Financial Derivatives/Introduction

A financial derivative is a financial instrument whose value is derived from the price of an asset (or a number of assets).

We live in a world where commodity prices can increase dramatically and then collapse, property prices can reach vertiginous levels and uncertainty is prevalent in all facets of economic life. But if we look more closely at this dynamic economic picture "risks" are not equally shared or perceived: for an airline company rising oil prices mean increased costs and the risk of reduced profits; for an investor looking for higher returns than a treasury bond, investment in oil is rewarded with high returns. Although for both parties the future is equally uncertain, each party has different exposure to the same set of future scenarios.

One can therefore see a role for a financial institution that offers to protect a party against a set of future scenarios—for a price of course. For the airline company, the financial institution can offer to sell jet fuel at a fixed price if the price of jet fuel at some future date is above this fixed price. In a way, the financial institution has created a derivative world for the airline company in which the price of jet fuel cannot go above the fixed price specified in the contract.

For the investor the financial institution can offer protection against his bet turning sour. Having invested $50 mm in oil contracts the prospect of oil prices falling by the time he liquidates his investment is not a welcome prospect. But the investor can pay a premium (the value of the derivative contract) and in return the financial institution can guarantee that if the price of oil drops below a fixed price at a future date, the investor will receive his original investment of $50 mm as if nothing happened.

As an intermediator, the financial institution has combined two opposing views of risk to create derivative contracts that are beneficial to both parties. In this example, it is almost as if the financial institution created a new market on "future events" and has sold the asset "price of oil above $50 in one year" for one price and the asset "price of oil below $50 in one year" for another price. This the financial institution achieved by offering transparent financial contracts, that specify the payoff at a future time as a mathematical function of the price of oil at that time.

The dramatic expansion of derivatives markets since the late seventies is in large part the result of the pioneering work in the field of neoclassical finance [Ross 2004]. The concepts that led to the historic breakthrough of Black-Scholes-Merton and the arbitrage-free pricing of options, were based on a new paradigm. Economics treats the value of asset as a function of supply and demand. The price of oil is determined by the worldwide demand for oil and the current production of oil. In the same way, one can assume that the value of a derivative will be determined by the number of companies that are exposed to the risk it covers and the number of financial institutions that are willing to take exposure to this risk. In this way of thinking, a derivative is just another asset albeit one that at some future date has a payoff determined by a formula.

This approach is valid if one assumes that a financial institution sells a derivative to an airline company and then moves on to the next deal. But if we look at the previous example, where it has guaranteed to sell oil in one year at a fixed price of $50 if the oil price is above $50, the financial institution runs the huge risk of buying oil whose price is $100 and then selling it for $50. Whatever the premium paid by the airline it will not be enough to cover it for such a disastrous scenario.

If the price of this derivative was determined by supply and demand then it is unlikely that a viable market would exist. Although the perception would be that the probability of an oil price of $100 in one year is less than 1%, the enormity of the losses that would occur under this scenario would force the financial institution to ask for a high premium. But the high cost of purchasing this derivative would mean that the airline company would not fare better by covering its exposure to high oil prices.

If the financial institution was able to hedge the derivative it sold to the airline company this would mean that its exposure to high oil prices would be neutralised. The definition of a hedging strategy is a trading strategy that during the life of a derivative contract neutralises the exposure of the resulting portfolio to changes in the price of the underlying asset. Note that by hedging a derivative contract one does not neutralise the exposure of the derivative to the price of the underlying asset. It is the combined portfolio of the derivative and the instruments used for the hedging strategy that has no exposure to variations in the price of the underlying asset. Under this new paradigm, the price of a derivative is not independent of the hedging strategy. And given a hedging strategy, there exists a unique, risk-neutral price for the derivative contract that is independent of supply and demand.

A simple illustration of this concept can be provided by the following example: suppose that an oil company has built an oil platform. For the project to be viable it must be able to sell oil in one year at least for $50. The financial institution guarantees that it will buy oil from the oil company at $50 in one year. Under the terms of this contract the income of the financial institution depends on the value of oil prices in one year.

