Derivatives Futures

Derivatives: Principles & Practice McGraw-Hill/Irwin

©Rangarajan K. Sundaram Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 1. Introduction Outline Introduction Forward Contracts Futures Contracts Options

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Objectives

This segment Introduces the following major classes of derivative securities Forwards Futures Options Discusses their broad characteristics and points of distinction. Discusses their uses at a general level. The objective is introductory: to lay the foundations for the detailed analysis of derivative securities.

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Derivatives

A derivative security is a financial security whose value depends on (or derives from) other, more fundamental, underlying financial variables such as the price of a stock, an interest rate, an index level, a commodity price or an exchange rate. In this module we focus on the following classes of derivative securities: Futures & forwards. Options

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Basic Distinctions - I

 Forward contracts are those where two parties agree to a specified trade at a specified point in the future.  Defining characteristic: Both parties commit to taking part in the trade or exchange specified in the contract.  Futures are variants on this theme:  Futures contracts are forward contracts where buyers and sellers trade through an exchange rather than bilaterally.

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Basic Distinctions – II

 Options: Characterized by optionality concerning the specified trade.  One party, the option holder, retains the right to enforce or opt out of the trade.  The other party, the option writer, has a contingent obligation to take part in the trade.  Call option: Option holder has the right, but not the obligation, to buy the underlying asset at the price specified in the contract.  Option writer has a contingent obligation to participate in the specified trade as the seller.  Put option: Holder has the right, but not the obligation, to sell the underlying asset at the price specified in the contract.  Option writer has a contingent obligation to participate in the specified trade as the buyer. Derivatives: Principles & Practice

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Derivatives are BIG Business ... BIS estimates of market size (in trillions of USD):

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... and a Rapidly Growing One BIS estimates of market size (in trillions of USD):

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Risk-Management Roles - I

These classes of derivatives serve important, but different, purposes. Futures and forwards enable investors to lock in cash flows from future transactions. Thus, they are instruments for hedging risk. "Hedging" is the offsetting of an existing cash-flow risk. Example A company that needs to procure crude oil in one month can use a one-month crude oil futures contract to lock in a price for the oil.

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Risk-Management Roles - II

 Options provide one-sided protection.  The option confers a right without an obligation. As a consequence:  Call  Protection against price increase.  Put  Protection against price decrease.  Example Suppose a company needs to procure oil in one month.  If the company buys a call option, it has the right to buy oil at the "strike price" specified in the contract.  If the price of oil in one month is lower than the strike price, the company can opt out of the contract.  Thus, the company can take advantage of price decreases but is protected against price increases.  In short, options provide financial insurance.

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Outline for Remaining Discussion

The rest of the material defines these classes of instruments more formally. Order of coverage: Forwards Futures Options

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Forward Contracts

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Forward Contracts

 The building block of most other derivatives, forwards are thousands of years old.  A forward contract is a bilateral agreement  between two counterparties  a buyer (or "long position"), and  a seller (or "short position")  to trade in a specified quantity  of a specified good (the "underlying")  at a specified price (the "delivery price")  on a specified date (the "maturity date") in the future.  The delivery price is related to, but not quite the same thing as, the "forward price." The forward price will be defined shortly.

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Forwards: Characteristics

 Important characteristics of a forward contract:  Bilateral contract Negotiated directly by seller and buyer.  Customizable Terms of the contract can be "tailored."  Credit Risk There is possible default risk for both parties. Futures & forwards differ on precisely these characteristics as we see shortly.

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The Role of Forwards: Hedging  Forwards enable buyers and sellers to lock-in a price for a future market transaction.  Thus, they address a basic economic need: hedging.  Demand for such hedging arises everywhere. Examples:  Currency forwards: lock-in an exchange rate for a future transaction to eliminate exchange-rate risk.  Notional outstanding in Dec-2008: $24.6 trillion.  Interest-rate forwards (a.k.a. forward-rate agreements): lock-in an interest rate today for a future borrowing/investment to eliminate interest-rate risk.  Notional outstanding in Dec-2008: $39.3 trillion.  Commodity forwards: lock-in a price for a future sale or purchase of commodity to eliminate commodity price risk.  Notional outstanding in Dec-2008: $2.5 trillion.

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BUT...

 An obvious but important point: The elimination of cash flow uncertainty using a forward does not come "for free."  A forward contract involves a trade at a price that may be "off-market," i.e., that may differ from the actual spot price of the underlying at maturity.  Depending on whether the agreed-upon delivery price is higher or lower than the spot price at maturity, one party will gain and the other party lose from the transaction.

