Chapter5

Transcript Chapter5

Determination of Forward and Futures Prices

Chapter 5 (Pages 103-123)

Consumption vs Investment Assets

 Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver)  Consumption assets are assets held primarily for consumption (Examples: copper, oil)

Short Selling

(Page 104-105)

 Short selling involves selling securities you do not own  Your broker borrows the securities from another client and sells them in the market in the usual way

Short Selling

(continued)

 At some stage you must buy the securities back so they can be replaced in the account of the client  You must pay dividends and other benefits the owner of the securities receives

Notation

0 : Spot price today

0 : Futures or forward price today

: Time until delivery date

: Risk-free interest rate for maturity

1. Gold: An Arbitrage Opportunity?

 Suppose that:  The spot price of gold is US$1000  The quoted 1-year futures price of gold is US$1100  The 1-year US$ interest rate is 5% per annum  No income or storage costs for gold  Is there an arbitrage opportunity?

2. Gold: Another Arbitrage Opportunity?

 Suppose that:  The spot price of gold is US$1000  The quoted 1-year futures price of gold is US$990  The 1-year US$ interest rate is 5% per annum  No income or storage costs for gold  Is there an arbitrage opportunity?

The Futures Price of Gold

If the spot price of gold is

& the futures price is for a contract deliverable in

years is

, then

F = S

(1+

)

where

is the 1-year (domestic currency) risk free rate of interest.

In our examples,

=1000,

=1, and

=0.05 so that

= 1000(1+0.05) = 1050

When Interest Rates are Measured with Continuous Compounding

= S

e rT

This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs

e = 2.71828 and is explained in Chapter 4 pages 82-84. It represents the exponential function to determine interest rates based on continuous (or daily) compounding.

If Short Sales Are Not Possible..

Formula still works for an investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low

When an Investment Asset Provides a Known Dollar Income

(page 110, equation 5.2)

0 = (

0 –

)

e rT

where

is the present value of the income during life of forward contract Practice Example 5.2 Page 110

When an Investment Asset Provides a Known Yield

(Page 111, equation 5.3)

= S

(

–

)

where

is the average yield during the life of the contract (expressed with continuous compounding) and as % of the underlying asset’s price at the time income is paid (See example 5.3 on top pg.111) Also See Page 78 in Chapter 4 for example to convert semiannual to daily (or continuous) compounding

Valuing a Forward Contract

Page 112

   Suppose that

is delivery price in a forward contract &

0 is forward price that would apply to the contract today The value of a long forward contract, ƒ , is ƒ = (

– K

)

–

Similarly, the value of a short forward contract is ƒ = (

–

0 )

–

Discuss Business Snapshot 5.2 on Page 113 – Valuing the difference between futures and forwards

Stock Index

(Page 115)

 Can be viewed as an investment asset paying a dividend yield  The futures price and spot price relationship is therefore

0 =

(

–

)

where

is the

dividend yield

on the portfolio represented by the index during life of contract

Stock Index

(continued)

 For the formula to be true it is important that the index represent an investment asset  In other words, changes in the index must correspond to changes in the value of a tradable portfolio  The Nikkei index viewed as a dollar number does not represent an investment asset (“Quanto” – see business snapshot 5.3 on Page 109, where underlying asset is in one currency and payoff is in another)

Index Arbitrage

 When

> S

e (r-q)T

an arbitrageur buys the stocks underlying the index and sells futures  When

< S

e (r-q)T

an arbitrageur buys futures and shorts or sells the stocks underlying the index

Index Arbitrage

(continued)

  Index arbitrage involves simultaneous trades in futures and many different stocks Typically a computer is used to generate the trades

Futures and Forwards on Currencies

(Page 117-119)

 A foreign currency is analogous to a security providing a dividend yield  The continuous dividend yield is the foreign risk-free interest rate  It follows that if

r f

rate is the foreign risk-free interest 

–

r f

has the effect of netting the interest rate effect on the investment in foreign currency

0 

S e

0 ( 

)

Why the Relation Must Be True

Figure 5.1, page 117

r f

1000

r f

dollars at time

T S

dollars at time

Futures on Consumption Assets

(Page 122)

0 

0 e (

r+u

)

where

is the storage cost per unit time as a percent of the asset value.

Alternatively,

0  (

)

e rT

where

is the present value of the storage costs.

The Cost of Carry

(Page 122-123)

   The cost of carry,

, is the storage cost plus the interest costs less the income earned U = present value of storage costs Therefore

= U + r - q     For an investment asset

0 For a consumption asset

= S

e cT



e cT

The convenience yield on the consumption asset,

, is defined so that

0 =

(

–

) (See pgs 122 &123 – assigning value to benefits not obtained by futures contracts)

Practice Problems:

 Problem 5.9

 Problem 5.10  Problem 5.13

 Problem 5.14

 Problem 5.15

 Problem 5.23

Fundamentals of Futures and Options Markets

Chapter5

Transcript Chapter5

Determination of Forward and Futures Prices

Chapter 5 (Pages 103-123)

Consumption vs Investment Assets

Short Selling

Short Selling

Notation

1. Gold: An Arbitrage Opportunity?

2. Gold: Another Arbitrage Opportunity?

The Futures Price of Gold

When Interest Rates are Measured with Continuous Compounding

If Short Sales Are Not Possible..

When an Investment Asset Provides a Known Dollar Income

When an Investment Asset Provides a Known Yield

Valuing a Forward Contract

Stock Index

Stock Index

Index Arbitrage

Index Arbitrage

Futures and Forwards on Currencies

Why the Relation Must Be True

Futures on Consumption Assets

The Cost of Carry

Practice Problems:

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