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Old Friday, March 23, 2007
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Default 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). All financial derivatives are based on contracts in which the payoff of the derivative is described as a mathematical function of the value of an asset (or a number of assets) either at a specific instance or a collection of observation points. The value of the derivative contract once these observations are made is therefore unambiguous -- unlike the value of oil or any asset which at any point does not have a value based on a mathematical formula but a value based on market dynamics.

Based on this definition one can question the utility of derivative contracts. After all what could be the use of a contract that is simply a mathematical function of the price of an asset that is already actively traded? This misunderstanding arises from an erroneous concept of risk.

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 ω for the price of oil one year from now the income of company i is Ci(ω). The bias of a specific company i to a set of scenarios Ωi 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 pi(ω) all possible scenarios. Therefore in purchasing a derivative contract, company i will value this contract as,

where Ωv is the set of scenarios for which the derivative offers a non-zero payoff and Cv(ω) is the payoff of the derivative under scenario ω. Denoting the risk neutral probabilities as π(ω), if,

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

For a specific event ω we can rank those companies that have Ci(ω) > 0 in ascending order; this constitutes a "supply" curve for this event. For those companies that have Ci(ω) < 0 we can rank them in descending order in terms of ; this constitutes a "demand" curve for this event. Companies with Ci(ω) > 0 can cover the negative exposure of companies with Ci(ω) < 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 [1]. 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.

Basic Derivative Contracts

Forwards

Spot markets allow the purchase and sale of an asset today. By contract a forward contract specifies the price at which an asset can be purchased or sold at some future date. Although a forward contract is classified as a derivative in many markets it is difficult to distinguish between the underlying and the forward contract. Large trading volumes in OTC forwards can in fact make them more significant than spot markets.

A forward contract does not require upfront payment. It is simply the purchase or sale of an asset at some future date at a fixed price (the forward price). Therefore the assumption is that the forward price reflects the value of this asset on this date. If this assumption is based on a market view, characterising a forward contract as a derivative is misleading.

The primary reason for the classification of a forward contract as a derivative is that in many cases its price can be derived through a no-arbitrage argument that relates the forward price of an asset to its spot price. For assets like oil this is not possible; given the spot price of a barrel of oil it is not possible to construct an arbitrage argument that relates it to the forward price. In the oil markets forwards or futures are effectively the underlying and cannot be understood as derivatives. In these markets the forward price of oil is similar in nature to the price of a stock: it reflects the current consensus of the market and has nothing to do with risk-neutral valuation.

In financial markets forwards can be determined through a no-arbitrage argument. Consider for example a forward on the USD vs EUR exchange rate. If today one euro can be exchanged for 1.3 dollars (FXspot) then in order to determine the forward exchange rate one year from now we can look at the following set of trades,

We buy a one year forward that guarantees an exchange rate of FXoneyear dollars per euro.
We borrow one dollar today.
We exchange it for (1/1.3) euros and invest this amount in a deposit account.
After one year we withdraw the principal and the interest earned and exchange them into dollars at FXoneyear.
The net cashflow of this trade at expiry is,

In the absence of arbitrage opportunities the net cashflow of this trade should be zero and therefore,

Another example is a forward contract on a zero coupon one year bond , one year from now. Given the price of a one year bond P1year and a two year bond P2year we look at the following set of trades,

Sell a one year zero coupon bond one year from now at forward price P1,1.
Buy a one year zero coupon bond today.
Sell a two year zero coupon bond today.
Since P1year > P2year we must borrow the difference. After one year we receive $1 from the one year bond and pay interest and principal on the amount borrowed. The two year bond has one year to maturity and we transfer it to the buyer of the forward in return for P1,1. Therefore the net cashflow in one year is,

− (P1year − P2year)(1 + r1year) + 1 + P1,1

In the absence of arbitrage opportunities this cashflow must be zero. Since,

we conclude that,

P1,1 = P2year / P1year

It is interesting to note that the formula,

is based on a "no-arbitrage" argument itself and the one year bond can be viewed as the "forward contract" for one dollar received in one year. Given the value of r1year, if the price of the one year bond was different from 1 / (1 + r1year) one could sell a one-year bond at a price P * > P1year. At expiry one dollar would be paid to the buyer of the bond but since the proceeds from the sale would have earned P * (1 + r1year) they would cover this payment and leave a clear profit. Only if P * = 1 / (1 + r1year) = P1year the condition of no arbitrage holds.

Futures

Futures contracts, like forward contracts, specify the delivery of an asset at some future date. Futures contracts, unlike forward contracts,

Require the buyer or the seller of the futures contract to post margin.
Have minimum margin requirements; these requirements are achieved through a margin call.
Use the process of mark-to-market.
There three requirements in practise are not unique to futures contracts. The best way to understand them is by looking at a specific futures contract.

