Building Intuition for the Derivative How do you wish the derivative was explained to you? The derivative is the heart of calculus, buried inside this definition: But what does it mean?
With differing values of national currencies, international traders needed a system to account for these differences.
Today, derivatives are based upon a wide variety of transactions and have many more uses. There are even derivatives based on weather data, such as the amount of rain or the number of sunny days in a region. There are many different types of derivatives that can be used for risk management or for speculation.
For example, imagine a European investor who purchases shares of a U. This investor is exposed to exchange-rate risk while holding that stock. If the value of the euro rises relative to the dollar, the investor's profits in dollar terms are less valuable when those profits are converted back into euro once the stock is sold.
To hedge this risk, the investor could purchase a currency derivative to lock in a specific exchange rate. Derivatives that could be used to hedge this kind of risk include futures and currency swaps.
A speculator who expects the euro to appreciate compared to the dollar could profit by using a derivative that rises in value An understanding of the overall concepts of derivative the euro. When using derivatives to speculate on the price movement of an underlying asset, the investor does not need to have an interest in the underlying asset.
Many derivative instruments are leveraged. That means a small amount of capital is required to have an interest in a large amount of value in the underlying asset. This gives the futures investor a leverage ratio greater than 8: The required margin to hold a futures or derivative position changes depending on market conditions and broker requirements.
Common Forms of 'Derivative' Futures Futures contracts are one of the most common types of derivatives. A futures contract or simply, futures is an agreement between two parties for the purchase and delivery of an asset at an agreed upon price at a future date.
Futures are exchange traded, and the contracts are standardized.
Traders will use a futures contract to hedge their risk or speculate on the price of an underlying asset. For example, on Nov. The company wants to do this because it needs oil in December and is concerned that the price will rise before the company actually needs to make the purchase.
Company-A can accept delivery of the oil from the seller of the futures contract, but if it no longer needs the oil, it can also sell the contract before expiration and keep the profits.
In this example, it is possible that both the futures buyer and seller were hedging risk. Company-A needed oil in the future and wanted to offset the risk that the price may rise in December with a long position in an oil futures contract.
The seller could be an oil company that was concerned about falling oil prices and wanted to eliminate that risk by selling or " shorting " a futures contract that fixed the price it would get in December.
It is also possible that the seller or buyer or both of the oil futures contract were speculators with the opposite opinion about the direction of oil in November and December.
If the parties involved in the futures contract were speculators, it is unlikely that either of them would want to make arrangements for delivery or shipment of crude oil. Speculators can end their obligation to purchase or deliver the underlying commodity by closing their contract before expiration.
Not all futures contracts are settled at expiration by delivering the underlying asset. Many derivatives are cash-settledwhich means that the gain or loss in the trade is a positive or negative cash flow to the trader. Forwards Forward contracts are an important kind of derivative similar to futures.
Unlike futures, forward contracts or "forwards" are not traded on an exchange, only over-the-counter. The buyers and sellers of forward contracts also have counterparty risks. Counterparty risks are a kind of credit risk in that the buyer or seller may not be able to live up to the obligations outlined in the contract.
If one party of the contract becomes insolventthe other party may have no recourse and could lose the value of its position. Once created, the parties in a forward contract can offset their position with other counterparties, which can increase the potential for counterparty risks as more traders become involved in the same contract.
Swap Swaps are another common type of derivative that is often used to swap one kind of cash flow with another.
For example, one might use an interest rate swap to switch from a variable interest rate loan to a fixed interest rate loan, or vice versa. Imagine that InvestCo, Inc. InvestCo may be concerned about rising interest rates that will increase the costs of this loan or encounter a lender that is reluctant to extend more credit while InvestCo has this variable rate risk.
Regardless of how interest rates change, the swap has achieved InvestCo's original objective of turning a variable rate loan into a fixed rate loan. Swaps related to the cash flows and potential defaults of mortgage bonds are an extremely popular kind of derivative. The counterparty risk of swaps like this eventually spiraled into the credit crisis of Options Options are another common form of derivative.PS: I should add that understanding of this point should also be considered more important that technical details when calculus is taught to physics majors or math majors, etc.
But if the latter have even a shred of competence, one need not explain that to them. (B). The derivative of the composition of two functions is given by the chain rule. Question 3: lim [e x-1] / x as x approaches 0 is equal to (A) 1 (B) 0 (C) is of the form 0 / 0 and cannot be calculated.
The definition of the derivative at x = a is given by f '(a) = lim [f(x) - f(a)] / (x - a) as x approaches a. The derivative of is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative). It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).
the tangent equation is the derivative function, the derivative at a point is the tangent equation, and, finally, the derivative at a point is the value of the tangent equation at that point (Ubuz, , p.
). Good to see students get something here but calculus needs a unified approach and the understanding of the derivative begins with a strong foundation in algebra (coordinate geometry) and pre-calculus.
The Derivative as a Function Notice that there is nothing in the development above that restricts the choice of the point a.
Since a is arbitrary and the deﬁnition of f0(a) does not use the symbol t, there is no loss of generality in rewriting the deﬁnition so that the derivative is a function of t, rather than merely the slope at a particular point.