above.

• The underlying asset of the put option is the XYZ stock.
• The strike price is \$55.
• The option will expire in one year.
• The option is assumed to be a European option, i.e. it can be exercised only at expiration.

By the put-call parity, the following gives the price of the put option.

\displaystyle \begin{aligned} P(55,1)&=C(55,1)-50+55 \ e^{-0.02} \\&=4.316821227-50+55 \ e^{-0.02} \\&=\ 8.2277 \end{aligned}

The next post will calculate the price of the same put using the binomial model.

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Remarks

We would like to comment that even though the example here may seem like an extreme simplification, the example has great value. First of all, this is an excellent introduction to the subject of option pricing theory. Secondly, the one-period example can be extended to a multi-period approach to describe far more realistic pricing scenarios. For example, we can break a year into many subintervals. We then use the 2-state method to describe above to work backward from the stock prices and option values of the last subinterval to derive the value of the replicating portfolio.

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Practice problems

Practice problems can be found in the companion problem blog via the following links:

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$\copyright \ \ 2015 \ \text{Dan Ma}$

## Put-Call Parity, Part 2

Put-call parity is a key idea in option pricing theory. It provides a tool for constructing equivalent positions. The previous post gives a general discussion of the put-call parity. In this post, we discuss the put-call parity for various underlying assets, i.e. the parity relations in this post are asset specific. The following is one form of the general put-call parity. This is the version (0) discussed in the previous post.

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$
$\text{ }$

The put-call parity has four components – the price of the call, the price of the put, the present value of the strike price and the present value of the forward price. In the general form of the put-call parity, the present value of the forward price completely take the dividends and time value of money into account. For a specific type of underlying asset, in order to make the put-call parity more informative, we may have to take all the interim payments such as dividends into account. Thus in the parity relations that follow, the general forward price is replaced with the specific forward price for that asset. Synthetic assets can then be created from the asset-specific put-call parity that is obtained.

The notations used here are the same as in the previous posts. The notation $F_{0,T}$ is the forward price. All contracts – forward and options and other type of contracts – are set at time 0 (today) and are to end at time $T$. The strike price for the options is $K$. The letter $r$ denotes the risk-free annual continuous interest rate. If the strike price $K$ is paid for an asset at time $T$, its present value at time 0 is $PV(K)=e^{-r T} K$. All options discussed here are European options, i.e. they can be exercised only at expiration.

All the parity relations that follow will obviously involve a call and a put. To make this extra clear, the call and the put in these relations have the same strike price and the same time to expiration. Thus whenever we say buying a call and selling a put, we mean that they are compatible in this sense.

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Put-call parity for stocks

Forward prices for stocks are discussed here. For a non-dividend paying stock, the forward price is $F_{0,T}=S_0 e^{r T}$, i.e. the price to pay for the stock in the future is the future value of the time 0 stock price. The following is the put-call parity of a non-dividend paying stock.

$\text{ }$
Put-Call Parity – non-dividend paying stock
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S1)$
$\text{ }$

The parity (S1) says that there are two ways to buy a non-dividend paying stock at time 0. One is the outright stock purchase (the left side). The other way (the right hand side) is to buy a call, sell a put and lend the present value of the strike price $K$. By buying a call and selling a put, it is certain that you will buy the stock by paying $K$, which is financed by the lending of $PV(K)=e^{-r T} K$ at time 0. In both ways, you own the stock at time $T$. There is a crucial difference. In the outright stock purchase, you own the stock at time 0. In the “options” way, the stock ownership is deferred until time $T$. For the non-dividend paying stock, an investor is probably indifferent to the deferred ownership in the right hand side of (S1). For dividend paying stock, deferred ownership should be accounted for the parity equation.

$\text{ }$
Put-Call Parity – dividend paying stock (discrete dividend)
$\displaystyle S_0-PV(\text{Div})=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S2)$
$\text{ }$

In (S2), $\text{Div}$ refers to the dividends paid during the period from time 0 to time $T$ and $PV(\text{Div})$ refers to the time 0 value of $\text{Div}$. The deferred stock ownership on the right hand side of (S2) does not have the dividend payments while the outright stock ownership has the benefit of the interim dividend payments. Thus the cost of deferred stock ownership must be reduced by the amount of the dividend payments. This is why the dividend payments are subtracted on the left hand side. The next parity relation is for a stock or stock index paying continuous dividend.

