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Faculty of Business and Economics

University of Hong Kong

Dr. Huiyan Qiu

MFIN6003 Derivative Securities

Lecture Note One

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Outline

Course Overview

Introduction to Derivatives: in general• What is a derivative?

• Derivatives markets

Technical preparation• Time value of money

• Basic transaction including short-selling

• No-arbitrage principle

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Overview of the Course The course is about:• the concept, the use, the pricing of

derivatives.

1. Introduction to derivatives in general

2. Introduction of forwards and options and risk management using forwards and options

3. Option spread, collars, and other option strategies

4. Pricing of forward and futures and futures trading

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Overview of the Course (cont’d)

5. Currency forward / futures, interest rate forward / futures

6. Swaps

7. Parity and other option relationships

8. Binomial option pricing model

9. Black-Scholes formula and delta-hedging

10. Financial engineering and security design, structured products, exotic options and credit derivatives

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What are Derivatives?

A derivative security is a financial instrument whose value derives from that of some other underlying asset or assets whose price are taken as given.

We examine how to use derivative contracts to deal with financial risks related to:

– Interest rates

– Commodity prices

– Exchange rates

– Stock prices

2009 ISDA Derivatives Usage Survey

Types of Risk Managed using Derivatives (%)

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Types of Derivatives Forward contracts and futures contracts

are agreements to buy or sell an asset at a certain future time T for a certain price K.

Swaps are similar to forwards, except that the parties commit to multiple exchanges at different points in time.

A call option gives the holder the right to buy the underlying asset by a certain date T for a certain price K .

A put option gives the holder the right to sell the underlying asset by a certain date T for a certain price K .

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A Concrete Example

You enter an agreement with a friend that says:• If the price of a bushel of corn in one year is

greater than $7, you will pay him $1

• If the price is less than $7, he will pay you $1

This agreement is a derivative

Questions:• What happens one year later? (outcome,

carry-out)

• Why do you or your friend want to enter this agreement at the first place?

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Uses of Derivatives

Risk management

• Hedging: where the cash flows from the derivative are used to offset or mitigate the cash flows from a prior market commitment.

Speculation

• Where derivative is used without an underlying prior exposure; the aim is to profit from anticipated market movements.

Reduce transaction costs

Regulatory arbitrage

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Observers

End user

End user

Intermediary

EconomicObservers

• Regulators

• Researchers

Three Different Perspectives

End users

• Corporations

• Investment managers

• Investors

Intermediaries

• Market-makers

• Traders

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Derivatives Markets

The over-the-counter or “OTC” market: where two parties find each other then work directly with each other to formulate, execute, and enforce a derivative transaction. • Forward contracts, most swaps including CDS,

structured products

The exchange market: where buyer and seller can do a deal without worrying about finding each other. • Futures contracts, most options

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Measures of Market Size and Activity

Four ways to measure a market • Open interest: total number of contracts that

are “open” (waiting to be settled). An important statistic in derivatives markets.

• Trading volume: number of financial claims that change hands daily or annually.

• Market value: sum of the market value of the claims that could be traded.

• Notional value: the value of a derivative product's underlying assets at the spot price.

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Exchange Traded ContractsContracts proliferated in the last three decades

Examples of futures contracts traded on the three derivatives market

What were the drivers behind this proliferation?

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Increased Volatility…

Oil prices: 1947–2006

Dollar/Pound rate: 1947–2006

Figure 1.1 Monthly percentage change in the producer price index for oil, 1947–2006.

Figure 1.2 Monthly percentagechange in the dollar/pound($/£) exchange rate, 1947–2006.

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…Led to New and Big Markets

Exchange-traded derivatives

Over-the-counter traded derivatives: even more!

Figure 1.3 Millions of futures contracts traded annually at the Chicago Board of Trade (CBT), Chicago Mercantile Exchange (CME), and the New York Mercantile Exchange (NYMEX), 1970–2006. The CME and CBT merged in 2007.

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Derivatives Products in HKExchange-traded derivatives products in HKEX include:• Equity Index Products (futures and options on

Hang Seng Index, H-shares Index, Mini-Hang Seng Index, Mini H-shares Index, and Dividend futures)

• Equity Products (stock futures and stock options)

• Interest Rate and Fixed Income Products (HIBOR futures and Three-year exchange fund note futures)

• Gold Futures

OTC market products: numerous

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Hong Kong Mercantile Exchange

HKMEX: an electronic commodities exchange• “… HKMEx seeks to become the preferred

platform where international and mainland market participants come together to trade commodity contracts for investment, hedging and arbitrage opportunities.”