The corn futures contract trades at the Chicago Board of Trade (CBOT). The specifications of the contract are very strict and require the delivery of "no. 2 yellow" corn; if other grades of corn are delivered instead the price paid is adjusted. The contract size can be in multiples of 5,000 bushels of corn. Futures can be purchased for delivery of corn in months December, March, May, July and September only. Trading this contract ceases on the business day nearest to the 15th calendar date of the delivery month. Delivery takes place two business days after the 15th calendar date of the delivery month.

Assume that one lot (5,000 bushels) of the Jul-07 contract was bought at 418 cents/bushel on 24th January 2007. The exchange would require the buyer to post initial margin of $900. If the buyer does not post this amount of money in her account with the exchange, her order cannot be executed. For this contract the maintenance margin is the same; during the life of this futures contract the balance of the account cannot go below this level; if for any reason the balance of the account falls below the maintenance margin, the buyer of this contract will receive a margin call.

On the date on which the trade was executed the mark-to-market of the futures contract is zero. Assume that on the next trading date, the settlement price of the futures contract is 418 3/4 cents/bushel (settlement price is the price traded for a futures contract at the close of the trading session). The mark-to-market of the Jul-07 corn futures is,

MtM = 5,000 * (4183 / 4 − 418) = $37.50

The balance on the buyer's account will now be $937.50. The account is like a normal deposit account and earns interest on its balance.

If the market price of the Jul-07 corn contract drops in the following day, the mark-to-market could drop from $37.50 to $12.50. In this case $25 are withdrawn from the buyer's account and the balance is now $912.50.

If on the last trading date of this contract (13th July 2007) the settlement price is 420 1/4 cents/bushel then the mark-to-market is $112.50. The final balance of the buyer's account is $1012.50 plus interest earned. Since the corn that will be delivered on the 17th July 2007 is worth $21,012.50, the buyer will pay this amount to the clearing house. The clearing house acts as counterparty in the transaction between the corn producer and the buyer and makes sure payments are made and corn is delivered to the warehouse nominated by the buyer.

Since the trader has earned $112.50 (plus interest) in effect the net payment for delivery of corn is $20,900. This is equivalent to paying 418 cents/bushel on the corn delivered. The futures contract has therefore enabled the buyer to purchase corn at the original price of 418 cents/bushel and hedge against price changes.

In order to compare the price of a forward contract F0 and the price of a futures contract Φ0 we look at the following set of trades:

We sell a forward contract to deliver a specific quantity of corn at some future date for price F0.
For dates i = 0,1,2,...,N − 1 we purchase a quantity qi of the futures contract so that the following conditions are satisfied:
q0(1 + r1)N − 1 = 1
(q0 + q1)(1 + r2)N − 2 = 1
...
(q0 + q1 + ... + qN − 1)(1 + rN − 1) = 1
where ri is the daily interest rate applicable for period starting on date i and ending on date N. This set of equations can be solved recursively. The value of the margin account on date N will be,

To undestand the last equation, we know that on date i the total quantity of futures purchased is q0 + q1 + ... + qi − 1. By the end of date i the mark-to-market change is equal to (q0 + q1 + ... + qi − 1)(Φi − Φi − 1). Depending on the direction of the change Φi − Φi − 1 this is a gain or a loss and earns or requires the payment of principal plus interest at the expiry date of the contract.

Given the conditions that give rise to the solutions for qi, the last equation is equal to ΦN − Φ0. Since at expiry the price of the futures is equal to the spot price of the asset and therefore FN = ΦN, if Φ0 is different from F0 a risk-free profit can be generated. Note that there is no cost in entering into the series of futures contracts and depending on the sign of the difference F0 − Φ0 the strategy can be reversed. Therefore the forward price must be equal to the futures price.

The strategy used in this analysis assumes that when we purchase an additional quantity qi of the futures contract we know the interest rate for the period i + 1 to N. Since in practise the actual value of the interest rate is not known assume that we can lock in a forward rate. However since we cannot predict the change in the mark-to-market of the futures contract in the period i to i + 1 we do not know the notional amount we must purchase.

Assume that on date i we make the assumption that there will be no change in the mark-to-market of the futures and therefore there is no need to lock in a forward rate for the period i + 1 to N. Since the most likely scenario is that we will be wrong, we will have to borrow or deposit the the actual change in the mark-to-market at the spot rate for the period ri + 1.

As long as the error in our estimate of the mark-to-market change is independent of the spot rate we can expect that the costs/benefits will balance to zero. But if the mark-to-market change of the futures contract is a function of spot rate the costs/benefits will not balance to zero and the futures strategy described above will not be able to replicate the payoff of the forward. We conclude that when the futures contract is a function of the interest rate the futures price will not be equal to the forward price.

