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

## Introducing options

This post is an introductory discussion of options. After introducing the basic concepts and terminology, we use payoff and profit diagrams to highlight and summarize the risk and benefit characteristics of call and put options.

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Motivating options with forward contracts

Suppose you need to buy an asset at the future time $T$. The price of the asset right now is $S_0$. You are of course concerned that prices of the asset will rise at or before time $T$. You can buy the asset now by paying $S_0$. If for some reasons buying the asset now is not a practical approach, you can lock in a price $F_{0,T}$ now to pay for the asset at time $T$. In other words, you can purchase a forward contract today to buy the asset at time $T$. For a more detailed discussion on forward contracts, see the following three posts:

There are two parties in a forward contract, the long position (the buyer) and the short position (the seller). The forward contract obliges the buyer (holder of the long position) to pay the forward price $F_{0,T}$ at expiration for the asset (of course the other side is obliged to sell at the forward price). The buyer is obliged to buy even when it is not advantageous for him to do so (i.e. when the price of the asset tumbles below the forward price $F_{0,T}$). Could there be a contract in which the buyer can buy when it is advantageous to do so and the buyer can walk away when it is not advantageous to buy?

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An example of a call option

The answer to the above question is yes. This is a contract where the buyer (the long position) has all the upside potential but essentially has limited downside risk. This is a call option, which is a contract where the buyer has the right to buy as asset at a price set ahead of time but not the obligation to buy.

To make the idea of a call option clear, let’s look at an example. Suppose that an investor buys a call option contract by agreeing to pay $40 for a share of XYZ company in 3 months. After 3 months, if the XYZ stock price is$50, then the investor will pay $40 for a share of XYZ, reaping a payoff of$10 per share. If the price in 3 months is $30, then the investor will walk away because he does not want to pay$40 for a share of stock that is worth less.

A call option contract has two parties, the buyer (the long position) and the seller (the short position). In 3 months, if the XYZ stock price is $50 per share, then the seller will sell to the buyer a share of XYZ for$40 because the buyer will choose to buy. In this case, the seller is selling a stock that is worth $10 more than the selling price and thus has a loss of$10 a share. If in 3 months, the stock price is $30, then buyer will not buy so the seller is not obliged to sell. When the contract expires, the seller will have a payoff of$0 if the XYZ stock price is at or below $40 per share. On the other hand, the seller’s payoff is negative (i.e. has a loss) if the stock price at expiration is above$40 per share.

The call option buyer buys the asset only when he makes money. The buyer’s payoff is potentially unlimited (when the stock price is through the roof) and is at worse $0. On the other hand, the seller’s payoff is at best$0 and is potentially unlimited when prices rise. Who would want to enter into such a contract as a seller (the short position)? There is definitely an imbalance in the example just described. To remove this imbalance, the seller must be compensated for taking the disadvantageous position of being a seller of a call option. The compensation is in the form of a call option premium. For example, the buyer of the option will have to pay the seller $2.78 for the privilege of being in the long position. When the XYZ share price is above$40, the seller has the obligation to sell a share at a loss. But the seller will be able to keep $2.78 when the share price is below$40. How to price options will be a subject in many subsequent posts. For now we just take the option premium as a given.

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Call options and put options

The above example is for a call option. It is a contract where the buyer has the right but not the obligation to buy the underlying asset (this means that the counter party, the call option seller, has the obligation to sell). Another type of options is a put option. It is a contract where the buyer has the right but not the obligation to sell the underlying asset (this means that the counter party, the put option seller, has the obligation to buy). To make the discussion easier, let’s look at some standard terminology of options.

• Strike price (or exercise price). For a call option, it is the price at which the buyer will buy the underlying asset. For a put option, it is the price at which the buyer will sell the underlying asset. The strike price can be set at any value.
• Exercise. For a call option, this is the act of paying the strike price to receive the underlying asset. For a put option, this is the act of receiving the strike price to deliver the underlying asset. Regardless of the type (call or put), an option is exercised only when it is advantageous for the buyer to do so.
• Expiration. This is the date that is the deadline for the option being exercised by the buyer of the option. After the expiration date, the option will be worthless.
• Exercise style. The exercise style has to do with the timing of the exercise of the option. If the option can only be exercised on the expiration date, it is said to be a European-style option. If the option can be exercised at any time on or before the expiration date, it is said to be an American-style option. The distinction of “American” and “European” for options is not based on geography. Both types of options are traded worldwide.

In the above example, the call option has a strike price of $40. It has an expiration of 3 months. After 3 months, if the stock price is over the strike price of$40, the buyer can choose to exercise the option. Since the buyer can only exercise on the expiration date, it is a European-style option.

A call option is a contract in which the buyer has the right but not the obligation to buy the underlying asset at the strike price on or before the expiration (depending on the exercise style). This means that the counter party of the call option buyer (the call option seller) has the obligation to sell the underlying asset when the buyer decides to exercise. The call option seller is also called the call option writer.

A put option is a contract in which the buyer has the right but not the obligation to sell the underlying asset at the strike price on or before the expiration (depending on the exercise style). This means that the counter party of the put option buyer (the put option seller) has the obligation to buy the underlying asset when the buyer decides to exercise. The put option seller is also called the put option writer.

The assets that are transacted in option contracts can be financial assets (e.g. stock, stock index, currencies, and interest rates) or commodities (both physical commodities and commodity futures contracts).

In the remainder of the post, we focus on European-style options.