Formally began trading on May 18, 2011

Products• 32 troy ounce gold futures: May 18, 2011

• 1,000 troy ounce silver futures: July 22, 2011

Website: http://www.hkmerc.com

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Technical Preparation

Time value of money, future value, present value, APR, EAR

Continuous compounding (Appendix B)

Basic transaction: short-selling (§1.4)

No Arbitrage Principle

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Time Value of Money

Time value of money refers to a dollar today is different from a dollar in the future

Time value of money is measured by the interest rate for the period concerned.

To compare money flows, we must convert them to the same time point.

Which one is more valuable?

$100 $110

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Future Value and Present Value

nr /m )( 1P VF V

where FV = future value

PV = present value

r = the quoted annual interest rate

m = the number of times interest is compounded per year

n = the number of compounding periods to maturity

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A Simple Example

$100 is deposited for a year at quoted annual percentage rate (APR) of 12% with monthly compounding.

Given 12% APR, the monthly interest rate is 1%. At the end of each month, interest is calculated and added to the principle to earn more interest. • End of month 1: $100(1+1%)

• End of month 2: $100(1+1%)(1+1%) = 100(1+1%)2

• :

• End of month 12: $100(1+1%)12 = $100(1+12.68%)

12.68% is the effective annual rate (EAR).

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APR and EAR

APR: annual percentage rate

EAR: effective annual rate

Compounding Frequency n

EAR

(% p.a.)

Annually 1 12.0000

Quarterly 4 12.5509

Monthly 12 12.6825

Weekly 52 12.7341

Daily 365 12.7475

Continuously ∞ 12.7497

n

n

APR1EAR1

APR = 12%

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Continuously Compounding

Continuously compounding: n → ∞ (infinity)

by definition of e.

APR = 12% EAR = 12.75%

• At 12% continuously compounding annual interest rate, the future value of $100 is $112.75.

rn

ne

n

r

1lim

1275.112.0

1limEAR1 12.0

e

n

n

n

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Continuous Dividend Payment

Consider a stock (in general, an asset) paying continuous dividend with annual rate of δ. Claim: The present value of 1 share at time T is then S0e-δT.

Reason: One share at time T

is equivalent to

e-δT shares at time 0 !

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Continuous Dividend Payment

Annual dividend yield is . Let’s first assume daily

compounding, then daily dividend yield is .

At day t, per share, there is dividend in

cash, which is equivalent to unit of

shares.

In stead of keeping cash dividend (varying), we reinvest to accumulate more shares.

Starting with one share at day 0, at the end of

the year, total number of shares is .

If continuous compounding shares.

365

tS

365

3651

e

365/

365/

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Continuous Dividend Payment

That it, one share today will result in

shares one year later.

To result in one share T years later, number

of shares needed today is thus . Or one

share T years later is equivalent to

shares today.

Therefore, the present value of 1 share at

time T is S0e-δT.

e

Te

Te

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Basic Transactions

Buying and selling a financial asset (cost)

• Brokers: commissions

• Market-makers: bid-ask (offer) spread

Example: Buy and sell 100 shares of XYZ

• XYZ: bid = $49.75, offer = $50, commission = $15

• Buy: (100 x $50) + $15 = $5,015

• Sell: (100 x $49.75) – $15 = $4,960

• Transaction cost: $5,015 – $4,960 = $55

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Short-SellingWhen price of an asset is expected to fall

• First: borrow and sell an asset (get $$)

• Then: buy back and return the asset (pay $)

• If price fell in the mean time: Profit $ = $$ – $

What happens if price doesn’t fall as expected?

If the asset pays dividend in between, who gets the dividend payment?

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Short-SellingExample: Cash flows associated with short-selling a share of HSBC for 90 days.

Note that the short-seller must pay the dividend, D, to the share-lender. In other words, the lender must be compensated for the dividend.

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Short-Selling (cont’d)

Why short-sell?• Speculation• Financing• Hedging

Credit risk in short-selling• Collateral and “haircut”

Interest received from lender on collateral • Scarcity decreases the interest rate• The difference between this rate and the

market rate of interest is another cost to your short-sale

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ExampleAssume that you open a 100 share position in Fanny, Inc. common stock at the bid-ask price of $32.00 - $32.50.

When you close your position the bid-ask prices are $32.50 - $33.00.

You pay a commission rate of 0.5%.

What is your profit or loss if

• Case 1: you purchase the stock then sell;

• Case 2: you short-sell the stock then close the position.

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Example (cont’d)

You pay ask price when you purchase a stock and you get bid price when selling a stock.

