FINA3204 02 Futures

The Hong Kong University of Science and Technology

FINA 3204: Derivative Securities Andrew Chiu

Derivative Securities FINA 3204

Forwards & Futures

Andrew Chiu, PhD [email protected]

mailto:[email protected]



Course Overview

Forwards & Futures

Market Mechanics

Hedging Strategies

Pricing

Options

Market Mechanics

Properties Trading

Strategies Pricing

Binomial Tree

Black-Scholes

Greeks

Other Derivatives

Warrants, CBBC Swaps Convertible

Bonds Structured Products



Forward & Futures

A forward contract is an agreement to buy or sell a certain quantity of an asset at a future maturity date for a specified delivery price.

The delivery price is chosen so that the initial value of the contract is zero

No money is exchanged when contract is written

Potentially infinite return? (Return=Profit/Cost)



Basic Specifications:

• Type of underlying asset

• Contract Size

• Delivery Price

• Delivery Arrangement

• Physical delivery to somewhere (or Cash Settlement)

• Maturity Date

Actual contracts are more sophisticated and depends on the underlying asset

• www.cmegroup.com

• www.hkex.com.hk/eng/prod/drprod/DMProducts.htm

Futures Contract Specification

http://www.cmegroup.com/

http://www.hkex.com.hk/eng/prod/drprod/DMProducts.htm



Physical Delivery Example

• www.shfe.com.cn/docview/docview_211231034.htm

• Some brokers do not allow physical delivery, and they may

liquidate your position prior to expiry

• www.interactivebrokers.com/en/?f=deliveryexerciseactions

Cash Settlement

• More and more futures are cash settled. There are also cash-

settled commodities futures.

Settlement Price

• Specified on contract

• HSI Index Futures Settlement Price

• http://www.hkex.com.hk/eng/prod/drprod/hkifo/fut.htm

Futures Settlement

http://www.shfe.com.cn/docview/docview_211231034.htm

http://www.interactivebrokers.com/en/?f=deliveryexerciseactions

http://www.hkex.com.hk/eng/prod/drprod/hkifo/fut.htm



A margin is cash or marketable securities deposited by an investor with his or her broker

The balance in the margin account is adjusted to reflect daily settlement

Margins minimize the possibility of a loss through a default on a contract

Example: Hong Kong Exchange Margin Requirements

• http://www.hkex.com.hk/eng/market/rm/rm_dcrm/riskdata/margin_hkcc/margin.htm

• The broker’s margin requirement may differ

Margins

http://www.hkex.com.hk/eng/market/rm/rm_dcrm/riskdata/margin_hkcc/margin.htm

http://www.hkex.com.hk/eng/market/rm/rm_dcrm/riskdata/margin_hkcc/margin.htm



Long Trader

Broker

Clearing House Member

Clearing House

Clearing House Member

Broker

Short Trader

Clearing House

Reduces default risk by requiring members to post collaterals and meet daily margin requirements.

flow of $ when

futures price rises



In the US, the regulation of futures markets is primarily the responsibility of the Commodity Futures and Trading Commission (CFTC)

Regulators try to protect the public interest and prevent questionable trading practices

Article:

• “CFTC Charges High-Speed Trader Under New Powers” Wall Street Journal (July 22, 2013)

Regulation



Forwards vs Futures

Forwards Futures

Private contract between 2 parties (OTC market)

Exchange-traded

Non-standard Contract Standard Contract

No cash up front Margin required

Some default risk No default risk

Settle at maturity Daily settlement (marking to market)

Usually 1 delivery date Range of delivery dates

Delivery or final cash settlement usually occurs

Contract usually closed out prior to maturity

Little regulation Regulated



Source: Bank for International Settlements.