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Payoff and profit diagrams

The payoff of a derivative contract is the value of the contract at a point in time. Knowing the payoff is not the entire story. To know how profitable the contract is, we need to subtract the cost of acquiring the position from the payoff. Thus the profit of a position is the payoff less the future value of the original cost of acquiring the position.

A payoff diagram for a derivative contract is a graph in which the value of the derivative at one point in time is plotted against the price of the underlying asset. A profit diagram for a derivative contract is a graph in which the profit of the derivative at one point in time is plotted against the price of the underlying asset. Payoff diagrams and profit diagrams are important graphical tools to summarize the risk of the derivative contracts. We are only interested in the payoff value and profit at the time of expiration.

We present 12 diagrams below. There are 4 positions to consider:

1. Long call option
2. Short call option
3. Long put option
4. Short call option

We show three diagrams for each position – payoff of option, profit of option and the third one combining the option and a matching forward.

Comment
Going through the payoff and profit diagrams below will go a long way to solidify your understanding of the definitions and risk and benefit characteristics of four option positions listed above. It is very easy to recite the basic definition of call option and put option. For someone new (or fairly new) to options, it can be confusing in going through these 4 option position. For example, even though the position of the short put should be a short position, it can actually be regarded as a long position in the sense that it represents an obligation of the put seller to buy the underlying asset and that the holder of the short put benefit from rising prices. Therefore, a good way of sorting out these 4 positions is to draw these diagrams on paper. Then compare your diagrams to the ones given here.

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Payoff and profit diagrams of call options

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Option Position 1 – Long Call Option
First, the payoff diagram of a long call option.

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Figure 1

To the left of the strike price $K$, the payoff graph in Figure 1 coincides with the x-axis. To the right of the strike price $K$, the payoff graph is above the x-axis and increasing. The level part of the graph shows that this is an option – the right to walk away from the contract. The payoff is 0 on the left side of the strike price $K$ since the call buyer has no incentive to pay the strike price for an asset that is worth less than the strike price. To the right of the strike price $K$, the payoff is the spot price less the strike price. The following summarizes the payoff of a long call where $S_T$ is the spot price of the underlying asset at expiration.

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$\displaystyle \text{payoff of a long call}=\left\{\begin{matrix} \displaystyle 0&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ S_T - K&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

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The following is the profit diagram of a long call.

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Figure 2

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The profit graph of a long call option has the same shape of the long call payoff graph. The profit graph is the result of adjusting the payoff graph downward by the amount of the call premium. When considering profit, it is necessary to subtract the call option premium from the payoff. The call option premium is paid at time 0. The payoff is at time $T$. So the amount subtracted is the future value of the call option premium.

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$\displaystyle \text{profit of a long call}=\left\{\begin{matrix} \displaystyle -\text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ S_T - K-\text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

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Clearly the holder of the long call position benefit from rising prices. So does the holder of a long forward position. The following diagram compares a long call option with a long forward.

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Figure 3

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The dotted line in Figure 3 is the profit graph of a long forward. Note that it has no “flat” part. There is no optionality in a forward. Regardless of price movements, the forward contract holder (long or short) has the obligation to buy or sell. A long forward can be used to hedge against a price increase in the underlying asset. But the long forward position is exposed to risk if prices fall. When the asset price is below the strike place, the long forward position has the obligation to buy the underlying asset while the long call position has the right to walk away. The price of the right to walk away is the call option premium the long call buyer pays to call seller.

One more comment about the long forward. For a forward contract, the profit and the payoff are identical since there is no initial cost of acquiring the forward contract.

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Option Position 2 – Short Call Option
The counter party to the call option buyer is the call option seller. The following shows the payoff diagram and the profit diagram of a short call.

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Figure 4

Figure 5

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Comparing Figure 4 and with Figure 1, it is clear that the call seller loses money where the call buyer makes money (to the right of the strike price $K$). In fact the loss for the short call is unlimited as the spot price increases as the gain is unlimited for the call buyer. To the left of the strike price, the call option buyer has no incentive to exercise. Thus the payoff to the short call is 0 to the left of the strike price. Figure 4 shows that the highest payoff for the short call option position is 0. As enticement, the payoff graph is lifted up by the amount of the call option premium to form the profit graph in Figure 5. The following further summarizes the payoff and profit of a short call option.

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$\displaystyle \text{payoff of a short call}=\left\{\begin{matrix} \displaystyle 0&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ K-S_T&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\displaystyle \text{profit of a short call}=\left\{\begin{matrix} \displaystyle \text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ K-S_T+\text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

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Both the short call position and the short forward position have similar risk characteristics. The following diagram compares a short call option with a short forward.

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2

Figure 6

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The dotted line in Figure 6 is the profit of the short forward contract. Both the short forward position and the short call position benefit from falling prices, except that the profit of the short call position is capped at the call option premium. When price rises, both positions are similar in that the losses are unlimited, except that the loss for the short call is reduced by the amount of the call option premium.

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Payoff and profit diagrams of put options

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Option Position 3 – Long Put Option
The following shows the payoff diagram and the profit diagram of a long put option.

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Figure 7

Figure 8

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The dynamics of the put option are the opposite of the call option. Let’s compare Figure 7 with Figure 1. For a call option (Figure 1), the contract has positive payoff to the right of the strike price. For a put option (Figure 7), the contract has positive payoff to the left of the strike price. Thus a long call position makes money when prices go up and a long put position makes money when prices go down. Recall that a put option grants the right to its buyer to sell the underlying asset at the strike price. Thus when prices are low at expiration, the long put position can sell the asset at a price higher than what the asset is worth.