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An Example

 A US-based exporter anticipates €200 million of exports and hedges against fluctuations in the exchange rate by selling €200 million forward at $1.30/€.  Benefit? Cash-flow certainty: receipts in $ are known.  Cost? Exchange-rate fluctuations may lead to ex-post regret.  Exchange rate at maturity is $1.40/€.  Exporter loses $0.10/€ for a total loss of $20 million on the hedging strategy.  Exchange rate at maturity is $1.20/€.  Exporter gains $0.10/€ for a total gain of $20 million on the hedging strategy.

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Forward Contracts: Payoffs  Forward to buy XYZ stock at F = 100 at date T.  Let ST denote the price of XYZ on date T.

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Forwards are "Linear" Derivatives

ST : Spot price at maturity of forward contract. F : Delivery price locked-in on forward contract. Derivatives: Principles & Practice

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The Forward Price

 We have seen what is meant by the delivery price in a forward contract.  What is meant by a forward price?  The forward price is a breakeven delivery price: it is the delivery price that would make the contract have zero value to both parties at inception.  Intuitively, it is the price at which neither party would be willing to pay anything to enter into the contract.

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The Forward Price and the Delivery Price

 At inception of the contract, the delivery price is set equal to the forward price.  Thus, at inception, the forward price and delivery price are the same.  As time moves on, the forward price will typically change, but the delivery price in a contract, of course, remains fixed.  So while a forward contract necessarily has zero value at inception, the value of the contract could become positive or negative as time moves on.  That is, the locked-in delivery price may look favorable or unfavorable compared to the forward price on a fresh contract with the same maturity.

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The Forward Price Is the forward price a well-defined concept?  Not obvious, a priori.  It is obvious that If the delivery price is set too high relative to the spot, the contract will have positive value to the short (and negative value to the long). If the delivery price is set too low relative to the spot, the situation is reversed.  But it is not obvious that there is only a single breakeven price. It appears plausible that two people with different information or outlooks about the market, or with different risk-aversion, can disagree on what is a breakeven price. This question is addressed in Chapter 3 on forward pricing.

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Futures Contracts

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Futures Contracts

A futures contract is like a forward contract except that it is traded on an organized exchange. This results in some important differences. In a futures contract:  Buyers and sellers deal through the exchange, not directly.  Contract terms are standardized.  Default risk is borne by the exchange, and not by the individual parties to the contract.  "Margin accounts" (a.k.a "performance bonds") are used to manage default risk.  Either party can reverse its position at any time by closing out its contract.

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Forwards vs. Futures

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The Futures Price

 As with a forward contract, there is no up-front payment to enter into a futures contract.

 Thus, the futures price is defined in the same way as a forward price: it is the delivery price which results in the contract having zero value to both parties.  Futures and forward prices are very closely related but they are not quite identical.  The relationship between these prices is examined in Chapter 3.

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Options

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Basic Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date.  Buy = Call option. Sell = Put option  On/before: American. Only on: European  Specified price = Strike or exercise price  Specified date = Maturity or expiration date  Specified asset = "underlying"  Buyer = holder = long position  Seller = writer = short position

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Broad Categories of Options

 Exchange-traded options: Stocks (American). Futures (American). Indices (European & American) Currencies (European and American)  OTC options: Vanilla (standard calls/puts as defined above). Exotic (everything else—e.g., Asians, barriers).  Others (e.g., embedded options such as convertible bonds or callable bonds).

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Options as Financial Insurance

 As we have noted above, option provides financial insurance.  The holder of the option has the right, but not the obligation, to take part in the trade specified in the option.  This right will be exercised only if it is in the holder's interest to do so.  This means the holder can profit, but cannot lose, from the exercise decision.

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Put Options as Insurance: Example  Cisco stock is currently at $24.75. An investor plans to sell Cisco stock she holds in a month's time, and is concerned that the price could fall over that period.  Buying a one-month put option on Cisco with a strike of K will provide her with insurance against the price falling below K.  For example, suppose she buys a one-month put with a strike of 22.50.

