Gallery of equations. 1. Introduction

Transcription

1 Gallery of equations. Introduction Exchange-traded markets Over-the-counter markets Forward contracts Definition.. A forward contract is an agreement to buy or sell an asset at a certain future time for a certain price Definition.2. The forward price for a contract is the delivery price that would be applicable to the contract if it were negotiated today (i.e., it is the delivery price that would make the contract worth exactly zero). Futures contracts Definition.3. A futures contract is a standardized contract, traded on a futures exchange, to buy or sell a certain underlying instrument at a certain date in the future, at a specified price. The future date is called the delivery date or final settlement date. The pre-set price is called the futures price. The price of the underlying asset on the delivery date is called the settlement price. The settlement price, normally, converges towards the futures price on the delivery date. Options Types of traders Hedgers Speculators Arbitrageurs

2 2 Ian Buckley Dangers 2. Mechanics of Futures Markets Background Specification of a futures contract Specification by exchange what where when trading hours quote conventions max price movements Convergence of futures price to a spot price S t,f t 3 S t,f t t t F t S t, b 0, F t S t, b 0, Figure.: Relationship between future price and spot price as delivery period is approached. E.g. gold (lhs) and oil (rhs). If at delivery futures price is above below spot, i.e. F T S T then arb opp is F T S T short future (zero cost) go long buy sell asset (for S T) deliver receive underlying ( earning at a cost of F T) Eventually, futures price will fall to match spot rise Prior to expiry, spot can be below above future,

3 CMFM03 Financial Markets 3 i.e. basis, e.g. gold, which are investment oil consumption assets F S rt prices related by F S ruyt Daily settlement and margins Definition Definition.4. A margin is cash or marketable securities deposited by an investor with his or her broker. Operation The balance in the margin account is adjusted to reflect daily settlement marking to market Types Initial margin amount deposited when contract entered Maintenance margin trigger level for margin call to restore balance to initial margin Differenceis variation margin 0 maintenance margin initial margin Investor can withdraw balance in excess of initial margin Table.. Operations of margins for a long position in two gold futures contracts. Fut price Daily gain Cum gain Mgn ac bal Mgn call

4 4 Ian Buckley Newspaper quotes Delivery Types of traders and types of orders Regulation Accounting and tax Forward vs. futures contracts Private vs exchange Single vs multiple delivery date Non-standard Final vs daily settlement 3. Hedging Strategies Using Futures Basic principles Arguments for and against hedging Hedges lose money What are competitors doing? Basis risk Definition.5. A basis,, is the extent to which the spot price of the asset to be hedged exceeds the futures price of the contract used for hedging. basis b spot price of asset to be hedged S futures price of contract used F (.) gain on futures effective S 2 terminal S stock price 2 F2 F F b 2 (.2) Cross hedging Proposition.6. The optimal hedge ratio is given by h S F.

5 CMFM03 Financial Markets 5 h S F (.3) Proof t t 2 time at which choice is made to hedge time at which asset is to be sold N A number of units of asset to sell at time t 2 N F number of futures contracts to short at time t h Y S i,f i S, F hedge ratiois h:n F N A total amount realized for the asset when the profit or loss on the hedge is taken into account asset prices and futures prices at time i, i,2 change in asset and futures prices over interval, i.e. S:S 2 S, F:F 2 F v variance of Y Total amount realized for asset and hedge: S, F, standard deviations of asset and future, and correlation coefficient between them Y S 2 N A F 2 F N F S N A S 2 S N A F 2 F N F S N A N A S h F S and N A are known at time t Minimising the variance of Y, corresponds to minimising the variance of S h F Derivative w.r.t. h v VarS h F S 2 h 2 F 2 2h S F v h 2h F 2 2 S F is zero (and second derivative +ve) when Number of futures contracts h S F Proposition.7. The number of futures contracts required is given by N h N A Q F.