The profit graph in Figure 8 is obtained by lowering the payoff graph in Figure 7 by the amount of the put option premium. This is due to the fact that the put option buyer has to pay a premium to the seller to gain the privilege of walking away from the contract when the prices are higher than the strike price. The following further summarizes the payoff and profit of a long put option.

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$\displaystyle \text{payoff of a long put}=\left\{\begin{matrix} \displaystyle K-S_T&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ 0&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\displaystyle \text{profit of a long put}=\left\{\begin{matrix} \displaystyle K-S_T-\text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ -\text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

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Interestingly, the long put position and the short forward position have similar characteristics. The following diagram compares a long put option with a short forward.

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Figure 9

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The dotted line in Figure 9 is the profit graph of the short forward. While the long put position can walk away when prices rise, the short forward position is still obligated to sell at a loss. Of course, the long put has to pay a premium for the right to walk away when price rises.

One interesting observation is that the long put position can be regarded in some sense a short position despite the word long in its name. For example, the long put is a right to sell the underlying asset and the long put position benefits from falling prices. Note the similarity with the short forward position, which has an obligation to sell the underlying asset and which also benefits from falling prices.

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Option Position 4 – Short Put Option

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Figure 10

Figure 11

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Recall that the short put option position (the seller of the put) has the obligation to buy the asset when the put buyer chooses to exercise. When the spot price at expiration is less than the strike price, the put seller will have to pay the strike price for an asset that is worth less than the strike price. This explains the downward slope to the left of the strike price in Figure 10 and Figure 11. The flat part of both Figure 10 and Figure 11 reflects the optionality – the right of the put buyer not to sell an asset that is worth more than the strike price. For this right to walk away, the put buyer will have to pay a premium to to the put seller. This explains that the profit graph in Figure 11 is the payoff graph lifted up by the amount of the put option premium.

The following further summarizes the payoff and profit of a short put option.

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$\displaystyle \text{payoff of a short put}=\left\{\begin{matrix} \displaystyle S_T-K&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ 0&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\displaystyle \text{profit of a short put}=\left\{\begin{matrix} \displaystyle S_T-K+\text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ \text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

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The short put position and the long forward position are similar in some sense. The following diagram compares a short put option with a long forward.

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Figure 12

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The dotted line in Figure 12 is the profit graph of the long forward. Figure 12 shows that both the long forward position and the short put position have substantial losses when price falls. The only difference is that the downside loss for the short put position is mitigated somewhat by the put option premium.

One interesting observation is that the short put option position can be regarded as a long position despite its name has the word short. It can be regarded as long in the sense that a short put position is obligated to buy asset from the put buyer and that a short put position benefit from rising prices. If prices rise, then the put buyer will not exercise and the put seller can pocket the put option premium. Note the similarity with a long forward. A long forward is an obligation to buy the underlying asset and also benefit from rising prices. The only difference is that the gain for a long forward is unlimited when prices rise, while the gain for a short put is only the option premium.

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Remarks

The risk and benefit characteristics discussed in all the above profit and payoff diagrams indicate that call options and put options can be used as insurance against loss in an investment position.

For example, put option is an insurance against a fall in the price of an asset. Let’s say you hold an asset and are planning on selling it at some point in the future. The risk is that the price may fall at the time of sales. To insure against such an adverse outcome, buy a put option with the same underlying asset and with strike price and expiration that match your need. At expiration of the put option, you will have a guarantee of a minimum sale price of your asset, which is the strike price. For this reason, the purchase of a put option is called a floor. A put option is a protection against falling prices of a long asset position.

On the other hand, a call option is insurance for an asset that we plan to own in the future. If we plan to buy shares of a stock at some future date, buying a call option on that stock now will guarantee a minimum purchased price. For this reason, buying a call option is called a cap. In general, a call option is a protection against rising prices for a short asset position.

This subsequent post discusses the insurance strategies of protective put and protective call.

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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:

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

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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.

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

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

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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).

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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

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

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

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

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

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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.

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

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

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

$\text{ }$

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.

$\text{ }$

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

## Putting a price on a forward contract

This post is a continuation of this previous post on forward contracts. The previous post discusses the basic features of forward contracts. How to price forward contracts is the subject of this post.

Suppose that you need to purchase a financial asset or commodity at time $T$ in the future. The price of the asset is $S_0$ right now (at time 0). The price at time $T$ is $S_T$, which is not known at time 0. You can wait until time $T$ to buy the asset by paying $S_T$. Waiting could be risky since the price could increase substantially. So waiting would exposure you to the risk of price uncertainty and as a result profit uncertainty. You can then buy the asset at time 0 (now) and hold it to time $T$. Due to business reasons or other reasons, this may not always be practical. An alternative is to lock in a price $F_{0,T}$ today to pay for the asset at time $T$.

The dynamics described above can apply to selling too. Suppose you have a financial asset or commodity that will be available for sales at time $T$. You can sell it at time $T$ for the price $S_T$, which is unknown at time 0. Or you can lock in a price $F_{0,T}$ today to sell the asset at time $T$.

The above scenario is in essence what a forward contract is. In this post, we discuss how to derive the forward price $F_{0,T}$. The focus here is on financial assets, in particular stocks, stock index and currencies.

As discussed in this previous post, a forward contract is a contract between two parties to buy or sell an asset at a specified price (called the forward price) on a future date. The forward price, the quantity $F_{0,T}$ introduced above, is set today by the two parties in the contract for a transaction that will take place in a future date, at which time the buyer pays the seller the forward price and the seller delivers the asset to the buyer.