K=

 If the price falls below $22.50, the put can be exercised and the stock sold for $22.50.  If the price increases beyond $22.50, the put can be allowed to lapse and the stock sold at the higher price.  In general, puts provide potential sellers of the underlying with insurance against declines in the underlying's price.  The higher the strike (or the longer the maturity), the greater the amount of insurance provided by the put. Derivatives: Principles & Practice

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Call Options as Insurance: Example  Apple stock is currently trading at $218. An investor is planning to buy the stock in a month's time, and is concerned that the price could rise sharply over that period.  Buying a one-month call on Apple with a strike of K protects the investor from an increase in Apple's price above K.  For example, suppose he buys a one-month call with a strike of K = 225.  If the price increases beyond $225, the call can be exercised and the stock purchased for $225.  If the price falls below $225, the option can be allowed to lapse and the stock purchased at the lower price.  In general, calls provide potential buyers of the underlying with protection against increases in the underlying's price.  The lower the strike (or the longer the maturity), the greater the amount of insurance provided by the call. Derivatives: Principles & Practice

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The Provider of this Insurance

 The writer of the option provides this insurance to the holder: The writer is obligated to take part in the trade if the holder should so decide.  In exchange, writer receives a fee called the option price or the option premium.  Chapters 9-16 are concerned with various aspects of the option premium including the principal determinants of this price and models for identifying fair value of an option.  Chapter 17 discusses how to measure the risk in an option or a portfolio of options.  Chapters 18 and 19 extend the pricing analysis to "exotic" options.  Chapter 21 studies hybrid securities such as convertible bonds that have embedded optionalities.

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Chapter 2. Futures Markets

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Outline Introduction Standardized Contract Terms Default Risk and Margin Accounts Futures Pricing

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Introduction

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Objectives

Futures contracts are the exchange-traded counterparts of forward contracts. Key features distinguishing futures contracts from forward contracts: The standardization of futures contracts. The ability to unilaterally reverse positions. The use of margin accounts or \performance bonds" to control default by investors in the market.

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The Origins

As economic mechanisms, forward markets are very old. Futures Industry Association traces the origin of forward trading to India around 2,000 BC. Substantial evidence of forward markets in Greco-Roman and medieval Europe. World's first futures market was quite possibly the Dojima Rice Market (Osaka, 1730).

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The 19th Century

 Modern futures markets are most associated with 19th century America, particularly the grain markets of Chicago.  The Chicago Board of Trade (CBoT) was established in 1848.  Swiftly followed by a number of other exchanges (New York, Milwaukee, St. Louis, Kansas City, ... ).  Over a thousand commodity exchanges sprang up in the US by the end of the 19th century.

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Volume of Trading on CBoT  Financial Futures were introduced in 1972.  Trading volume on the CBoT in millions of contracts in the first two decades since that point:

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The Top 15 Futures Contracts Worldwide: 2008

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Standardized Contract Terms

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Standardization

 Contract terms must be standardized, since buyer and seller do not interact directly.  Standardization is perhaps the most important task performed by the exchange.  Essential in promoting liquidity and improving quality of hedge.  Apart from contract maturity dates, standardization involves three components: 1. Quantity (size of contract). 2. Quality (standard deliverable grade). 3. Delivery options (other deliverable grades + price adjustment mechanism).

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Example: Commodity Futures Contracts

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Example: The Corn Futures Contract

 Exchange: Chicago Board of Trade (CBoT).  Contract months: March, May, July, September, December.  Size: 5,000 bushels.  Quality: No. 2 Yellow Corn.  Delivery options:  No. 1 Yellow and No. 3 Yellow may also be delivered.  If No. 1 Yellow is delivered, the short position receives 1.5 cents per bushel more than the contract price.  If No. 3 Yellow is delivered, the short position receives 1.5 cents per bushel less than the contract price.

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Example: The Yen Futures Contract

 Exchange: Chicago Mercantile Exchange (CME)  Contract months: March, June, September, December.  Size: 12,500,000 Yen.  Quality: Not relevant.  Delivery options: None.

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Consequences of Delivery Options

 Why provide delivery options in futures contracts?  It makes corners and squeezes more difficult.  Enhance market liquidity. •

However, there is an important cost: the quality of the hedge is degraded. The delivered grade is not the standard grade, but the cheapest-to-deliver grade.

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Default Risk and Margin Accounts

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Margin Accounts

 Since buyers and sellers do not interact directly, there is an incentive for either party to default if prices move adversely.  To inhibit default, futures exchanges use margin accounts. This is effectively the posting of collateral against default.  The level at which margins are set is crucial for liquidity. Too high levels eliminate default, but inhibit market participation. Too low levels increase default risk.  In practice, margin levels are not set very high.