The financial institution now sells a derivative that guarantees the airline company a fixed price for oil in one year if prices are above $50. Since the financial institution will pay $50 for the oil produced it agrees to do that for a small premium. For all scenarios where the future price of oil is above $50 it will have an income equal to the premium (premium A).

The financial institution also agrees with an investor that if the oil price in a year is below $50 it will sell oil to the investor at this fixed price. For this the financial institution pays the investor a premium (premium B).

Given these three transactions, the income of the financial institution in one year is equal to the difference between premium A and premium B under all future price scenarios. The financial institution has no exposure to increases or decreases in the oil price one year from now. Therefore it is able to value the two derivative contracts from a risk-neutral perspective rather than from its subjective perception of the impact of future price scenarios to its future income.

(One issue with this example is why would the investor and the airline company be convinced to do these deals at risk-neutral prices. After all both the airline company and the investor derive their income precisely because of their exposure to some form of risk. The answer is that because of their specific exposure to different sources of risk they tend to overvalue risk from the perspective of a risk-neutral financial institution. As a result the premiums they pay for reducing their exposure to specific price scenarios seem low given their valuation of this exposure. The "paradox" of derivatives existing in a market where the majority of participants do not use risk-neutral pricing is no paradox; in fact if market participants valued their exposure to risk in a risk-neutral framework this would mean that they were already hedged and there would be no need for a derivatives market.)

In a formal setting, given a set of possible scenarios $$ \omega $$ for the price of oil one year from now the income of company $$ i $$ is $$ C_i(\omega) $$. The bias of a specific company $$ i $$ to a set of scenarios $$ \Omega_i $$ can be understood as its business strategy. For all scenarios in this set, the income of the company is positive while for all scenarios outside this set the income of the company is negative. Company $$ i $$ will assign a subjective probability $$ p_i(\omega) $$ all possible scenarios. Therefore in purchasing a derivative contract, company $$ i $$ will value this contract as,

$$ V_i = \sum_{\omega \in \Omega_v} p_i(\omega) C_v(\omega) $$

where $$ \Omega_v $$ is the set of scenarios for which the derivative offers a non-zero payoff and $$ C_v(\omega) $$ is the payoff of the derivative under scenario $$ \omega $$. Denoting the risk neutral probabilities as $$ \pi(\omega) $$, if,

$$ V = \sum_{\omega \in \Omega_v} \pi(\omega) C_v(\omega) $$

is less that $$ V_i $$ then company $$ i $$ perceives an economic benefit from the purchase of this derivative.

For a specific event $$ \omega $$ we can rank those companies that have $$ C_i(\omega)>0 $$ in ascending order; this constitutes a "supply" curve for this event. For those companies that have $$ C_i(\omega)<0 $$ we can rank them in descending order in terms of $$ \mid C_i(\omega)\mid $$; this constitutes a "demand" curve for this event. Companies with $$ C_i(\omega)>0 $$ can cover the negative exposure of companies with $$ C_i(\omega)<0 $$ but the higher their profit from this event the higher the price they require. If a market existed for trading event-specific risk then an equilibrium price of risk could be obtained. The derivatives market is the real-world version of this principle. Financial institutions can offer the liquidity for hedging large exposures; by combining derivative transactions that cover companies with opposite exposure to the same payoff they can offer a price that seems too low for companies with adverse exposure and too high for companies for which added exposure is beneficial. The rapid growth of the derivatives market is testament to this fact. But the most beneficial effect of this growth is that it leads to a more stable economic climate. As companies are able to hedge against adverse events changes in macroeconomic variables are less likely to cause insolvency. And since most economic recessions are caused by the adverse impact of economic variables like oil prices or interest rates to the performance of corporations the severity of business cycles is greatly reduced.

Studies from the Bank for International Settlements show that on September 2006 there were  $26 trillion of notional amounts outstanding in futures and $50 trillion in options all exchange traded. For over-the-counter derivatives (derivatives traded outside exchanges through bilateral agreements) on June 2006 there were $370 trillion of notional amounts outstanding.