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

## Creating synthetic forwards

When a customer buys a forward contract from a market maker, the market maker can create an offsetting position to protect against the risk of holding a short forward position. In this post, we explain how to create a synthetic forward contract to hedge a forward position. This post is a continuation of these two previous posts on forward contracts: An introduction to forward contracts and Putting a price on a forward contract.

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Synthetic forward contracts

Let’s say the market maker has sold a forward contract to a customer and the contract allows the customer to buy a share of stock at expiration. The customer has the long forward position and the market maker is holding the short forward position. To offset the risk of the short forward, the market maker can create a synthetic long forward position.

In this discussion, we assume that the stock in question pays annual continuous dividends at the rate of $\delta$. Thus the forward price is $F_{0,T}=S_0 \ e^{(r-\delta) T}$ (see equation (5) in this previous post). The following is the payoff of the long forward position:

$\text{ }$

$\text{Long forward payoff at expiration} = S_T-F_{0,T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

$\text{ }$

The market maker that is in a short forward position will need to offset the long forward position in (1). To do that, the market maker can borrow the amount $S_0 \ e^{-\delta T}$ to buy $e^{-\delta T}$ shares of the stock at time 0. The stock purchase is financed by the borrowed amount. So there is no upfront cost to the market maker at time 0.

Now let’s look at what happens at time $T$. The $e^{-\delta T}$ shares will become 1 share at time $T$. The market maker can sell the 1 share to the customer at time $T$, thus receiving $S_T$. The market maker will also have to repay $S_0 \ e^{(r-\delta) T}$ to the lender, leaving the market maker with the amount $S_T-S_0 \ e^{(r-\delta) T}$. The following table summarizes the cash flows in these transactions.

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Table 1 – Borrowing to buy shares replicates the payoff to a long forward

$\left[\begin{array}{llll} \text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ \text{Buy } e^{-\delta T} \text{ shares of stock} & \text{ } & -S_0 \ e^{-\delta T} & +S_T \\ \text{ } & \text{ } \\ \text{Borrow } S_0 \ e^{-\delta T} & \text{ } & +S_0 \ e^{-\delta T} & -S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ \text{Total} & \text{ } & \text{ } \ \ 0 & \ \ S_T-S_0 \ e^{(r-\delta) T} \end{array}\right]$

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In the above table, the payoff to the market maker is $S_T-S_0 \ e^{(r-\delta) T}$, which is exactly the long forward payoff indicated in equation (1). This means that the process of borrowing to buy shares of stock replicates the payoff to a long forward and thus is a synthetic forward contract. We have the following relationship.

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$\text{Long forward} = \text{Long Stock} + \text{Short zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

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If a market maker is holding a long forward position, then he can offset the risk of holding the long forward by creating a synthetic short forward contract. The cash flows in the synthetic short forward contract is simply the reverse of (2). Thus we have the following relationship.

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$\text{Short forward} = \text{Short Stock} + \text{Long zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

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Before we discuss how a market maker can use the strategies of (2) and (3) to hedge, we discuss other synthetic positions that can be obtained from relationship (2).

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Other synthetic positions

By manipulating the synthetic forward in the relationship (2), we can create a synthetic stock as well as a synthetic bond.

$\text{ }$

$\text{Long stock} = \text{Long Forward} + \text{Long zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$

$\text{Long zero-coupon bond} = \text{Long Stock} + \text{Short Forward} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$

$\text{ }$

If relationship (2) is understood, then (3), (4) and (5) are obtained by rearranging (2). For example, moving a long asset to the other side of the equation becomes a short. To further illustrate the idea of synthetically creating assets, we describe the cash flows for the transactions behind (4) and (5).

$\text{ }$

Table 2 – A long forward plus lending creates a synthetic share of stock

$\left[\begin{array}{llll} \text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ \text{Long a forward} & \text{ } & \ \ 0 & \ \ S_T-F_{0,T} \\ \text{ } & \text{ } \\ \text{Lend } S_0 \ e^{-\delta T} & \text{ } & -S_0 \ e^{-\delta T} & \ \ S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ \text{Total} & \text{ } & -S_0 \ e^{-\delta T} & \ \ S_T \end{array}\right]$

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Table 3 – Buying shares of stock and shorting a forward creates a synthetic bond

$\left[\begin{array}{llll} \text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ \text{Buy } e^{-\delta T} \text{ shares of stock} & \text{ } & -S_0 \ e^{-\delta T} & \ \ S_T \\ \text{ } & \text{ } \\ \text{Short a forward } & \text{ } & \ \ 0 & \ \ F_{0,T}-S_T \\ \text{ } & \text{ } \\ \text{Total} & \text{ } & -S_0 \ e^{-\delta T} & \ \ F_{0,T} \end{array}\right]$
$\text{ }$

Looking at the Total row in table 2, the end result is that the market maker pays the time 0 price of $e^{-\delta T}$ shares and obtain the time $T$ value of one share. Thus the cash flows in Table 2 create a synthetic share of the stock.

The Total row of Table 3 tells us that the end result of Table 3 can be described in this way: the market maker lends out the amount $S_0 \ e^{-\delta T}$ at time 0. At time $T$, the market maker receives the future value of the loan, which is $F_{0,T}=S_0 \ e^{(r-\delta) T}$. Thus the cash flows in table 3 create a synthetic zero-coupon bond.

