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}<S_0 \ e^{rT}. 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.

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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]

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

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