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Financial Risk Management Fall 2007 Week 1: Introduction to Derivatives About This Semester What do we want to achieve? • Understand various types of derivative instruments such as options, forward & futures, and swaps – Similarities and distinctions – Contractual cash flows – Pricing • Develop hedging skills based on derivative instruments: – Measurement of financial risk – Transformation of cash flows – Management of risk of derivatives How do I get a good grade? • 60% of the course grade is determined by your performance in the midterm and the final exams. – Similar problems as in homework assignments will show up in those exams. Put sincere efforts into your homework assignments. – Ask questions - in class, call me, or drop by my office if you don’t understand anything that is covered in the lectures. Asking questions is an excellent way of learning things. Never hesitate. • 40% of the course grade is determined by your attendance & class participation and homework assignments. – Attendance requirements will be strictly enforced. Don’t miss classes. – I will frequently ask casual questions or prompt you to express your opinion on certain topics during lectures. Try to answer those questions or participate in discussions actively. If I do not remember you from class participation, you are losing points. – Homework problems will be assigned after each lecture. Important notice for class hour change • October 3rd is a national holiday. – Hence there will be no class on that day. • A make-up class is temporarily scheduled on October 5th. 1. Derivative Markets and Instruments 1. Derivative Markets • Definition of derivatives – A derivative is a financial instrument whose return is derived from the return on another instrument. – The Underlying Asset • A derivative derives its value from another instrument. • The instrument that decides the value of a derivative is called the underlying asset. – An alternative definition of a derivative instrument is that it is a contract that determines the price at which the designated underlying assets are exchanged between two parties. • Derivative Markets – Established exchanges – Over-The-Counter Markets • Size of the OTC derivatives market at year-end 2005 – $285 trillion notional principal – $9.1 trillion market value 2. Types of Derivatives • Option: a contract between two parties that gives one party, the buyer, the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today • Forward Contract: a contract between two parties for one party to buy something from the other at a later date at a price agreed upon today – Exclusively over-the-counter • Futures: a contract between two parties for one party to buy something from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default – Exclusively traded on a futures exchange • Option on Future: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today – also known as commodity options or futures options – Exclusively traded on a futures exchange • Swap: a contract in which two parties agree to exchange a series of cash flows – Exclusively over-the-counter • Other Derivatives – Other types of derivatives include swaptions and hybrids. – Their creation is a process called financial engineering. 3. The Role of Derivative Markets • Risk Management – Hedging vs. speculation – Setting risk to an acceptable level • Price Discovery • Operational Advantages – Transaction costs – Liquidity – Ease of short selling • Market Efficiency 4. Theoretical Value of a Derivative Instrument • Market Efficiency – Efficient market: A market in which the price of an asset equals its true economic value. – An efficient market is a consequence of rational and knowledgeable investor behavior. • Theoretical Fair Value – Theoretical fair value of a derivative instrument is the true economic value of the contractual cash flows. • Arbitrage and the Law of One Price – Arbitrage is a type of profit-seeking transaction where the same good trades at two prices. • Example: See Figure 1.2, p. 10 – The Law of One Price tells us that there should be no arbitrage opportunity in an efficient market. Currently asset 1 is selling at a price higher than twice the price of asset 2. In each state in the future, the cash flow from asset one is exactly twice that from asset two. If one purchases two units of asset 2 and sell one unit of asset 1 now, he/she can earn arbitrage income of $3. 2. Introduction to Options 1. Basic Option Terminology • price/premium • call vs. put • exercise price/strike price/striking price • expiration date • Moneyness concepts – In-the-money – Out-of-the-money – At-the-money 2. Organized Options Trading • Market Maker – Bid, ask, and bid-ask spread – Scalpers, position traders, spreaders • Floor Broker – Designated primary market makers (DPMS) • Order Book Official (Board broker) – Limit orders • Other Option Trading Systems – Specialists – Registered options traders – Electronic trading systems • Off-Floor Option Traders 3. Mechanics of Trading • Placing an Opening Order – Types of orders: market, limit, good-till-canceled, day, stop, all-or-none, all-or-none-same-price • Role of the Clearinghouse – Options Clearing Corporation (OCC) – Clearing firms • Placing an Offsetting Order – In the exchange-listed options market – In the over-the-counter options market • Exercising an Option – European vs. American style – Assignment – Cash settlement 4. Types of Options • Stock Options • Index Options • Currency Options • Other Types of Traded Options – interest rate options – currency options – options attached to bonds – exotic options – warrants, callable bonds, convertible bonds – executive options • Real Options 5. Transaction Costs in Option Trading • Floor Trading and Clearing Fees • Commissions • Bid-Ask Spread • Other Transaction Costs 3. Characteristics of Option Prices 1. Cash Flows when Exercised Long Position Short Position Cash Flow Cash Flow Call Option 0 0 Price of Exercise Price Underlying Asset Exercise Price Cash Flow Cash Flow Put Option 0 Price of 0 Exercise Price Underlying Asset Exercise Price 2. Intrinsic Value and Time Value • Intrinsic Value – Intrinsic value is the cash flows to the buyer if exercised immediately – Also called minimum value, parity value, parity, or exercise value • Time Value – Actual option price is normally higher than the intrinsic value. – Time value of an option is the difference between the price and the intrinsic value. – Also called speculative value 3. Basic Notation and Terminology • Symbols – S0 (stock price) – X (exercise price) – T (time to expiration = (days until expiration)/365) – r (risk-free rate) – ST (stock price at expiration) – C(S0,T,X), P(S0,T,X) 4. Computation of Risk-Free Rate • Example – Date: May 14. Option expiration: May 21 • Remaining maturity is 7 days – T-bill rate for the same maturity • bid discount = 4.45, ask discount = 4.37 • Computing the risk-free rate – Average T-bill discount = (4.45+4.37)/2 = 4.41 – T-bill price = 100 - 4.41(7/360) = 99.91425 – T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 – So 4.57 % is risk-free rate for options expiring May 21 5. Characteristics of Call Option Price Example 5-1. Minimum Value of a Call • C(S0,T,X) 0 (for any call) • For American calls: – Ca(S0,T,X) Max(0,S0 - X) • Intrinsic value: Max(0,S0 - X) 5-2. Maximum Value of a Call Maximum Value: S0 C(S0,T,X) S0 Maximum Value:S0 C(S0,T,X) S0 5-3. Value of a Call at Expiration Value of a Call at Expiration: C(ST,0,X) = Max(0,ST - X) 5-4. Effect of Time to Expiration • Two American calls differing only by time to expiration, T1 and T2 where T1 < T2. • Ca(S0,T2,X) Ca(S0,T1,X) • Deep in- and out-of-the-money – Time value maximized when at-the-money • Concept of time value decay • Cannot be proven (yet) for European calls 5-5. Effect of Exercise Price • Two European calls differing only by strikes of X1 and X2. • Construct portfolios A and B. • Portfolio A has non-negative payoff; hence, Ce(S0,T,X1) Ce(S0,T,X2) 5-5. Effect of Exercise Price (continued) • Limits on the Difference in Premiums – We must have (X2 - X1)(1+r)-T Ce(S0,T,X1) - Ce(S0,T,X2) X2 - X1 Ce(S0,T,X1) - Ce(S0,T,X2) X2 - X1 Ca(S0,T,X1) - Ca(S0,T,X2) 5-6. Lower Bound of a European Call • Construct portfolios A and B. • B dominates A. – This implies that (after rearranging) Ce(S0,T,X) Max[0,S0 - X(1+r)-T] – This is the lower bound for a European call 5-7. American Call Versus European Call • In general, Ca(S0,T,X) Ce(S0,T,X) – Hence Ca(S0,T,X) Max(0,S0 - X(1+r)-T) • If there are no dividends on the stock, an American call will never be exercised early. – It will always be better to sell the call in the market. • Early Exercise of American Calls on Dividend-Paying Stocks – If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. 6. Characteristics of Put Option Price 6-1. Minimum Value of a Put • P(S0,T,X) 0 (for any put) – Intrinsic value: Max(0,X - S0) • For American puts: Pa(S0,T,X) Max(0,X - S0) 6-2. Maximum Value of a Put • Pe(S0,T,X) X(1+r)-T • Pa(S0,T,X) X 6-3. Value of a Put at Expiration • P(ST,0,X) = Max(0,X - ST) 6-4. Effect of Time to Expiration • Two American puts differing only by time to expiration, T1 and T2 where T1 < T2. • Pa(S0,T2,X) Pa(S0,T1,X) • Cannot be proven for European puts 6-5. Effect of Exercise Price • Two European puts differing only by X1 and X2. • Construct portfolios A and B. • Portfolio A has non-negative payoff; hence Pe(S0,T,X2) Pe(S0,T,X1) 6-5. Effect of Exercise Price (continued) • Limits on the Difference in Premiums – We must have (X2 - X1)(1+r)-T Pe(S0,T,X2) - Pe(S0,T,X1) X2 - X1 Pe(S0,T,X2) - Pe(S0,T,X1) X2 - X1 Pa(S0,T,X2) - Pa(S0,T,X1) 6-6. Lower Bound of a European Put • Construct portfolios A and B. • A dominates B. – This implies that (after rearranging) Pe(S0,T,X) Max(0,X(1+r)-T - S0) – This is the lower bound for a European put. 6-7. American Put Versus European Put • Pa(S0,T,X) Pe(S0,T,X) • Early Exercise of American Puts – There is always a sufficiently low stock price that will make it optimal to exercise an American put early. – Dividends on the stock reduce the likelihood of early exercise. 7. Put-Call Parity • Form portfolios A and B where the options are European. • The portfolios have the same outcomes at the options’ expiration. – Thus, it must be true that S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T – This is called put-call parity. 7. Put-Call Parity (continued) • Put-Call parity for American options can be stated only as inequalities: N C a (S'0 , T, X) X D j (1 r) t j j1 S0 Pa (S'0 , T, X) C a (S'0 , T, X) X(1 r) T where N S S0 D j (1 r) ' t j 0 j1 Homework Assignments No. 1 • Selected from end of chapter problems: Chapter 1: #4, #15, #16 Chapter 2: #1, #8, #17 Chapter 3: #1, #13, #20 • Due date: September 19th

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