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The Margining Procedure

 Initial margin.  Amount initially deposited by investor into margin account.  Marking-to-market.  Daily adjustment of margin account balances to reflect gains/losses from futures price movements over the day.  Maintenence margin.  Floor level of margin account.  If balance falls below this, customer receives margin call.  If the margin call is not met, account is closed out immediately.

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Margining: An Example

 Investor takes long position in 10 wheat futures contracts at a futures price of $3.60 per bushel.  Size of one futures contract: 5,000 bushels.  Thus, futures price: $18,000 per contract.  Suppose that  Initial margin = $878 per contract.  Maintenence margin = $650 per contract.

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Example: Marking-to-Market

 Settlement price on day 1: $3.58 per bushel = $17,900 per contract.  "Loss" from holding long contract at $3.60: $100/contract.  Total "loss" = $1,000: debited from the margin account.  Margin account balance: $7,780.  Effectively:  Original contract at $3.60/bushel has been replaced with new contract at $3.58/bushel.  Difference is debited from the margin account.  Of course, margin account of corresponding short position would increase by $1,000.

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Example: Marking-to-Market

 Settlement price on day 2: $3.54 per bushel = $17,700 per contract.  "Loss" per contract: $(17, 900 — 17, 700) = $200 per contract.  Total loss = $2,000; debited from the margin account.  New margin account balance: $5,780.  Since balance is less than maintenance margin, margin call results.  If account is topped up (to initial margin levels), situation continues.  If not, investor's position is closed out.

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Margining: Summary  To reiterate: marking-to-market essentially involves:  Rewriting the customer's futures contract at the current settlement price, and  settling immediately the gains or losses to the customer from the rewiting.  Basic idea: If an investor is not able to meet "small" losses (from price movements over a day), it is unlikely he will be able to meet larger losses that might result.  Historically, margining has worked very well in inhibiting default.  Exchanges also retain right to change margin at any time.  Used to defuse potentially market-threatening situations (e.g., 1980silver crisis).

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Margin Levels: Examples

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Futures Pricing

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Futures vs. Forward Prices

 Analytical valuation of futures contracts complicated by two considerations: 1. Delivery options provided to the short position. 2. Margining which creates uncertain interim cash flows.  These features will have an impact on futures prices compared to another wise identical forward contract.  The question is: how much of an effect? Is it quantitatively significant?  For most futures contracts (especially short-dated ones), it turns out the effect is minimal: such contracts can be priced as if they are forward contracts.

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Chapter 3

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Outline Introduction Pricing Forwards by Replication Numerical Examples Currency Forwards Stock Index Forwards Valuing Forwards

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Introduction

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Objectives

 This chapter introduces students to the pricing of derivatives using noarbitrage considerations.  Key points: 1. The cost-of-carry method of pricing forward contracts. 2. The role of interest rates and holding costs/benefits in this process. 3. The valuation of existing forward contracts.

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The Forward Price

 Forward price: The delivery price that makes the forward contract have "zero value" to both parties.  How do we identify this zero-value price?  Combine The Key Assumption: No arbitrage with The Guiding Principle: Replication.

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Pricing Forwards by Replication

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The Key Assumption: No Arbitrage  Maintained Assumption: Market does not permit arbitrage.  What is "arbitrage?" A profit opportunity which guarantees net cash inflows with no net cash outflows. Such an opportunity represents an extreme form of market in efficiency where two identical securities (or baskets of securities) trade at different prices.  Assumption is not that arbitrage opportunities can never arise, but that they cannot persist.  This is a minimal market rationality condition: it is impossible to say anything sensible about a market where such opportunities can persist.

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The Guiding Principle: Replication  Replication: Fundamental idea underlying the pricing of all derivative securities.  The argument:  Derivative's payoff is determined by price of the underlying asset.  So, it "should" be possible to recreate (or replicate) the derivative's pay offs by directly using the spot asset and, perhaps, cash.  By definition, the derivative and its replicating portfolio (should one exist) are equivalent.  So, by no-arbitrage, they must have the same cost.  Thus, the cost of the derivative (its so-called "fair price") is just the cost of its replicating portfolio, i.e., the cost of manufacturing its outcomes synthetically.  Key step: Identifying the replicating portfolio. Derivatives: Principles & Practice

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Replicating Forward Contracts

 Forward contracts are relatively easy to price by replication.  Consider an investor who wants to take a long forward position.  Notation: S : current spot price of asset. T : maturity of forward contract (in years). F : forward price for this contract (to be determined).  We will let 0 denote the current date, so T is also the maturity date of forward contract.

