1. 1-1 Faculty of Business and Economics University of Hong Kong Dr. Huiyan Qiu MFIN6003 Derivative Securities Lecture Note One

2. 1-2 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

3. 1-3 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

4. 1-4 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

5. 1-5 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

6. 2009 ISDA Derivatives Usage Survey Types of Risk Managed using Derivatives (%) 1-6

7. 1-7 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.

8. 1-8 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?

9. 1-9 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

10. 1-10 Observers End user End user Intermediary Economic Observers Regulators Researchers Three Different Perspectives End users Corporations Investment managers Investors Intermediaries Market-makers Traders

11. 1-11 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

12. 1-11 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.

13. 1-13 Exchange Traded Contracts Contracts proliferated in the last three decades Examples of futures contracts traded on the three derivatives market What were the drivers behind this proliferation?

14. 1-14 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 percentage change in the dollar/pound ($/£) exchange rate, 1947–2006.

15. 1-15 …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.

16. 1-16 Derivatives Products in HK Exchange-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

17. 1-17 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.comhttp://www.hkmerc.com

18. 1-18 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

19. 1-19 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

20. 1-20 Future Value and Present Value 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

21. 1-21 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 ).

22. 1-22 APR and EAR APR: annual percentage rate EAR: effective annual rate Compounding Frequency n EAR (% p.a.) Annually112.0000 Quarterly412.5509 Monthly1212.6825 Weekly5212.7341 Daily36512.7475 Continuously ∞ 12.7497 APR = 12%

23. 1-23 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.

24. 1-24 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 S 0 e -δT. Reason: One share at time T is equivalent to e -δT shares at time 0 !

25. 1-25 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.

26. 1-26 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 S 0 e -δT.

27. 1-27 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

28. 1-28 Short-Selling When 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?

29. 1-29 Short-Selling Example: 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.

30. 1-30 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

31. 1-31 Example Assume 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.

32. 1-32 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

33. 1-33 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?

34. 1-34 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

35. 1-35 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.

36. 1-36 Law of No Arbitrage Can 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?)

37. 1-37 Two Examples Example 1: the effect of dividend payment on stock price change Example 2: how to make arbitrage profit

38. 1-38 Cum-Dividend/Ex-Dividend Prices A stock that pays a known dividend of d t dollars per share at date t S t c = the cum-dividend stock price at date t S t e = 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 S t c = S t e + d t

39. 1-39 No Arbitrage Argument Suppose that S t c < S t e + d t buy the stock cum-dividend receive the dividend sell the stock ex-dividend reap the arbitrage profits (S t e + d t ) – S t c > 0 Suppose that S t c > S t e + d t sell the stock at the cum price buy it back immediately after the dividend is paid reap the arbitrage profits (S t c – S t e ) – d t > 0

40. 1-40 No-Arbitrage Pricing Method Example: Current stock price S 0 = $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 F 0, 6 = $26.00 (forward price) Is there arbitrage profit to make? (Is the forward contract fairly priced?)

41. 1-41 Example (cont ’ d) How to generate a portfolio (synthetic contract) which duplicates the cash flows and value of the contract under consideration Cash flows of the contract: Time 0: Zero Time 6: Outflow of $26 and inflow of S 6 at time 6 (value of the contract: S 6 – 26.) Synthetic contract: borrow $25.00 to buy the stock Time-0 cash flow: Zero

42. 1-42 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!

43. 1-43 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

44. 1-44 Learn from the Example Arbitrage-free forward price: F 0, T = S 0 e rT Forward price is the deferred value of the spot price The deferred rate is the risk-free rate Exercise: S 0 = $25.00; F 0, 6 = $25.50 The continuously compounded risk-free annual interest rate = 7.00% What arbitrage would you undertake? How to make profit?

45. 1-45 Something is worth whatever it costs to replicate it Derivatives 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.

46. 1-46 End of the Notes!