Copying and distributing are prohibited without permission of the publisher. # Is A Forward Contract Always A Delta One Trade?

10 Aug 2003

Many derivatives professionals will immediately call a forward contract a delta one trade. That's because they believe that a forward sale position can always be perfectly hedged by buying the same amount of the underlying asset at the spot price and vice versa. This is also known as a static hedge because once it is executed it does not have to be altered for the duration of the contract.

But is that always true? Below is a quick example that illustrates that a forward contract does not necessarily have a delta equal to one. To simplify the calculations, let's assume interest rates are 0 (r=0%).

Stock ABC has a spot price of USD100 and it pays no dividend. The one-year forward sale price on a share of ABC stock, based on simple cash-and-carry formula, will be USD100 (with no interest nor dividend, there is no cost of carry). To hedge your position, you borrow USD100 today and buy one share of ABC stock. After a year, you deliver the share of ABC stock and receive USD100, which is then used to pay down the USD100 loan. You have closed out both the forward contract and loan contract, and the delta one position does create a risk-free hedge.

The same still holds if the ABC stock pays a fixed amount of dividend, for example USD1 per year. The one-year forward sale price, again using the cash-and-carry formula, will be USD99. To put on a delta-hedge, you borrow USD100 today and buy one share of ABC stock. During the year, you receive USD1 dividend income. After a year, you deliver the share and receive USD99 dollars. With USD100 in your hands, you repay your loan completely (USD99 from the forward contract and USD1 dividend income). The delta one strategy still works.

But things become different if instead of paying USD1 dividend, ABC stock pays out a dividend with a 1% yield. Similar to the above example, the forward price will remain USD99 and a delta one hedge involves borrowing USD100 today and buying one share of ABC stock. However, assume during the year, the stock price drops to USD50 when it pays out the dividend, and the dividend is USD0.50 per share (USD50 *1% = USD0.50). At maturity, you deliver the share and receive USD99. You are 50 cents short to repay the loan. The cash flows are as follows: For sure, if the stock price goes to USD150 instead of USD50, the delta one position will earn 50 cents instead (dividend income will then be USD1.50). But that is speculation, not hedging.

The right hedge, for the forward contract on a 1% dividend yield, should be .99 (99%). So instead of buying one share today, you borrow USD99 and buy a .99 share of ABC stock. When the stock drops to USD50 and pays a dividend of 50 cents, you can use the dividend income to buy .01 share to increase your holding to 1 share. At maturity, you deliver one share and receive USD99, which repays the original loan completely. The cash flows are as follows: The above examples demonstrate that the delta for forward contracts is not necessary equal to one and the right hedge is not always static. In the real world, the risk-free interest rate will not be zero, but it could be shown that, following similar analysis, the right initial hedge should still be, approximately, .99 (similar to 0.99/1, it's 1.04/1.05 = .9905).

Analysis

The above example could be explained from an over-hedge perspective. With the risk free rate at 5% and dividend rate 1%, the forward contract accretes at 4% (5%-1%, since the dividend element is priced out from the forward contract price), while the hedge position grows at 5% (the hedge position receives the stock dividend). The right hedge is .99 (= 1.04/1.05). Putting on a delta one position causes an over-hedge of a .01 long position in the underlying stock.

That's why the combined position results in a net loss when the stock price drops (loss on the over-hedge) and a net gain when stock price grows (gain on the over-hedge). When the stock price drops from 100 to 50, the loss on the over-hedge is USD0.50, i.e., .01*(50-100) = USD.50; similarly, when the stock price rises from 100 to 150, the gain on the over-hedge is USD0.50 (01* (150-100) = USD.50). They are the same exact numbers shown in the above example.

Fundamentally, we can view the dividend yield term as another instrument with a non-zero delta. The dividend, expressed in yield term, is a factor that increases in value when the stock price goes up and decreases when the stock price drops. This type of price sensitivity with respect to stock price is exactly the definition of delta.

So what's the delta on the dividend term? Not surprisingly, it's .01:

Delta = change of dividend value/change of stock price
(S) = 1% * S/S = .01

Since the forward contract priced out the dividend term (a negative carry for the forward contract's cash and carry pricing), our hedge position should short an amount equivalent to the delta of the dividend term (.01), which, it follows, reduces the overall delta for the forward contract from 1 to .99.

More generally, the above analysis reminds us that we have to take into account all the factors that move with the stock price when we determine the right delta for a forward contract. Delta is defined as the ratio of the change in the price of a derivative instrument to the change in the price of the underlying asset. So it's quite natural that any factor with value sensitivity regarding the underlying stock will have an impact on the delta of the forward contract, making it different from one. In addition to dividends, borrowing cost for hedging and balance sheet usage costs are all examples of such factors.

Conclusion

The right hedge for forward contracts does not always have a delta equal to one. Dividends, borrow cost and balance sheet usage costs are examples that could potentially make delta on the forward contracts different. In determining the right delta for a forward contract, derivatives professionals should take into account all factors that have price sensitivity with respect to the underlying asset.

This week's Learning Curve was written byWinston Wenyan Ma, associate in the equity derivatives products and solutions group atJPMorgan Securitiesin New York.

10 Aug 2003
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