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Replicating Forwards

 At maturity of the contract, the investor pays $F and receives one unit of the underlying.  To replicate this final holding: Buy one unit of the asset today and hold it to date T.  Both strategies result in the same final holding of one unit of the underlying at T.  So, viewed from today, they must have the same cost.  What are these costs?

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Cost of Forward Strategy

 The forward strategy involves a single cash outflow of the delivery price F at time T  So, cost of forward strategy: PV (F ).

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Cost of Replicating Strategy

 To replicate, we must Buy the asset today at its current spot price S. "Carry" the asset to date T. This involves:  Possible holding/carrying costs (storage, insurance).  Possible holding benefits (dividends, convenience yield).

 Let

M = PV (Holding Costs) — PV (Holding Benefits).  Net cost of replicating strategy: S + M.

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The Forward Pricing Condition

 By no-arbitrage, we obtain the fundamental forward pricing condition:

 Solving this condition for F, we obtain the unique forward price consistent with no-arbitrage.

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Violation of this Condition  Arbitrage

 If PV (F ) > S + M, the forward is overvalued relative to spot. Arbitrage profits may be made by selling forward and buying spot. "Cash and carry" arbitrage. Forward contract has positive value to the short, negative value to the long.  If PV (F ) < S + M, the forward is undervalued relative to spot. Arbitrage profits can be made by buying forward and selling spot. "Reverse Cash and carry" arbitrage. The contract has positive value to the long and negative value to the short.

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Determinants of the Forward Price

 The forward price is completely determined by three inputs: Current price S of the spot asset. The cost M of "carrying" the spot asset to date T. The level of interest rates which determine present values.  This is commonly referred to as the cost-of-carry method of pricing forwards.  Two comments: Forward and spot prices are tied together by arbitrage: they must move in "lockstep." To what extent then do (or can) forward prices embody expectations of future spot prices?

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Pricing Formulae I: Continuous Compounding

 Fundamental pricing equation: PV (F) = S + M.  Let r be the continuously-compounded interest rate for horizon T.  So PV (F ) = e —rT F.  Therefore, e —rT F = S + M, so

 When there are no holding costs or benefits (M = 0),

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Numerical Examples

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Example 1  Consider a forward contract on gold. Suppose that: Spot price: S = $1, 140/ oz. Contract length: T = 1 month = 1/12 years. Interest rate: r = 2.80% (continuously compounded). No holding costs or benefits.  Then, from the forward pricing formula, we have

 Any other forward price leads to arbitrage. If F > 1, 142.66, sell forward and buy spot. If F < 1, 142.66, buy forward and sell spot.

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Abrbitrage with an Overpriced Forward  Suppose that F = 1, 160, i.e., it is overpriced by 17.34.  Arbitrage strategy: 1. Enter into short forward contract. 2. Buy one oz. of gold spot for $1,140. 3. Borrow $1,140 for one month at 2.80%.

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Abrbitrage with an Underpriced Forward

 Suppose that F = 1, 125, i.e., it is underpriced by $17.66.  Arbitrage strategy: 1. Enter into long forward contract. 2. Short 1 oz. of gold. 3. Invest $1,140 for one month at 2.80%.

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Holding Costs and/or Benefits

 Holding costs are often non-zero.  With equities or bonds, there are often holding benefits such as dividends or coupons.  With commodities, there are often holding costs such as storage and insurance.  Such interim cash flows affect the total cost of the replication strategy and should be taken into account in pricing.  Our second example deals with such a situation.

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Example 2

 Consider a forward contract on a bond.  Suppose that:  Spot price of bond: S = 95.  Contract length: T = 6 months.  Interest rate: r = 10% (continuously compounded) for all maturities.  Coupon of $5 will be paid to bond holders in 3 months.  What is the forward price of the bond?

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Example 2: The Forward Price

 The coupon is a holding benefit.  So M is minus the present value of $5 receivable in 3 months:

 Thus, the arbitrage-free forward price F must satisfy

so

 Any other forward price leads to arbitrage.

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Arbitrage with an Overvalued Forward

 For example, suppose F = 98.  Then, the forward is overvalued relative to spot, so we want to sell forward, buy spot, and borrow.  Buying and holding the spot asset leads to a cash outflow of 95 today, but we receive a coupon of 5 in 3 months.  There are may ways to structure the arbitrage strategy. Here is one. We split the initial borrowing of 95 into two parts, with  one part repaid in 3 months with the $5 coupon, and  the balance repaid in six months with the delivery price received on the forward contract.