We also make the following simplifying assumptions:

1. Trading costs and taxes are ignored.
2. Individuals can always borrow or lend at a risk-free interest rate.
3. Arbitrage opportunities do not exist.

In this post, we focus on the pricing of forward contracts on stock, stock indexes and currencies. Assume that the annual risk-free interest rate that is available for investors is $r$.

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Forward price on a stock – nondividend paying

The first case is that the stock pays no dividend. This simple case will help derive the case of paying dividends. Suppose that you want to own a share of a stock at time $T$ in the future. Just like the scenarios described above, there are two ways to do this.

• Buy a share at time 0 and hold it until time $T$.
• Enter into a forward contract to buy one share of the stock at time $T$.

In the first way, you pay $S_0$ at time 0 to own the stock. In the second way, you pay $F_{0,T}$ at time $T$ to own the stock. In either way you own a share of the stock at time $T$. In the second way, in order to have the amount $F_{0,T}$ available at time $T$, you can invest $F_{0,T} \ e^{-rT}$ at time 0 at the risk-free interest rate compounded continuously. So at time 0, the cost outlay for the first way is $S_0$. At time 0, the cost outlay for the second way is $F_{0,T} \ e^{-rT}$. If there is to be no arbitrage, the two would have to be the same.

$F_{0,T} \ e^{-rT}=S_0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$

$F_{0,T}=S_0 \ e^{rT} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

Thus equation (1) gives the forward price on a stock that pays no dividend. The forward price in this case is simply the future value of the price of the asset at time 0.

If equation (1) is violated, then there would be arbitrage opportunities. Then traders can buy low and sell high to generate risk-free profit. To see how this work, suppose that $F_{0,T}>S_0 \ e^{rT}$. Then you can buy low and sell high. At time 0, borrow the amount $S_0$ and buy a share of the stock. At time 0, also sell a forward contract (i.e. enter into a short forward contract) at the forward price $F_{0,T}$. At time $T$, sell the share of the stock and obtain the forward price $F_{0,T}$ and pay $S_0 \ e^{rT}$ to the lender, producing a sure and positive profit $F_{0,T}-S_0 \ e^{rT}$.

On the other hand, suppose $F_{0,T}. This time the arbitrage strategy is still to buy low and sell high. You can buy a forward contract at the forward price $F_{0,T}$ and simultaneously borrow a share and sell it at the price $S_0$. Invest the amount $S_0$ at the risk-free rate to obtain $S_0 \ e^{rT}$ at time $T$. At time $T$, buy a share of the stock at the price $F_{0,T}$ and then return it to the lender. The amount that remains is $S_0 \ e^{rT}-F_{0,T}$, which is a risk-free profit.

The above two arbitrage examples establish equation (1) as the correct forward price of a non-dividend paying stock.

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Forward price on a stock – discrete dividends

We now consider the case that the stock pays dividends in known amounts at known times during the life of the forward contract. In other words, this is the case that the frequency, the timing of the dividends and the amounts of the dividends are known ahead of time. To determine the forward price $F_{0,T}$, we still consider the two ways to own a share at time $T$.

• Buy a share at time 0 and hold it until time $T$.
• Enter into a forward contract to buy one share of the stock at time $T$.

However, there is now an important difference between these two ways. It is that the owner of the stock in the first way receives the dividends during the contract period while the owner of the forward contract is not entitled to receive dividends. By the time the forward contract owner receives the share at time $T$, she has missed out on all the dividend payments. So the forward contract owner must be compensated for the missed dividend payments. Consequently the forward contract owner should pay less than the price for an outright purchase at time 0. How much less? By the amount of the dividends. So we need to subtract the value of the dividends from the stock price.

The price of the first way (outright stock ownership at time 0) accumulated to time $T$ is $S_0 \ e^{rT}$. Thus we need to subtract the cumulative value of the dividends from this price.

$\displaystyle F_{0,T}=S_0 \ e^{rT}-\text{CV of the dividends} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

Here CV means cumulative value. To be more specific, suppose that during the contract period, there are $n$ dividend payments $d_1,d_2,\cdots,d_n$ received at times $t_1,t_2,\cdots,t_n$. Then the forward price should be $S_0 \ e^{rT}$ subtracting the future values of the dividends at time $T$.

$\displaystyle F_{0,T}=S_0 \ e^{rT}-\sum \limits_{j=1}^n \ d_j \ e^{r (T-t_j)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$

We illustrate the forward prices discussed thus far with an example.

Example 1
Suppose that the stock for XYZ company is $50 today. The annual continuously compounded risk-free rate is 3%. Calculate the following: • The price for a forward contract to deliver 500 shares of XYZ 6 months from now, assuming that the stock pays no dividends. • The price for a forward contract to deliver 500 shares of XYZ 6 months from now, assuming that the stock pays quarterly dividend of$1.50 with the first one occurring 3 months from now.

First consider the no dividend case. The forward price for one share is:

$F_{0,0.5}=50 \times e^{0.03(0.5)}=50 \times e^{0.015}=50.75565$

Then the forward price for the contract is $500 F_{0,0.5}=25377.83$.

Now consider the case with dividends. There are two dividend payments in the contract periods. The first one is accumulated forward for 3 months and the second one is assumed to be paid at expiration. The forward price for one share is:

$F_{0,0.5}=50 \times e^{0.03(0.5)}-1.50 e^{0.03(0.25)}-1.5=47.74436$

Then the forward price for the contract is $500 F_{0,0.5}=23872.18$.