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How market makers use synthetic forwards

If the market maker is holding a short forward position, he can use relationship (2) to create a synthetic long forward to offset the short forward position. On the other hand, if the market maker is holding a long forward position, then the market maker can use relationship (3) to create a short forward to offset the long forward.

The following table displays the cash flows involved in hedging using the idea in (2). For easier reference, equation (2) is repeated below.

$\text{ }$

$\text{Long forward} = \text{Long Stock} + \text{Short zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

$\text{ }$

Table 4 – A market maker offsetting a short forward with a synthetic long forward

$\left[\begin{array}{lllll} \text{ } &\text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ 1 & \text{Buy } e^{-\delta T} \text{ shares of stock} & \text{ } & -S_0 \ e^{-\delta T} & +S_T \\ \text{ } & \text{ } \\ 2 & \text{Borrow } S_0 \ e^{-\delta T} & \text{ } & +S_0 \ e^{-\delta T} & -S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ 3 & \text{Short forward} & \text{ } & \text{ } \ \ 0 & \ \ F_{0,T}-S_T \\ \text{ } & \text{ } \\ 4 & \text{Total} & \text{ } & \text{ } \ \ 0 & \ \ F_{0,T}-S_0 \ e^{(r-\delta) T} \end{array}\right]$

$\text{ }$

Row 2 in Table 4 is the short bond (borrowing cash has the effect of selling a bond). The borrowed cash is then used to buy stocks (the long stock in Row 1). Rows 1 and 2 form the synthetic long forward. Row 3 is the short forward position held by the market maker. Note that the total cash flow at time $T$ is $F_{0,T}-S_0 \ e^{(r-\delta) T}$, which is 0 assuming the no-arbitrage pricing principle. Thus the synthetic long forward neutralizes the actual short forward. All the ingredients of the last cash flow – forward price, spot price, risk-free interest rate and dividend yield – are known at time 0. Thus these transactions result in a risk-free position.

Table 4 illustrates a trading strategy that we want to highlight. A trading strategy in which an investor holds a long position in a security or commodity while simultaneously holding a short position in a forward contract on the same security or commodity is called a cash-and-carry. When using this strategy, the long position is held until the delivery date of the forward contract and is used to cover the obligation of the short position. Thus a cash-and-carry is risk-free.

Table 4 illustrates a cash-and-carry trade from the perspective of a market maker wishing to hedge the risk from a short position. When the cash-and-carry strategy is used by an arbitrageur, it is called a cash-and-carry arbitrage. The arbitrage strategy is to exploit the pricing inefficiencies for an asset in the cash (spot) market and the forward (or futures) market in order to make risk-less profits. The arbitrageur would try to carry the asset until the expiration date of the forward contract and the long asset position is used to cover the obligation of the short position. The strategy of cash-and-carry arbitrage is only profitable if the cash inflow from the short position exceeds the acquisition costs and carrying costs of the long asset position, i.e. $F_{0,T}>S_0 \ e^{(r-\delta) T}$, in which case an arbitrageur or market maker can use the strategy outlined in Table 4 to make a risk-free profit.

The following table displays the cash flows involved in hedging a long forward position using a synthetic short forward, i.e. using equation (3). For easier reference, equation (3) is repeated below.

$\text{ }$

$\text{Short forward} = \text{Short Stock} + \text{Long zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

$\text{ }$

Table 5 – A market maker offsetting a long forward with a synthetic short forward

$\left[\begin{array}{lllll} \text{ } &\text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ 1 & \text{Sell } e^{-\delta T} \text{ shares of stock} & \text{ } & +S_0 \ e^{-\delta T} & -S_T \\ \text{ } & \text{ } \\ 2 & \text{Lend } S_0 \ e^{-\delta T} & \text{ } & -S_0 \ e^{-\delta T} & +S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ 3 & \text{Long forward} & \text{ } & \text{ } \ \ 0 & \ \ S_T-F_{0,T} \\ \text{ } & \text{ } \\ 4 & \text{Total} & \text{ } & \text{ } \ \ 0 & \ \ S_0 \ e^{(r-\delta) T}-F_{0,T} \end{array}\right]$

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Row 1 in Table 5 is the short stock – borrowing the shares and sell them to receive cash. Then lend the cash from the sales of the borrowed stock (the long bond in Row 2). Rows 1 and 2 form the synthetic short forward. Row 3 in Table 5 is the long forward position held by the market maker. Note that the total cash flow at time $T$ is $S_0 \ e^{(r-\delta) T}-F_{0,T}$, which is 0 assuming the no-arbitrage pricing principle. Thus the synthetic short forward neutralizes the actual long forward. As in Table 4, all the ingredients of the last cash flow – forward price, spot price, risk-free interest rate and dividend yield – are known at time 0. Thus these transactions result in a risk-free position.

A reverse cash-and-carry is the reverse of cash-and-carry. Thus a reverse cash-and-carry is a trading strategy in which an investor holds a short position in a security or commodity while simultaneously holding a long position in a forward contract on the same security or commodity. Table 5 illustrates a reverse cash-and-carry from the perspective of a market maker. If the cash outflow from the long position is less than the selling proceeds and interest income of the short asset position, i.e. $F_{0,T}, then the market maker or an arbitrageur can use the strategy outlined in Table 5 to make a risk-free profit.

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