$\text{ }$
Put-Call Parity – dividend paying stock (continuous dividend)
$\displaystyle S_0 e^{-\delta T}=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S3)$
$\text{ }$

Continuous dividends are reinvested (as additional shares) where $\delta$ is the annual continuous compounded dividend rate. The forward price is $F_{0,T}=S_0 e^{(r-\delta) T}$. The present value of the forward price is $S_0 e^{-\delta T}$, which is the left hand side of (S3). The left side of (S3) is saying that $e^{-\delta T}$ shares at time 0 will accumulate to 1 share at time $T$. The right hand side is saying that buying a call, selling a put and lending out the present value of $K$ at time 0 will lead to ownership of 1 share at time $T$.

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Synthetic stocks and other synthetic assets

In this section, we consider synthetic assets that can be created from the parity relations on stocks. These synthetic assets are parity relations. The left side of each of these relations is an asset that exists naturally in the financial market place. The right hand side is the synthetic asset – a portfolio that is an alternative asset that has the same cost and payoff, thus a portfolio that mimics the natural asset. For example, a synthetic stock is a combination of put and call and a certain amount of lending that will replicate the same payoff as owning a share of stock. In the next section, we will resume the discussion of put-call parity on underlying assets.

Each of the parity relation in this section is derived from an appropriate stock put-call parity by solving for the desired asset. For a synthetic stock, we put the stock on the left hand side by itself.

$\text{ }$
Synthetic stock – non-dividend paying
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn1)$
$\text{ }$
Synthetic stock – discrete dividend paying
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K+PV(\text{Div}) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn2)$
$\text{ }$
Synthetic stock – continuous dividend paying
$\displaystyle S_0 =(C(K,T)-P(K,T)+e^{-r T} K) \ e^{\delta T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn3)$
$\text{ }$

Note that (Syn1) is identical to (S1) since there is no dividend. The portfolio on the right hand side is the synthetic stock. For example, for (Syn2), the strategy of buying a call, selling a put, and lending out the present values of the strike price and the interim dividends is an alternative way to own a discrete dividend paying stock. There is a crucial difference between outright stock ownership on the left hand side and the deferred stock ownership on the right hand side. The synthetic stock pays no dividends. Thus the outright stock ownership is worth more than the synthetic stock. In other words, the cost of outright stock ownership exceeds the synthetic cost. By how much? By the present value of the interim dividends. This is why the present value of the dividend payments is added to the right hand side of (Syn2) and (Syn3).

Now we consider synthetic T-bills (or synthetic risk-free asset).

$\text{ }$
Synthetic T-bill – based on non-dividend paying stock
$\displaystyle e^{-r T} K=S_0-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T1)$
$\text{ }$
Synthetic T-bill – based on discrete dividend paying stock
$\displaystyle e^{-r T} K+PV(\text{Div})=S_0-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T2)$
$\text{ }$
Synthetic T-bill – based on continuous dividend paying stock
$\displaystyle e^{-r T} K=S_0 e^{-\delta T}-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T3)$
$\text{ }$

In (T1), (T2) and (T3), the right hand side is the synthetic way of creating a T-bill. Let’s look at (T3).

Relation (T3). In order to hold a synthetic T-bill, you buy $e^{-\delta T}$ shares of stock, sell a call and buy a put at time 0. At time $T$, the $e^{-\delta T}$ shares become 1 share, which will be used to meet the demand of either the call option or put option. If the stock price is more than $K$, the call buyer will want to exercise the call and you as a seller of the call will have to sell 1 share at the strike price $K$. If the stock price is less than $K$ at time $T$, you as the put buyer will want to sell 1 share of stock at the strike price $K$. So in either case, you have the amount $K$ at time $T$, precisely the outcome if you buy a T-bill with maturity value $K$.

Next we consider synthetic call options.

$\text{ }$
Synthetic call – based on non-dividend paying stock
$\displaystyle C(K,T)=S_0-e^{-r T} K+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C1)$
$\text{ }$
Synthetic call – based on discrete dividend paying stock
$\displaystyle C(K,T)=S_0-e^{-r T} K-PV(\text{Div})+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C2)$
$\text{ }$
Synthetic call – based on continuous dividend paying stock
$\displaystyle C(K,T)=S_0 e^{-\delta T}-e^{-r T} K+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C3)$
$\text{ }$

The right hand side of the above three equations are synthetic ways to buy a stock call option. They can be derived by solving for $C(K,T)$ in the put-call parity relation in respective stock. It also pays to think through the cash flows on both sides. The right hand side of each of (C1) through (C3) consists of a leveraged position (stock purchase plus borrowing) and a long put to insure the leveraged position. For example, in the right hand side of (C1), borrow $e^{-r T} K$ and buy one share of stock (the leveraged position). Then use a purchased put to insure this leveraged position.