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Forward price on a stock index – continuous dividends

We now consider an asset that pays dividends at an annual continuously compounded rate that is denoted by $\delta$. The dividends are paid continuously and reinvested back in the asset. So instead of receiving cash payment, the owner gets more shares. If the investor starts out with one share at time 0, she ends up with $e^{\delta \ T}$ shares at time $T$.

For a stock index containing many stocks, assuming a continuous compounded dividend rate will simplify the discussion.

The forward contract owner wishes to pay $F_{0,T}$ for one share of the stock index at time $T$. Again, there are two ways to do this.

• Buy a $e^{-\delta \ T}$ shares at time 0 and hold them until time $T$.
• Enter into a forward contract to buy one share of the stock index at time $T$.

Recall that the dividends come in the form of additional shares. To get one share at time $T$, we need to start with $e^{-\delta \ T}$ shares at time 0. So in the first way, we need to pay $S_0 e^{-\delta \ T}$ at time 0. In the second way, the value at time 0 of the forward price is $F_{0,T} \ e^{-rT}$. Again, to avoid arbitrage, the two present values must equal. We have $S_0 e^{-\delta \ T}=F_{0,T} \ e^{-rT}$, producing the following:

$F_{0,T}=S_0 \ e^{(r- \delta) \ T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$

In this case of continuous dividends, the dividends come in the form of additional shares. The forward contract owner misses out on the additional shares. So the forward contract owner must be compensated for not receiving the additional shares. Equation (5) indicates that the buyer of the forward contract is compensated by getting a smaller interest rate $r-\delta$. So in this sense, the dividend rate is like a negative interest rate.

We now illustrate the continuous case with an example.

Example 2
Suppose that a stock with current stock price of $50 pays a 10% continuous annual dividend. The annual continuously compounded risk-free rate is 4%. What is the price for a forward contract for the delivery of 100 shares of XYZ? The contract is to be expired 1 year from now. If you observe a forward price of$49 on a contract on the same stock with the same expiration date, what arbitrage strategy would you use?

The forward for price for one share is:

$F_{0,1}=50 \times e^{(0.04-0.10) \times 1}=50 \times e^{-0.06}=47.08822668$

The forward price for 100 shares is $100 F_{0,1}=4708.82$. If you observe a forward price of 49 instead of the true theoretical forward price of 47.088, do the following “buy low sell high” strategy.

$S_0=50$, $\delta=0.10$, $r=0.04$ and $T=1$.

Borrow $S_0 e^{- \delta T}$ to buy $e^{- \delta T}$ shares at time 0. Simultaneously sell a forward contract to buy one share at the forward price $49 one year from now. At the end of one year, $e^{- \delta T}$ becomes 1 share. As the party holding the short forward position, sell the one share at$49. Then repay $S_0 e^{- \delta T} e^{r T}=S_0 e^{(r- \delta) T}=47.088$ to the lender. This produces a risk-free profit of $19.12 per share. The profit for 100 shares is$191.2.

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Currency forward price

We use a dollar/euro example to illustrate. Suppose that we want to obtain one euro at time $T$ by paying dollars. There are two risk-free rates here as there are two currencies. Let $r$ be the risk-free rate of the domestic currency (US dollars) and let $r_f$ be the risk-free rate of the euro (here the subscript stands for foreign). Let $x_0$ be the exchange rate (dollar per euro) at time 0. Once again, there are two ways to obtain one euro at time $T$. The first way is to pay US dollars to buy euros now. Let’s work backward. To get one euro at time $T$, we need to have $e^{-r_f T}$ euro at time 0. Thus we need to have $x_0 e^{-r_f T}$ dollars at time 0. We have the following two ways.

• Exchange $x_0 \ e^{-r_f T}$ dollars into euros at time 0 and hold them until time $T$.
• Enter into a forward contract to buy one euro at time $T$.

In the first way, we need to have $x_0 \ e^{-r_f T}$ dollars ready at time 0. In the second way, we need to have $F_{0,T}$ ready at time $T$ or have $F_{0,T} \ e^{-r T}$ ready at time 0. Equating the two, we have $F_{0,T} \ e^{-r T}=x_0 \ e^{-r_f T}$, leading to the following:

$F_{0,T}=x_0 \ e^{(r- r_f) \ T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)$

Note that equation (6) is just like equation (5). So the risk-free interest rate for the foreign currency plays the role of a continuous dividend rate. One way to interpret the rate difference $r-r_f$ is that it is the cost of carry for a foreign currency. In this interpretation, we borrow at the domestic rate $r$ and invest the borrowed funds in a foreign risk-free account paying at the rate $r_f$. The earnings from the foreign account will offset the cost of the domestic borrowing.