OTC vs. Exchange



CME Group

• www.cmegroup.com

Hong Kong Exchange

• www.hkex.com.hk

China Financial Futures Exchange

• www.cffex.com.cn/en_new/

Shanghai Futures Exchange

• www.shfe.com.cn/Ehome/index.html

Dalian Commodity Exchange

• www.dce.com.cn/portal/cate?cid=1114494099100

Zhengzhou Commodity Exchange

• english.czce.com.cn

Futures Exchanges

http://www.cmegroup.com/

http://www.hkex.com.hk/

http://www.cffex.com.cn/en_new/

http://www.shfe.com.cn/Ehome/index.html

http://www.dce.com.cn/portal/cate?cid=1114494099100

http://english.czce.com.cn/



Many futures contracts are traded almost 24 hours from

Monday to Friday

Since April 8, 2013, HSI futures trading hours can be

traded between 5:00pm -11:00 pm

• http://www.hkex.com.hk/eng/prod/drprod/hkifo/fut.htm

Futures Trading Hours

http://www.hkex.com.hk/eng/prod/drprod/hkifo/fut.htm



1000

1020

1040

1060

1080

1100

1120

SPX ESU01

9/11 Terrorist Attack



840

850

860

870

880

890

900

910

920

930

Oct 24, 2008 Bloody Friday

SPX ESZ08

Subprime Mortgage Crisis



A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price

A short futures hedge is appropriate when you know you will sell an asset in the future and want to lock in the price

Hedging with Futures



Arguments in Favor of Hedging

Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables



Arguments against Hedging

Shareholders are usually well diversified and can make their own hedging decisions

• Example: Should gold miners hedge?

Don’t hedge when the price of the good does not affect the bottom line.

• Example: Gold jewelry manufacturers

Hard to explain a situation when the hedge loses money and the firm is better off not hedging.

• Example: Airlines hedging oil, but oil price falls.



Long Hedge for Purchase of Silver

-50

-40

-30

-20

-10

0

10

20

30

0 10 20 30 40

Pay

me

nt

for

Silv

er

in 1

Ye

ar

Price of Silver in 1 Year (S1)

S0=19 F0=20 F1=S1 in 1 year Unhedged Cashflow = -S1



Long Hedge for Purchase of Silver

-50

-40

-30

-20

-10

0

10

20

30

0 10 20 30 40

Cas

hfl

ow

in 1

Ye

ar

Price of Silver in 1 Year (S1)

Payment for Silver

Futures Payoff

Total

Hedged Cashflow = - S1 + (F1 - F0) = -F0

Futures Price converges to Spot Price: F1=S1

What if the purchase is made in 6 months?

S1

-S1 Buy

silver

F1 - F0

Futures Payoff

Net CF

5 -5 -15 -20

10 -10 -10 -20

20 -20 0 -20

30 -30 +10 -20



Basis Risk In the previous example, if the purchase is made in 6

months, the futures will need to be closed out before maturity. Then the hedged cashflow is:

Hedged Cashflow = - S0.5 + (F0.5 - F0) Basis = S0.5 - F0.5 which is usually not zero

Basis is defined as the spot price minus the futures price

There is no basis risk at maturity because the futures price converges to the spot price

Futures

Price

Futures

Price Spot Price

Spot Price



Hedging Using Index Futures

To hedge the risk in a portfolio the number of contracts that should be shorted is

where VA is the value of the portfolio, b is its beta, and VF is the value of one futures contract

Purpose: • For short-term protection, especially for large equity portfolios.

Remember, you’re also capping the upside!

• Locking in benefits of stock picking by hedging away market risk

But is beta constant? What happens during market downturns?

F

A

V

Vb



We can roll short-term futures contracts forward to hedge long-term risk.

Example: A firm needs to purchase 5000 troy ounces of silver (equivalent to 1 futures contract) for each of the next 2 months. It can hedge in 2 ways: 1. Long 2 silver futures contracts. One contract expires at the end of

the next month, the other expires in 2 months.

2. Long 2 futures contracts expiring in the next month. At the end of the next month, let one contract expire. Sell the other contract and long a new contract that expires in one month. This is stack and roll.