Another way to look at synthetic call is that the right hand side consists of a protective put and borrowing. A protective put is the combination of a long asset and a long put. For example, the right hand side of (C1) consists of $S_0+P(K,T)$ (a protective put) and the borrowing of $e^{-r T} K$, the present value of $K$.

Here’s the synthetic put options.

$\text{ }$
Synthetic put – based on non-dividend paying stock
$\displaystyle P(K,T)=C(K,T)-S_0+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P1)$
$\text{ }$
Synthetic put – based on discrete dividend paying stock
$\displaystyle P(K,T)=C(K,T)-S_0+e^{-r T} K+PV(\text{Div}) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P2)$
$\text{ }$
Synthetic put – based on continuous dividend paying stock
$\displaystyle P(K,T)=C(K,T)-S_0 e^{-\delta T}+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P3)$
$\text{ }$

The right hand side of each of (P1) through (P3) is a synthetic put, a portfolio that mimics the payoff of a put option. Note that the right hand side consists of a long call and a short stock position (this is a protective call) and the lending of the present value of $K$.

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Put-call parity for currencies

A previous post on forward prices shows that the currency forward price is $F_{0,T}=x_0 \ e^{(r-r_f) T}$ where $x_0$ is the exchange rate (units of domestic currency per unit of foreign currency, e.g. dollars per euro), $r$ is the domestic risk-free rate and $r_f$ is the foreign currency risk-free rate. The present value of $F_{0,T}$ is then $e^{-r T} \ F_{0,T}=x_0 \ e^{-r_f T}$, which is the number of units of the domestic currency (e.g. dollars) at time 0 in order to have one unit of foreign currency (e.g. euro) at time $T$. Substituting $e^{-r T} \ F_{0,T}=x_0 \ e^{-r_f T}$ into the parity relation of (0), we have:

$\text{ }$
Put-Call Parity – Currencies
$\displaystyle x_0 \ e^{-r_f T}=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (F1)$
$\text{ }$
$\displaystyle x_0 \ e^{-r_f T}-e^{-r T} K=C(K,T)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (F2)$
$\text{ }$

In (F1) and (F2), we assume that the call and the put are denominated in dollars, i.e. both the strike price $K$ and the put premium and call premium are denominated in dollars. For ease of discussion, let’s say the foreign currency is euro. The premium $C(K,T)$ discussed here is in dollars and grants the right to pay $K$ to get 1 euro. The premium $P(K,T)$ discussed here is in dollars and grants the right to pay 1 euro to get $K$. Thus the strike price $K$ is an exchange rate of USD per euro.

For example, let’s say $K=$ 0.80 USD/Euro at time 0. If at time $T$ the exchange rate is $x_T=$ 0.9 USD/Euro, the call buyer would want to exercise the option by paying 0.8 USD for 1 euro. If at time $T$ the exchange rate is $x_T=$ 0.7 USD/Euro, then the long put position would want to exercise the put by paying 1 euro to get 0.8 USD.

The relation (F1) indicates that the difference in the call and put premiums plus lending the present value of the strike price is the same as lending the present value of the amount in dollars (the domestic currency) that is required to buy 1 euro at time $T$.

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Put-call parity for bonds

For a zero-coupon bond, the forward price is simply the future value of the bond price. For a coupon paying bond, the future price has to reflect the value of the coupon payments. In the following parity relations, $B_0$ is the bond price at time 0. The amount $PV(\text{Coupons})$ is the present value of the coupon payments made during the life of the options.

$\text{ }$
Put-Call Parity – zero-coupon bond
$\displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (B1)$
$\text{ }$
$\displaystyle B_0-PV(\text{Coupons})=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (B2)$
$\text{ }$

Note that for the zero-coupon bond, the parity relation is similar to the one for non-dividend paying stock.

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Summary

The following is the list of all the asset specific put-call parity relations discussed in this post.

$\text{ }$
Forward/Futures
$\displaystyle e^{-r T} \ F_{0,T}=C(K,T)-P(K,T)+PV(K)$
$\text{ }$

Non-dividend paying stock
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Discrete dividend paying stock
$\displaystyle S_0-PV(\text{Div})=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Continuous dividend paying stock
$\displaystyle S_0 e^{-\delta T}=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Currency
$\displaystyle x_0 \ e^{-r_f T}=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Bond
$\displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

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$\copyright \ \ 2015 \ \text{Dan Ma}$

## Put-Call Parity, Part 1

Put–call parity is a relationship between the price of a European call option and European put option with the same strike price and time to expiration. It is one of the most important relationships in option pricing. It provides a tool for constructing equivalent positions. This post is a general discussion of put-call parity. In the next post, we discuss put-call parity in greater details for various underlying assets – e.g. stocks, treasuries and currencies.