Example 3
Suppose that a dollar denominated forward contract calls for the delivery of 10 million yens at the end of 6 months. Suppose that the annual continuously compounded risk-free rate for yen is 3% and the annual continuously compounded risk-free rate for dollars is 1%. Currently the dollar/yen exchange rate is $0.008 per yen. Calculate the forward price in dollars for this contract. Plugging in all the relevant inputs, the following dollar forward price per yen: $F_{0,0.5}=0.008 \ e^{(0.01- 0.03) \ 0.5}=0.008 \ e^{-0.01}=0.007920399$ Thus the dollar forward price per 10 million yens is 10000000 (0.007920399)=79203.9867. ___________________________________________________________________________________ Remarks One idea emerged in the derivation of the above forward prices is that dividends have the effect of a negative interest rate. The dividend payments are not received by the forward contract buyer since she only receives the stock at the expiration date. As a result, the forward contract buyer must be compensated for missing out on the dividends. In general, if the asset produces income before the forward contract buyer receives the asset, the effect of the missed income would be like a negative interest rate (to compensate for the missed income). This idea also applies to commodity forward pricing. If a commodity has income streams (e.g. it can be leased out), the forward price must reflect this negative interest rate. On the other hand, if a owning a commodity incurs expenses (e.g. storage costs), then the forward contract buyer will have to pay more for the commodity since the forward contract buyer has to compensate the commodity owner for the expenses. The following table summarizes all the forward prices discussed above. The price $F_{t,T}$ is the forward price of an asset set at time $t$ to be purchased at a future time $T$. All the other variables are as discussed above. $\text{ }$ $\left[\begin{array}{ll} \text{Underlying asset} & \text{Forward price} \\ \text{ } & \text{ } \\ \text{Non-dividend paying stock} & F_{t,T}=S_t \ e^{r(T-t)} \\ \text{ } & \text{ } \\ \text{Dividend paying stock (discrete)} & F_{t,T}=S_t \ e^{r(T-t)}-\text{CV of the dividends} \\ \text{ } & \text{ } \\ \text{Dividend paying stock (continuous)} & F_{t,T}=S_t \ e^{(r- \delta) \ (T-t)} \\ \text{ } & \text{ } \\ \text{Currency} & F_{t,T}=x_t \ e^{(r- r_f) \ (T-t)} \end{array}\right]$ ___________________________________________________________________________________ $\copyright \ \ 2015 \ \text{Dan Ma}$ ## An introduction to forward contracts Basic derivatives contracts are forward contracts, futures contracts, call options and put options. These are contracts between two parties with a payoff in a future date and the payoff is tied to the price of an underlying asset (hence the name derivative). Here are some natural questions. What are derivatives? How do they work? What do they cost? Why do investors or firms want to enter into such a contract? If you enter into such a contract, what are the obligations and benefits? This post is a brief introduction of forward contracts. In this post, we explain the basic features of forward contracts. In the next post, we discuss the prices of forward contracts. ___________________________________________________________________________________ Forward contracts – basic features Forward contracts allow firms or investors to lock into a price today for a purchase or sale of an asset that will take place at some point in the future. Thus forward contracts are a basic financial risk management tool. A forward contract is a customized contract between two parties to buy or sell an asset at a specified price on a future date. Each forward contract will specify the following: • The underlying asset. • The expiration date. • The forward price. • The contractual obligation. The underlying asset of a forward contract is the financial asset or commodity the seller must deliver to the buyer. A forward contract will specify the quantity and the type of assets to be transacted. The assets that are transacted in the future can be financial assets (e.g. stock, currencies, and interest rates) or commodities (e.g. gold, corn and natural gas). The expiration date is the time at which the contract is settled. The forward price is the price (per unit of the asset) that is agreed upon today by both parties of the contract to transact in the future (i.e. on the expiration date). On the expiration date, the buyer pays the seller the forward price and the seller delivers the asset to the buyer. The last bullet point refers to the obligation that the buyer must buy and the seller must sell on the date of settlement even if the price movement means a loss to one of the parties in the contract. The party who commits to buying the asset on the delivery date is said to take a long position (or entering into a long forward). The party who commits to selling the asset on the delivery date is said to take a short position (or entering into a short forward). The party in the long position is said to “buy” a contract and the party in the short position is said to “sell” a contract. Generally the long position is the party that makes money when the price of the asset goes up and the short position is the party that makes money when the price goes down. Note that the long position agrees to buy the asset at pre-agreed upon fixed price. So the long position profits when prices rise. The payoff to a forward contract is the value of the position at expiration. The following shows the payoff. Payoff to long forward = spot price at expiration – forward price Payoff to short position = forward price – spot price at expiration For a forward contract, the settlement can take place on a delivery basis (i.e. there is a physical transfer of the asset) or on a cash basis. If settlement is on a cash basis, the long position will receive the payoff to the long position (which can be positive or negative) and the short position will receive the payoff to the short position (which could be positive or negative). For an illustration of how this works, see the example discussed below. ___________________________________________________________________________________ Futures versus forwards Futures contracts are similar to forward contracts in that a futures contract is also a contract between two parties to buy or sell an asset at a specified price on a future date and that both parties have the obligation to do so on the delivery date. But there are important institutional and pricing differences. For examples, futures contracts are exchange traded while forward contracts are traded over the counter. Thus futures contracts tend to be standardized and forward contracts are customized contracts. The futures exchange standardizes the types of contracts that may be traded. For example, it establishes contract size, the acceptable grade of commodity (if it is a commodity exchange), contract delivery dates and so on. On the other hand, forward contracts can be tailored to meet the unique or unusual requirements of the party seeking the contract. Futures contracts also have the advantage of liquidity and a reduced level of credit risk. Standardization means that many traders will concentrate on a small set of contracts. Thus trading in futures contracts can be highly liquid. Many futures contracts can be liquidated through a broker rather than directly with the counter party of the contract. Another difference between futures and forwards is that the exchange requires the parties in a futures contract to settle any gains or losses on the contract on a daily basis (this is called the marking to market). The daily marking to market required by the exchange amounts to a guarantee of performance of the parties in a contract. Thus costly credit checks on the counter party of a futures contract are not necessary. On the other hand, for a forward contract, no money changes hand until the delivery date. So credit risk, the risk that the other trader will not perform, can be a much greater concern for traders entering into forward contracts. ___________________________________________________________________________________ Forward contracts – an example To see how forward contracts work and how they might be used to hedge risk, let’s look at an example. Suppose that a corn producer has 2 million bushels of corn to sell in 6 months. The corn producer is concerned about a substantial price decline of corn. A food company that uses corn has the opposite risk in that its profit will suffer due to a substantial price increase in corn. Both parties can offset this risk if they enter into a forward contract that obligates the corn producer to sell 2 million of bushels of corn in 6 months to the corn user at the forward price of$3.85 per bushel with settlement on a cash basis. The corn producer is the short position in this contract and the food company (the corn user) is the long position. The corn producer is obligated to sell 2 million of bushels of corn to the corn user at the agreed upon price of $3.85 per bushel. The corn user is obligated to buy the corn from the producer at$3.85 per bushel.