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The Arbitrage Strategy

 So the full arbitrage strategy is:  Enter into short forward with the delivery price of 98.  Buy the bond for 95 and hold for 6 months.  Finance spot purchase by  borrowing 4.877 for 3 months at 10%  borrowing 90.123 for 6 months at 10%.  In 3 months:  receive coupon $5  repay the 3-month borrowing.  In 6 months:  deliver bond on forward contract and receive $98  repay 6-month borrowing.

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Cash Flows from the Aribtrage

 Question: What is the arbitrage strategy if F = 91.50?

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Currency Forwards

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Forwards on Currencies & Related Assets

 Forwards on currencies need a slightly modified argument.  For example, suppose you want to be long £1 on date T.  Two strategies: Forward contract: Pay $F at time T, receive £1. Replicating strategy: Buy £x today and invest it to T, where x = PV (£1).  PV(£1) is the amount that when invested at the sterling interest rate will grow to £1 by time T. The "£" inside the PV expression is to emphasize that present values are being taken with respect to the £ interest rate.

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Currency Forwards: Replication Costs

Cost of the forward strategy in USD: PV ($ F ) = F x PV ($1). Cost of the spot (or replicating) strategy in USD: S x PV (£1) As usual, S denotes the spot price of the underlying in USD. Here, the underlying is GBP, so S is the spot exchange rate ( $ per £).

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Currency Forwards: The General Pricing Expression

 By no-arbitrage, we must have

S x PV (£1) = F x PV ($1).  Solving we obtain the fundamental forward pricing expression for currencies:

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Currency Forwards with Continuous Compounding

 Let r represent the T-year USD interest rate and d the T-year GBP interest rate, both expressed in continuously-compounded terms.  Then, PV ($ 1) = e—rT and PV (£1) = e—dT .  Using these in the general currency forward pricing expression and simplifying, we obtain

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Stock Index Forwards

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Stock Index Forwards

 We can also price forwards on stock indices using this approach.  A stock index is essentially a basket of a number of stocks.  If the stocks pay dividends at different times, we can approximate the dividend payments well by assuming they are continuously paid.  Dividend yield on the index plays the role of the variable d in the formula.  Literally speaking, the idea of continuous dividends is an unrealistic one, but, in general, the approximation works very well.  Computationally, much simpler than calculating cash value of dividend payments expected over contract life and using the known-cash-payouts formula.

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Example 4: Index Forwards

 Data:  Current level of S&P index: 1,343  One-month interest rate (continuously-compounded): 2.80%  Dividend yield on the S&P 500: 1.30%  What is the price of a one-month (= 1/12 year) futures contract?  In our notation: S = 1343, r = 2.80%, d = 1.30%, and T = 1/12.  So the theoretical futures price is

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S&P 500 Futures Prices: Jan 15, 2010

Spot: 1136.03 (S&P on Jan 15, 2010)

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Valuing Forwards

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Valuing Existing Forwards

 Consider a forward contract with delivery price K that was entered into earlier and now has T years left to maturity.  What is the current value of such a contract? We answer this question for the long position. The value of the contract to the short position is just the negative of the value to the long position.  So suppose we are long the existing contract.  Suppose also that the current forward price for the same contract (same underlying, same maturity date) is F.

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Offsetting the Existing Forward

 Consider offsetting the existing long forward position with a short forward position in a new forward contract.  Original portfolio:  Long forward contract with delivery price K and maturity T.  New portfolio:  Long forward contract with delivery price K and maturity T.  Short forward contract with delivery price F and maturity T.  Value of original portfolio = Value of new portfolio (why?).

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Valuation by Offset

 What happens to the new portfolio at maturity?  Physical obligations in the underlying offset.  Net cash flow: F — K.  So new portfolio - certainty cash flow of F — K at time T.  This means: Value of New Portfolio = PV (F — K ).  Therefore: Value of Long Forward = PV (F — K ). and Value of Corresponding Short Forward = PV (K — F ).

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Forward Pricing: Summary

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Forward Pricing: Summary

 A forward contract is a commitment by buyer and seller to take part in a fully specified future trade.  The commitment to the trade makes forward payoffs linear.  The forward price is that delivery price that would make the contract have zero value to both parties at inception.  The forward price can be determined by replication, and depends on the cost of buying and "carrying" spot.  The value of a forward contract is the present value of the difference between the locked-in delivery price on a contract and the current forward price for that maturity.

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