Stack and roll is desirable if long-term contracts are unavailable or illiquid.

Stack and Roll



Holding futures positions will result in profit or loss even if the underlying is not moving.

For a long (short) futures position,

• roll yield is negative (positive) in contango

• roll yield is positive (negative) in backwardation

Contango means contracts with longer maturities have higher prices

Backwardation means contracts with longer maturities have lower prices

http://www.ipathetn.com/us/product/vxx/#/overview

Roll Yield

http://www.ipathetn.com/us/product/vxx/



Price is determined using the “no-arbitrage” argument

Assumptions:

• No transaction costs

• Same tax rate on all net trading profits

• Same interest rate for saving and borrowing

• Arbitrageurs exist to take advantage of arbitrage opportunities

For derivatives we often use continuously compounded interest rates

Forward and Futures Prices



In math, e is just a number:

Consider an interest rate problem:

2.71828... an irrational numbere

4

$120 $100 (1 ) 20% stated annualized rate

20%$121.6=$100 1 interest paid quarterly

4

20%? =$100 1 interest paid at every instant

r r

Continuous Compounding



It turns out that:

How much is $100 continuously compounded at 20%?

In real life, such a product does not exist. The smallest compounding period is usually daily.

When the compounding period is infinitesimally small, we’re said to be working in continuous time.

1 2.71828...r re

r

0.20$122.14 = $100 e

Continuous Compounding



Recall:

But in Finance we only use the natural log (base e):

Beware: on your calculator the function for natural logarithm is ’ln’, not ‘log’

In Finance, wherever you see ‘log’ without the base, you can safely assume that it is referring to base ‘e’.

3

10

3

10

10 1000

log 1000 3

log 10 3

0.20 0.20log ln 0.20e e e

Natural Logarithm



Some useful properties:

ln

( ) 0 ( )

( ) ( )

( ) ln( )

( ) ln(1) 0 Note: ln( negative number ) is illegal

( ) ln( ) 1

( ) ln( ) ln( ) ln( )

( ) ln ln( ) ln( )

yx xy

x y x y x

x

i e ii e e

iii e e e iv e x

v e x

vi

vii e

viii xy x y

xix x y

y

e and Natural Logarithm



Arbitrage is possible when:

1. The same asset does not trade at the same price on all markets.

2. Two assets with the same cash flows do not have the same price.

3. An asset with a known future price does not trade at its future price

discounted by the risk-free rate.

No-arbitrage Argument:

There should be no arbitrage opportunities. Traders called

arbitrageurs will quickly take advantage of the opportunity and in the

process eliminate any mispricing.

“To make the parrot into a learned financial economist, he only needs to learn

the single word ‘arbitrage’”, Ross (1987)

Pricing of Derivatives



Notation

S0: Spot price today

F0: Futures or forward price today

T: Delivery Time

r: Risk-free interest rate for maturity T



Suppose that: • The spot price of a non-dividend-paying stock is $40

• The 3-month forward price (F0) is $43

• The 3-month US$ interest rate is 5% per annum

Arbitrage Example

Cashflows Now 3 months later

Short Forward 0 Sell stock +43

Borrow money to buy stock

Borrow +40 Buy stock -40 = 0

Repay loan -40 Pay interest -40 x (e0.05x0.25 – 1)= -0.503 = - 40.503

Arbitrage Profit = 43 – 40.503 = $2.497 To avoid arbitrage: F0 <= 40.503



Suppose that: • The spot price of a non-dividend-paying stock is $40

• The 3-month forward price (F0) is $39

• The 3-month US$ interest rate is 5% per annum

Arbitrage Example

Cashflows Now 3 months later

Long Forward 0 Buy stock -39

Short stock and Save money in bank

Short stock +40 Invest in Savings -40 = 0

Principle in Savings +40 Receive interest +40 x (e0.05x0.25 – 1)= +0.503 = + 40.503