There is no cash changing hand between the two parties at the time the contract is made. A forward contract is in essence a purchase or sale of an asset such that the delivery of the asset is deferred. The money is exchanged at the time of settlement but the price is set ahead of time when the contract is made. Such a contract locks in the price to be paid or received for the delivery of the asset or commodity in question. The effect is that the contract protects each party from future price fluctuation. Let’s look at a numerical example. After 6 months, considering the following 3 possibilities:

1. There is essentially no change in corn prices.
2. The spot price is lower than the forward price, say $3.35 per bushel. 3. The spot price is higher than the forward price, say$4.35 per bushel.

In the first case, there is no exchange of cash between the corner producer and the financial institution. If there is only a small difference between the spot price and the forward price, there is essentially no risk to either party.

In the second case, the payoff per bushel to the short position is $3.85 –$3.35 = $0.50. Thus the corn user owes the corn producer$1,000,000. Thus the corn producer is protected from a price decline.

In the third case, the payoff per bushel to the short position is $3.85 –$4.35 = -$0.50. Thus the corn producer owes the corn user$1,000,000. Here, the corner user is protected from a price upswing.

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Remarks

The above example shows that forward contracts (or futures contracts) can be used to hedge price risk. A previous post gives an example of how forward contracts can be used to hedge exchange rate risk for exporter (or investor investing abroad). These two examples illustrate one useful feature of forward contracts. Corporations use forward contracts is to offset their risk exposures and limit themselves from any fluctuations in price. When a company knows that it will have a need to purchase an asset or commodity in the future, it can take a long position in a contract to hedge its position. When a company knows that it will have a need to sell an asset or commodity in the future, it can take a long position in a contract to hedge its position. Forward contracts can be very useful in limiting the price risk exposure faced by a corporation. The main advantage of using forward contracts is that it removes the uncertainty about the future price of an asset or commodity, thus eliminating the uncertainty in profit due to unpredictable fluctuation in prices.

In the next post, we discuss the pricing of forward contracts.

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

## An example of a currency forward contract

In this post, we use an example to illustrate how a forward contract can be used to hedge exchange rate risk. Consider the following two examples.

1. An exporter in the United States sells high tech manufacturing equipment to a Canadian importer. The total amount of goods sold is 11 million Canadian dollars (CAD), to be received by the exporter one year from now.
2. An investor in the United States invests in one-year risk-free Canadian government bills that pay 10% annually. The US investor starts with US $8 million and exchanges the amount for C$10 million at the exchange indicated below. At the end of the year, the investor expects to receive C $11 million. In both examples, we assume that the current US-Canada exchange rate today is US$0.80 per C $1 (Canadian dollar, CAD). We use the two examples to illustrate the inherent risk when selling or investing abroad. Assume that the 11 million Canadian dollars received for the manufacturing equipment represent good profit. When the exchange rates move in the wrong way, the exporter may end up with a loss when the Canadian dollars are converted into US dollars. On the other hand, the risk-free government bills are risk-free investment for Canadian investors. But for US investors that have to convert the proceeds back into US dollars, the end result may be a loss. In this post, we use these two examples to illustrate the potential danger of exchange rate risk and discuss some of the ways to mitigate this risk. In some sense, the two examples are one example. The exporter or investor will receive C$11 million in one year (the same cash flow made in the same currency and then converted back to the same domestic currency). Thus both examples are subject to the same dynamics inherent in the movement of exchange rates. Even though both business contracts are profitable in terms of Canadian dollars, when the exchange rates move in an adverse direction, the parties in both examples could potentially experience a loss. Thus both examples can be discussed as one example.

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The exchange rate risk

Let’s look at how movements of exchange rates can impact the above example positively or negatively.

As mentioned at the beginning, the initial exchange rate is US $0.80 per C$1. Suppose that the exchange rate is US $0.9 = C$1 a year later. This means that the Canadian dollar has gained in value against the US dollar (1 Canadian dollar can now buy more US dollars). The Canadian fund to be received is now worth more in US dollars. The 11 million Canadian dollars are now $9.9 million US dollars. This amount is US$1.1 million over the US $8 million the investor spent in buying the Canadian bills. So this is a good outcome for the exporter or investor. On the other hand, suppose that the exchange rate is US$0.7 = C $1 one year later. This means that the US dollar has gained in value against the Canadian dollar (1 Canadian dollar now buys less US dollars). So the Canadian fund to be received is worth less in US dollars. The 11 million Canadian dollars would only be worth US$7.7 million. This amount is US $1.1 million less than the US$8 million the investor spent in buying the Canadian bills. For the investor in Canadian government bills, this represents a negative rate of return (investing US $8 million and receiving US$7.7 million a year later).