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Synthetic forward – buying a call and selling a put

Suppose you follow the strategy of buying a call and selling a put (at time 0) where both options have the same underlying asset, the same strike price $K$ and the same time $T$ to expiration. At time $T$, it is certain that you will buy the underlying asset by paying the strike price $K$. Too see this, if at expiration of the options, the asset price is more than $K$, then you, as a call buyer will want to exercise the call option and pay $K$ to buy the asset. If the asset price at expiration is less than $K$, then you as a call buyer will not want to exercise but the put buyer that bought from you will want to exercise the put option. As a result, you will also buy the asset by paying the strike price $K$. Thus by entering into a long call and a short put (on the same underlying asset, with the same strike and same time to expiration), you will end up buying the underlying asset at time $T$ at the strike price $K$. What is being described sounds very much like a forward contract – a contract in which you can lock in a price today to pay for an asset a time $T$ in the future. For this reason, the strategy of buying a call and selling a put is called a synthetic forward contract.

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Put-call parity

The above discussion on synthetic forward suggests that there are two ways to buy an underlying asset (e.g. a stock) at time $T$ in the future. They are:

1. Enter into a forward contract to buy the underlying asset by paying the forward price $F_{0,T}$ at time $T$.
2. Buy a call and sell a put today (on the same underlying asset, with the same strike price $K$ and the same time $T$ to expiration).

The two different strategies generate the same payoff. Hence they must have the same cost. Otherwise there would be arbitrage opportunities. By the “no-arbitrage pricing” principle, the net cost of the two strategies must equal. The cost at time 0 of the “buy call sell put” strategy is $C(K,T)-P(K,T)$, plus the present value of the strike price $K$, where $C(K,T)$ and $P(K,T)$ represent the call option premium and put option premium, respectively. The cost at time $T$ of the forward contract strategy is the forward price $F_{0,T}$. Thus cost at time 0 of the forward contract strategy is the present value of $F_{0,T}$. We can now equate the costs of the two strategies.

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$
$\text{ }$

The notation $PV(\cdot)$ denotes the time 0 value of an amount at the time $T$. Equation (0) is one form of the put-call parity, which is a statement that buying a call and selling a put is equivalent to a synthetic forward contract. It also tells us that buying a call and selling a put plus lending the present value of the strike price is equivalent to buying the underlying asset.

Other versions can be derived by algebraically rearranging equation (0), some of which have interesting interpretations. The following is one of them.

$\text{ }$
Put-Call Parity
$\displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
$\text{ }$

The left hand side of (1) is the net option premium – the premium paid for the call less the premium received for the put. When this amount is not zero, it is in effect the premium of the synthetic forward contract (this amount is the initial cash outlay for the synthetic forward contract). This is one difference between a synthetic forward and an actual forward. Note that an actual forward contract has zero premium (the initial cash outlay is zero). Another difference is that the “forward price” of the synthetic forward is the strike price $K$ of the options and while the forward price of the actual forward is $F_{0,T}$.

Suppose that the strike price $K$ is chosen to be less than the actual forward price $F_{0,T}$. Then the holder of the synthetic forward contract can buy the asset at a price lower than the forward price. This is certainly a benefit. In order to get this benefit, the holder of the synthetic forward contract has to pay the net option premium, which is the result of the call being more expensive than the put. In this scenario, the net payment is a little higher at time 0. As a result, the payment at time $T$ is a little less.

Suppose that the strike price $K$ is chosen to be more than the actual forward price $F_{0,T}$. Then the holder of the synthetic forward position is obliged to pay for the underlying asset at a price higher than the forward. It then makes sense for the holder of the synthetic forward position to be compensated by receiving a payment initially. This would occur if the put is more expensive than the call. In this scenario, the net payment is a little less at time 0, leading to a larger payment at time $T$.

If the strike price is chosen to be the same as the forward price $F_{0,T}$, then equation (1) suggests that the synthetic forward mimic exactly the actual forward (both have zero premium). For this to happen, premiums for the put and the call must be equal.

The right hand side of (1) is the value of the discount resulted from paying the strike price instead of the forward price. This version of the put-call parity says that the discount is identical to the net option premium.