Arbitrage Profit = 40.503 – 39 = $1.503 To avoid arbitrage: F0 >= 40.503



In general, if the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then:

F0 = S0erT

Forward Prices

Initial Action t = 0 Final Action t = T

Short Forward 0 Sell one share at F0 F0

Borrow money S0 Repay money with interest -S0erT

Buy one share -S0 (One share is worth ST)

Total 0 F0 - S0erT = 0



Let Dividend = PV(D) at time 0 or FV(D) at maturity date

F0 = [S0 - PV(D)]erT

When Underlying Pays a Dividend


Short Forward 0 Sell one share at F0 F0


Buy one share -S0 Receive dividends during period and invest in savings

FV(D) = PV(D) erT

Total 0 F0 – [S0 - PV(D)]erT = 0



The underlying pays a continuously compounded yield q (ex: stock index dividend, interest earned on currency)

F0 = S0 e(r-q)T

Assume that the income from the asset is reinvested into the asset. So 1 share grows to eqT shares.

When Underlying has a Known Yield


Short eqT Forward 0 Sell eqT share at F0 F0 eqT


Buy one share -S0 (grows into eqT shares)

Total 0 F0 eqT

– S0erT = 0



The underlying asset provides no income or dividend

F0 = S0erT

The underlying asset provides a known cash income

F0 = [S0 - PV(D)]erT

The underlying pays a continuously compounded yield q

F0 = S0 e(r-q)T

Essentially, we can generalize the formula in terms of cost of carry (c):

F0 = S0 ecT

Cost of carry is the net cost for owning the actual asset.

Summary



Treat the foreign currency like an asset that pays a known yield. The interest rate (rf) from the foreign currency is similar to the payout from an asset.

The forward price of a contract on a foreign currency is:

Forward on Foreign Currency

F S er r Tf

0 0( )



Unlike stocks, owning commodities require storage costs (u). This is adjusted into the forward price as a negative income.

On the other hand, sometimes owning the physical asset is beneficial because a manufacturer has the option to use it immediately. Thus holding a commodity provides a benefit referred to as the convenience yield (y).

The forward price of a contract on a commodity is:

F0 = [S0 +U]e(r-y)T OR F0 = S0 e(r+u-y)T

If the commodity cannot be sold (consumption asset):

F0 [S0 +U]e(r-y)T OR F0 S0 e(r+u-y)T

Forward on Commodities



Valuing a Forward Contract

A forward contract is worth zero (except for bid-offer spread effects) when it is first negotiated

Later it may have a positive or negative value

Banks and investors need to value the contract each day (marking to market)

Suppose that K is the original delivery price and F0 is the forward price for a contract that would be negotiated today, the value of a long forward contract is:

f = (F0 – K )e–rT

Similarly, for a short forward contract:

f = (K – F0)e–rT



Example

Let’s use a 1-year gold forward contract as an example. Price of gold is currently 1000, interest rate is 5%, and there is no convenience yield or storage cost.

When the contract is first established, the delivery price K is calculated as:

K = F0 = S0 erT = 1000e0.05(1) =1051.271

6 months later, the price of gold rises to 1200 and interest rate rises to 6%. The value of the long forward contract is:

F0 = S0 erT = 1200e0.06(0.5) =1236.545

f = (F0 – K )e–rT = (1236.545 – 1051.271 )e–0.06(0.5)

f = 179.799



Futures vs Forward Prices

The futures price is the delivery price that makes the futures contract worth zero.

When short-term interest rate is constant, the futures price and the forward price are the same. Otherwise they can be different.

This is due to the daily settlement requirement for futures. When the underlying correlates with interest rate, futures price is larger than forward price.

Ex: Underlying rises leading to a positive cash flow at the end of day. This is saved at a higher than average interest rate. Conversely, when the underlying falls, the loss is financed at a lower than average interest rate.



Arbitrage?

FINA3204 02 Futures

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