If the market interest rate is ignored,

• Case 1: loss of $32.50

• Case 2: loss of $132.50

If the effective market interest rate over your holding period is 2%,

• Case 1: loss of $97.825

• Case 2: loss of $68.82

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Discussion

Question 1: With zero interest rate, why the loss in short-selling is more than the loss in outright purchase?

Question 2: Interest rate seems to have positive effect on the profit/loss on short-selling but negative effect on the profit/loss on outright purchase. Reason?

Question 3: At what interest rate, profit/loss from short-selling or from outright purchase is the same?

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Pricing Approaches Much of this course will focus on the pricing of a derivative security. In general there are two approaches to price an asset (or a contract or a portfolio):

Pricing an asset using an equilibrium model: • Determine cash flows and their risk

• Use some theory of investor’s attitude towards risk and return (e.g. CAPM) to figure out the expected rate of return

• Conduct discounted cash flow analysis to find the present value of future cash flows

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Pricing Approaches Pricing an asset by analogy (using no-arbitrage):• Find another asset, whose price you know, that

has the same payoffs of the asset to be priced.

Arbitrage is any trading strategy requiring no cash input that has some probability of making profits, without any risk of a loss• Law of One Price: two equivalent things cannot

sell for different prices.• Law of No Arbitrage: a portfolio involving zero

risk, zero net investment and positive expected returns cannot exist.

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Law of No ArbitrageCan one expect to continually earn arbitrage profits in well functioning capital markets?

From an economic perspective, the existence of arbitrage opportunities implies that the economy is in an economic disequilibrium.

Assumptions: • No market frictions (transaction costs? bid/ask

spread? restriction on short sales? taxes?)

• No counterparty risk (credit risk? collateral requirements? margin requirements?)

• Competitive market (liquidity concern?)

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Two Examples

Example 1: the effect of dividend payment on stock price change

Example 2: how to make arbitrage profit

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Cum-Dividend/Ex-Dividend Prices

A stock that pays a known dividend of dt dollars per share at date t

Stc = the cum-dividend stock price at date t

Ste = the ex-dividend stock price at date t

Assumptions

• no arbitrage opportunities,

• no differential taxation between capital gains and dividend income

The following relation can be shown to hold

Stc = St

e + dt

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No Arbitrage ArgumentSuppose that St

c < Ste + dt

• buy the stock cum-dividend

• receive the dividend

• sell the stock ex-dividend

• reap the arbitrage profits (Ste + dt) – St

c > 0

Suppose that Stc > St

e + dt

• sell the stock at the cum price

• buy it back immediately after the dividend is paid

• reap the arbitrage profits (Stc – St

e) – dt > 0

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No-Arbitrage Pricing MethodExample:

• Current stock price S0 = $25.00, there is no dividends payment in the following 6 months

• The continuously compounded risk-free annual interest rate = 7.00%

• A contract (forward contract): agreement to buy the stock at time 6 for F0, 6 = $26.00 (forward price)

Is there arbitrage profit to make? (Is the forward contract fairly priced?)

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Example (cont’d)How to generate a portfolio (synthetic contract) which duplicates the cash flows and value of the contract under considerationCash flows of the contract:

• Time 0: Zero

• Time 6: Outflow of $26 and inflow of S6 at time 6 (value of the contract: S6 – 26.)

Synthetic contract: borrow $25.00 to buy the stock

• Time-0 cash flow: Zero

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Example (cont’d)At time 6,

• Synthetic contract: pay back the borrowed money and still have the stock. Payment:

25[ e(.07)(6/12) ] = 25.89

• Forward contract: pay $26.00 to have the stock

Conclusion: the contract is over-priced!

Sell it! (Short it!)

At the same time,

buy (long) the synthetic contract!

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Example (cont’d)At Time 0 (Cash)

• Borrow $25.00 at a 7.00% annual rate for 6 months

• Buy the stock at $25.00

• Write the forward at $26.00

Between 0 and 6 (Carry)

At time 6

• Pay back borrowed money: 25[ e(.07)(6/12) ] = 25.89

• Get $26.00 from the forward (and give up the stock)

• Net payoff: $0.11

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Learn from the ExampleArbitrage-free forward price: F0, T = S0 erT

Forward price is the deferred value of the spot price

The deferred rate is the risk-free rate

Exercise:

• S0= $25.00; F0, 6 = $25.50

• The continuously compounded risk-free annual interest rate = 7.00%

• What arbitrage would you undertake? How to make profit?

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Something is worth whatever

it costs to replicate itDerivatives securities are by definition those for which a perfect replica can be constructed from other better-known securities.

The role of models: find the replica.

Buying (selling) the replica is the same as buying (selling) the derivative.

Absence of arbitrage implies the two have the same price.

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End of the Notes!