## 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

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

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

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

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