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Derivatives: Principles & Practice ©Rangarajan K. Sundaram Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

2Derivatives: Principles & Practice ©Rangarajan K. Sundaram

Chapter 1. Introduction

Outline

Introduction

Forward Contracts

Futures Contracts

Options


Objectives

This segment

Introduces the following major classes of derivative securitiesForwardsFuturesOptions

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.


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


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.


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 are BIG Business ...

BIS estimates of market size (in trillions of USD):


... and a Rapidly Growing One

BIS estimates of market size (in trillions of USD):


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.


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.


Outline for Remaining Discussion

The rest of the material defines these classes of instruments more formally.

Order of coverage:

ForwardsFuturesOptions


Forward Contracts


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.


Forwards: Characteristics

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.

Important characteristics of a forward contract:


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.


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.


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.


Forward Contracts: Payoffs

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


Forwards are "Linear" Derivatives

ST : Spot price at maturity of forward contract.

F : Delivery price locked-in on forward contract.


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.


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.


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.


Futures Contracts


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.


Forwards vs. Futures


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.


Options


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


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).


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.


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 K = 22.50.

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.


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.


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.


Chapter 2. Futures Markets


Outline

Introduction

Standardized Contract Terms

Default Risk and Margin Accounts

Futures Pricing


Introduction


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.


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).


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.


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:


The Top 15 Futures Contracts Worldwide: 2008


Standardized Contract Terms


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).


Example: Commodity Futures Contracts


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.


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.


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.


Default Risk and Margin Accounts


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.


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.


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.


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.


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.


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).


Margin Levels: Examples


Futures Pricing


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.


Chapter 3


Outline

Introduction

Pricing Forwards by Replication

Numerical Examples

Currency Forwards

Stock Index Forwards

Valuing Forwards


Introduction


Objectives

This chapter introduces students to the pricing of derivatives using no-arbitrage 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.


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.


Pricing Forwards by Replication


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.


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.


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.


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?


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 ).


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.


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.


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.


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?


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),


Numerical Examples


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.


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%.


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%.


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.


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?


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.


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.


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.


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

Cash Flows from the Aribtrage


Currency Forwards


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.


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

£).


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:


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





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.


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


S&P 500 Futures Prices: Jan 15, 2010

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


Valuing Forwards


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.


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?).


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 ).


Forward Pricing: Summary


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.