6 6 Ian Buckley N h N A Q F (.4) Stock index futures Proposition.8. To hedge the risk in a portfolio the number of contracts that should be shorted is N P, where P is the value of the portfolio, is its beta, and A is the value of A the assets underlying one futures contract. N P A (.5) Changing Beta Proposition.9. To change the beta of a portfolio from to, a short position in P contracts is required A, a long position in P contracts is required. A Rolling the hedge forward 4. Interest Rates Types of rates Measuring interest rates Equations relating discrete and continuous compounding rates R c R m m ln R m m m R cm (.6) Zero rates Definition.0. A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T Bond pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate

7 CMFM03 Financial Markets 7 P i coupons n c R i T i principal R n T n n c R i T i c R n T n i (.7) Cashflows c are related to annual coupon with compounding frequency m by c m by c c m m Definition.. The bond yield for a bond is the discount rate that makes the present value of the cash flows equal to the market price. coupons n i principal c Y T i Y T n P (.8) Par yield Definition.2. The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. n 00 i c Y i T i c Y n T n 00 (.9) c Y n T n n i Y i T i (.0) If we name d Y n T n n and A i Y i T i, we obtain dm c m A (.) Determining Treasury zero rates Table.2. Data for 5 bonds, 2 paying a semi-annual coupon Principal Time to maturity Coupon Cash price

8 8 Ian Buckley $! " #$! % % & '# #! $ & & #!( % % & % )) '#!# )) ) % % & ) Forward rates Definition.3. The forward rate is the future zero rate implied by today s term structure of interest rates! " # $ % $& %& %' & $ ( #) $ +! % * $ % * Formula for forward rates R F R 2 T 2 R T R T 2 T 2 R 2 R T 2 T T (.2) Derivation T 2 R 2 T R T 2T R F If not, arbitrage opportunities, etc...

9 CMFM03 Financial Markets 9 Here compounding frequency superscript m reminds us that these are continuously compounded rates Instantaneous forward rate Definition.4. The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. It is R F R T R where R is the T-year T rate R F R T R T (.3) Cf. f t T Y t T T t Y t T T (.4) Forward rate agreements Definition.5. A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period. FRA valued by assuming that the forward interest rate is certain to be realized Payoffs LR K m R M m T 2 T (.5) Discrete compounding frequency m T 2 T Valuation FRA worth zero when R K R F Why buy an FRA for $$ if you can lock in price of forward borrowing Go long FRA with rate R K, short FRA with rate R F Costs are V FRA and 0 respectively Net (deterministic) payoff at T 2 is LR K R M R F R M T T 2 Value of FRA where a fixed rate R K will be received on a principal L between times T and T 2 is Cash flow at time T 2 R 2 T 2 V FRA LR m K R m F T T 2 (.6)

10 0 Ian Buckley Duration Definition.6. The duration of a bond that provides cash flow c i at time t i is n D t i c i y t i i B where B is its price and y is its yield (continuously compounded) B D y B (.7) Duration is the proportional change in the bond price per unit (parallel) shift in the yield curve Convexity C B 2 B y 2 n i t i 2 c i y t i B (.8) B B D y 2 Cy2 (.9) Theories of the term structure of interest rates 5. Determination of Forward and Futures Prices Summary Asset Forward / futures price Value of long forward contract No income S 0 rt S 0 K rt Income of present value I S 0 I rt S 0 IK rt Yield q S 0 rqt S 0 qt K rt Investment assets vs. consumption assets Definition.7. An investment asset is an asset that is held primarily for investment.

11 CMFM03 Financial Markets E.g. stocks, bonds, gold, silver Definition.8. A consumption asset is an asset that is held primarily for consumption. Short selling Assumptions and notation Forward price of an investment asset See Chapter Known income Instrument Holding Value at 0 Value at T Stock S 0 S T Bank/bond Left at end S 0 I Paid off by divs I S 0 II S 0 I rt Forward 0 S T F 0 Total 0 F 0 S 0 I rt Known yield Instrument Holding at t Value at 0 Value at t Value at T Stock qtt S 0 qt S t qtt S T Bank/bond S 0 qt rt S 0 qt S 0 qt rt S 0 rqt Forward 0 S T F 0 Total 0 F 0 S 0 rqt Valuing forward contracts Proposition.9. The value, f, of a long forward contract is given by f F 0 K rt, where... f F 0 K rt (.20)