So fluctuation in exchange rates has real financial consequences for anyone who is involved in the import/export business or investing abroad. The example in this post highlights the risk when the receivables in the investment are in a foreign currency. Any exporter who received foreign funds for the goods that they sell will fear the appreciation of their domestic currency, equivalently the depreciation of the foreign currency. The same can be said for any investor who has to convert the foreign investment proceeds into their domestic currency.

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Ways to mitigate the exchange rate risk

The example illustrates that a business that exports or imports goods and services and investors who invest internationally need to consider how to protect against adverse movements in exchange rates. Even a small variation in the exchange rate can result in a big financial loss.

One way to hedge the exchange rate risk is to use a forward contract. A foreign currency forward contract is a contract to buy or sell a specific amount of a currency at a fixed exchange rate at a specific time in the future.

If the exporter/investor worries that the Canadian dollar will depreciate in one year so that the Canadian dollars received will be worth less, they can hedge this risk by entering into a forward contract. Here’s how. At the time the export contract or investment contract is made, the exporter or investor would agree to sell 11 million Canadian dollars for US dollars at a fixed exchange rate (this is called the forward exchange rate) one year later. The counter party to the exporter/investor, usually a bank, will agree to buy 11 million Canadian dollars at the terms specified in the contract.

Suppose that the forward exchange rate is US $0.77 per Canadian dollar. When the exporter or investor receives the 11 million Canadian dollars a year later, he/she can exchange the Canadian dollars for 0.77 x 11 = 8.47 million US dollars. Entering into this forward contract has the effect of locking in a favorable exchange rate one year ahead of time. If one year later, the exchange rate is US$0.70 per C $1 (the Canadian dollar has weakened since one Canadian dollar can buy fewer US dollars), then the exporter/investor will only get US$7.7 million if there is no forward contract. With the forward contract, the amount of US dollars the exporter/investor will receive will still be 8.47 million US dollars. This would be a happy outcome for the exporter/investor.

If the exchange swings the other way and if there is no forward contract, it would be a profitable scenario. However, the forward contract is legally binding. That means the exporter/investor will have to honor the forward exchange rate set at the beginning. For example, if one year later, the exchange rate is US $0.90 per C$1 (the Canadian dollar has strengthened since one Canadian dollar can buy more US dollars), the exporter/investor will not be able to realize the profit and has to honor the forward exchange rate of US $0.77 per C$1. The amount of US dollars received will still be $8.47 million. There are other ways to hedge the same risk. For example, the exporter/investor can also enter into a foreign currency futures contract or buy a foreign currency option. Another possibility is to open a bank account denominated in the foreign currency that is received. In this particular example, the exporter/investor can open a bank account denominated in Canadian dollars (either in Canada or in the US). The Canadian funds can be deposited into such an account and then converted back to US dollars when the exchange rate is favorable. There are pros and cons for each option, and which method to use depends on the business needs of the exporter/investor. We would like to point out that the basic working of futures contracts is similar to forward contracts. Both futures contract and forward contract allow people to buy or sell a specific type of asset at a given price at a specific time in the future. One important difference is that futures contracts are exchange-traded and thus are standardized contracts. On the other hand, forward contracts are private agreements between two parties and can be tailored to meet the unique or unusual requirements of the party seeking the contract. There are other differences but we will not discuss them here. Our purpose here is to give a basic illustration on how to hedge the exchange rate risk. ___________________________________________________________________________________ What happens if the export proceeds are made in the domestic currency? In the exporter example discussed here, suppose that the business agreement is for the exporter to receive the export proceeds in US dollars (the domestic currency of the exporter). Then the exchange rate risk is transferred from the exporter to the importer. The exporter will not worry about getting less US dollars when the exchange rate moves in the wrong way, since he/she will be paid in US dollars. The importer will have to exchange Canadian dollars into US dollars to pay for the imported goods. Then it is the importer who bears the exchange rate risk and finds ways to hedge the risk. Let’s say in the above example, the exchange rate profile is switched. The total amount of goods is worth CAD$11 million, which is worth US $8.8 million based on the exchange rate at the time the business contract is made (US$0.80 per CAD). Let’s say the importer will have to pay US $8.8 million one year after the business contract is made. The importer will fear that the US dollar will get stronger, in which case, he/she will have to come up with more Canadian dollars to buy the same amount of US dollars. To hedge this risk, the importer can, at the time the business deal is made, enter into a foreign currency forward contract to buy US$8 million at a fixed exchange rate one year later.

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Remarks

The hedging in the example discussed above will completely eliminate the exchange rate risk for the exporter/importer or investor. Such perfect hedging may not be possible in all circumstances. In the above example, we know the exact amount of foreign currency to be received or paid. We are also certain of the timing of the payments. We also have the certainty of the payments – the investment is in risk-free bonds for example. If the investment is not in risk-free bonds but is instead in risky Canadian equity, we would know neither the ultimate values in Canadian dollars of the Canadian equity nor how many Canadian dollars to sell forward one year later. In the exporter/importer example above, we assume that there is no credit risk – that is, payments will always be made at the appointed time. If that is not the case, the hedging tool of the forward contract will not be effective.

The risk-free interest rate of 10% in the Canadian government bills in the above example is probably not a realistic rate. Currently both US and Canada are low interest rate environments. Both countries are well functioning economies and the exchange rate movements between US and Canada are typically not volatile. So the example here is only meant to serve the purpose of illustrating the hedging of exchange rate risk.

The next post gives an introduction to forward contracts.

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