Another exception occurs when the futures price can change by large amounts from one date to the next. The term "large amounts" here means that a one day move accounts for a large percentage of the difference between Φ0 and ΦN. In this case on the date when this large price change occurs the error in the notional locked in at the forward rate is large enough to magnify the error in our estimate of the change in the mark-to-market. Furthermore, all subsequent mark-to-market changes are much smaller and cannot balance this cost/benefit. Fortunately, most exchanges limit the maximum change in the futures price that can occur from one date to the next. But if such large price moves are possible then, even if the futures price is not a function of the interest rate, the assumption that it is equal to the forward price is wrong.

In general, the relation between the futures and the forward price cannot be derived through a static arbitarge strategy unless interest rates have a deterministic term-structure. The derivation of the relation between the futures and the forward price of an asset is one of the first applications of dynamic hedging.

Swaps

Primarily used as hedging instruments, against varying interest payments. The base concept is quite easy to follow; you swap a fixed rate for a floating rate or vice-versa. In the case of companies that offer Variable Rate Bonds, they can enter into a swap agreement with a broker/dealer; where the company pays the broker a fixed rate as per agreement and the broker provides them with the floating rate, which can be used to make periodic coupon payments. In essence, the company has hedged it's risk against a sudden rate increase, as it is locked in a fixed rate over time. Swaps may be terminated with one party paying it's counterpart a certain fee, which may have been determined at time of initial agreement or may be based on future payments if interest rates were to remain constant.

Options

There are two types of stock options:

Call Option: Call option gives the buyer a right to purchase the given stock at the strike price. Thus Call option is generally bought when the buyer is bullish about the underlying security.
Put Option: Similarly buying a put option gives you the right to sell the underlying stock at the strike price. Put option is bought when the buyer has bearish views about the underlying security.
Each option comes with an "Exercise Date". European options may only be exercised on the exercise date, whereas American options may be exercised at any time up till the exercise date.

Due to the put-call parity, it is possible to create artificial call or put options if the other is not available. Put options may also be used as a hedging instrument, against possible decline in value of the underlying stock.

While stocks with high volatility (modified duration) are high risk, options whose underlying stock have high volatility are actually better. They provide a possibility of a higher payout if the stock goes up in proportion to it's volatility and the same amount of loss.

Advanced Structures

Theoretically, the price of an option (or option premium)consists of two elements: the Intrinsic value and time value of an option. Therefore, Premium=Intrinsic value+Time value.

The price of an option consists of five things: Strike Price, Price of the underlying asset, Time to maturity, Risk-Free interest rate and Volatility. Since the first four can be read from the markets, the only unknown factor in the price of the option is volatility.

Pricing of Derivatives

Futures

There are two basic concepts in finance: time-value of money and uncertainty about expectations. The two concepts are the core of financial valuations, including futures contracts.

cost-of-carry model is the most widely accepted and used for pricing futures contract

Cost-of-carry Model

Cost-of-carry model is an arbitrage-free pricing model. Its central theme is that futures contract is so priced as to preclude arbitrage profit. In other words, investors will be indifferent to spot and futures market to execute their buying and selling of underlying asset because the prices they obtain are effectively the same. Expectations do influence the price, but they influence the spot price and, through it, the futures price. They do not directly influence the futures price. According to the cost-of-carry model, the futures price is given by Futures price = Spot Price + Carry Cost - Carry Return (1)

Carry cost (CC) is the interest cost of holding the underlying asset (purchased in spot market) until the maturity of futures contract. Carry return (CR) is the income (eg, dividend) derived from underlying asset during holding period. Thus, the futures price (F) should be equal to spot price (S) plus carry cost minus carry return. If it is otherwise, there will be arbitrage opportunities as follows

When F > (S + CC - CR): Sell the (overpriced) futures contract, buy the underlying asset in spot market and carry it until the maturity of futures contract. This is called "cash-and-carry" arbitrage. When F < (S + CC - CR): Buy the (under priced) futures contract, short-sell the underlying asset in spot market and invest the proceeds of short-sale until the maturity of futures contract. This is called "reverse cash-and-carry" arbitrage

Pricing via Duplication

Since for all options, put-call-parity must hold, if three of the terms are known, the last one can also be found using the formula:

Put-Call-Parity: C + PV(X) = P + S

where

C = Price of Call Option

PV(X) = Present Value of Strike Price

P = Price of Put Option

S = Current value of the underlying asset
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Thank you Rehman Bhai for providing us with these info, i will read it, its helpful for me as my specialization iz in Finance [Ist Semester MS(BA)], i m copying it to a word doc and will print it. thanks
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@Jamal..
its my pleasure dear.........

Quote:
i m copying it to a word doc and will print it
u can do anything u like with it....

Regards
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