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Protective put and covered call

The next two versions can be interpreted in terms of a protective put and a covered call. A protective put consists of a long asset position and a long put. It is the strategy of buying a put option to protect against the risk of falling prices of a long asset position. A covered call consists of a long asset position and a short call. The covered call uses the upside profit potential of the long asset to back up (or cover) the call option sold to the call buyer. First, the protective call version:

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
$\text{ }$

The left hand side of (2) is the time 0 cash outlay of buying the underlying asset and buying a put. The right hand side of (2) is time 0 cash outlay of buying a call option (with the same strike and time to expiration as the put) and buying a zero-coupon bond costing $PV(K)$. Thus equation (2) tells us that buying the underlying asset and buying a put on that asset (i.e. a protective put) have the same cost and generate the same payoff as the buying a call option and buying a zero-coupon bond. Adding a bond lifts the payoff graph but does not change the profit graph. Thus buying the asset and buying a put has the same profit as buying a call. Because of Equation (2), buying the underlying asset and buying a put is called a synthetic long call option. This point is also discussed in this previous post. Here’s the version of the put-call parity involving covered call.

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
$\text{ }$

The left hand side of (3) is the time 0 cash outlay of buying the underlying asset and selling a call on that asset (i.e. a covered call). The right hand side of (3) is the time 0 cash outlay of buying a zero-coupon bond costing $PV(K)$ and selling a put. Thus a covered call has the same cost and same payoff as buying a bond and selling a put. Once again, adding a bond does not change the profit. Thus a covered call has the same profit as selling a put. For this reason, a buying the underlying asset and selling a call is called a synthetic short put option. This point is also discussed in this previous post.

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Summary

As a summary, we gather the various versions of the put-call parity in one place along with their interpretations.

$\text{ }$
Versions of Put-Call Parity
$\text{ }$
$\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$
Interpretation: Time 0 cost of a long asset = Time 0 cost of (Long Call + Short Put + Long Bond).

$\text{ }$

$\displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
Interpretation: Net option premium (call option premium that is paid out less put option premium received) = the value of the discount as a result of paying the strike price instead of the forward price.
$\text{ }$

$\displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
Interpretation: Time 0 cost of (Long Asset + Long Put) = Time 0 cost of (Long Call + Long Bond).
The portfolio on the left (Long Asset + Long Put) is called a protective put.
Because of (2), a protective put is considered a synthetic long call option.
$\text{ }$

$\displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
Interpretation: Time 0 cost of (Long Asset + Short Call) = Time 0 cost of (Long Bond + Short Put).
The portfolio on the left (Long Asset + Short Call) is called a covered call.
Because of (3), a covered call is considered a synthetic short put option.
$\text{ }$

In each of the above versions of parity, the portfolio of investments on the left side is equivalent to the portfolio of investment on the right side. More specifically, each version equates the costs of obtaining the portfolios at time 0. The bond indicated in the interpretations is a zero-coupon bond. A long position on a bond means lending.

One comment about the four parity relations discussed here. We derive the first one, which is version (0) by comparing the cash flows of two equivalent investments. The other three versions are then derived by algebraically rearranging the first version. As a learning device, it is a good idea to think through the cash flows and payoff of versions (2) through (3) independently of version (0). Doing so is a great practice and will help solidify the understanding of put-call parity. Drawing payoff diagrams can make the comparison easier. It is also possible to just think through the cash flows of both sides of the equation. For example,

let’s look at version (2). On the right side, you lend $PV(K)$ and buy a call at time 0. Then at time $T$, you get $K$ back. If the price of the underlying asset at that time is more than $K$, then you exercise the call – using the $K$ that you receive to buy the asset. So on the right hand, side, the payoff is $S_T-K$ if asset price is more than $K$ and the payoff is $K$ if asset price is less than $K$ (you would not exercise the call in this case). On the left hand side, you lend $PV(F_{0,T})$ and buy a put at time 0. At time $T$, you get $F_{0,T}$ back and you use it to pay for the asset. So you own the asset at time $T$. If the asset price at time $T$ is less than $K$, you exercise by selling the asset you own and receive $K$. Thus the payoff on the left hand side is $S_T-K$ if asset price is more than $K$ (in this case you don’t exercise the put and instead you profit from holding the asset). The payoff is $K$ if the asset price at time $T$ is less than $K$ (this is the case where you exercise the put option). The comparison shows that both sides of (2) have the same payoff at time $T$. Then it must be the case that they also have the same cost at time 0. Otherwise, there would be an arbitrage opportunity by buying the side that is low and sell the other side.

The basic put-call parity relations discussed in this post can be used in a “cookbook” fashion to create synthetic assets. For example, version (0) indicates that buying a call, selling a put and lending the present value of the strike price $K$ has the same cost and payoff as buying a non-dividend paying stock. Thus version (0) is a basis for constructing a synthetic stock. In the next post, we discuss the put-call parity for different underlying assets.

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$\copyright \ \ 2015 \ \text{Dan Ma}$