12 2 Ian Buckley Proof Forward (delivery price) Holding Value at 0 Value at T Forward (K) f S T K Forward (F 0 ) 0 S T F 0 Total f F 0 K Are forward prices and futures prices equal? Argument that they are not when IRs and underlying are correlated Consider :S, r 0 Sr likely Long future, immediate gain due to mk-to-mkt Invested at higher than average rate Similarly when S Positive more correlation S, r, long future attractive than long forward Negative less Proof that foward and futures prices are equal when interest rates are constant Proposition.20. A sufficient condition for forward and futures prices to be equal is that interest rates be constant. Strategy: Profit take a long futures position of at the beginning of day 0 increase position to 2 at the beginning of day long futures position i at start of day i on day (end of day 0) is F F 0 on day i is F i F i i and is banked Compounded value from day i on day n is F i F i i ni F i F i n Table.3. Dynamic investment strategy in futures contracts

13 CMFM03 Financial Markets 3 Day 0 2 n n Futures price Futures posn F 0 F F 2 F n F n 2 3 n 0 Gain 0 F F 0 F 2 F 2 Gain comp'd to n 0 F F 0 n Value at day n of entire strategy F 2 F n F n F n n F n F n n n F i F i n F F 0 F 2 F F n F n n i F n F 0 n S T F 0 n Cost of each increment to the futures position is Combined strategy of dynamic strategy above (costs zero, payoff S T F 0 n ) invest F 0 in a risk-free bank account (costs F 0 at 0, pays off F 0 n at expiry) Total cost at 0 is F 0 ; total payoff at n is S T n Table.4. Combined investment strategy: dynamic futures strategy above + bank Description Cost (PV at 0) Payoff (at n) Dynamic futures strategy 0 S T F 0 n Bank account, holding F 0 F 0 F 0 n Total F 0 S T n Table.5. Investment strategy: long forward contract + bank Description Costat 0 or other times Payoff (at n) Long forward unit 0 S T G 0 n Bank account, holding G 0 G 0 G 0 n Total G 0 S T n Both strategies have the same payoff after n days, so must be worth the same at time 0 F 0 G 0

14 4 Ian Buckley Futures prices of stock indices Forward and futures prices on currencies F 0 S 0 rr ft (.2) Time Foreign FX Dollars 0 S 0 T r f T F 0 r f T S 0 rt Figure.2: Two ways of converting a single unit of a foreign currency to dollars at time T. *+,-.- /&'#+,0 *+0 (, -. /. / * ' ' + 0 ' ' ' *!* + + ' * * + ' Futures on commodities Proposition.2. The initial forward price F 0 and spot price S 0 for a consumption asset for which the present value of the storage costs are U satisfy F 0 S 0 U rt F 0 S 0 U rt (.22) Proposition.22. The initial forward price F 0 and spot price S 0 for a consumption asset for which the storage costs per unit time is u satisfy F 0 S 0 rut F 0 S 0 rut (.23) Consumption commodities Proposition.23. The initial forward price F 0 and spot price S 0 for a consumption asset for which the present value of the storage costs are U obey the inequality F 0 S 0 U rt F 0 S 0 U rt (.24)

15 CMFM03 Financial Markets 5 Storage costs proportional to commodity price Proposition.24. The initial forward price F 0 and spot price S 0 for a consumption asset for which the storage costs per unit time is u obey the inequality F 0 S 0 rut F 0 S 0 rut (.25) Convenience yields Definition.25. The convenience yield is the value of y such that when the storage costs are known and have present value U, then F 0 yt S 0 U rt. Similarly for storage costs that are a constant proportion u of the spot price: F 0 yt S 0 rut. F 0 yt S 0 U rt F 0 S 0 ruyt (.26) Convenience yield measures extent to which forward price of consumption assets falls short of the theoretical value for investment assets The cost of carry Definition.26. The cost of carry is the storage cost plus the interest costs less the income earned. c r u q (.27) Relationships between forward and spot prices in terms of the cost of carry Proposition.27. The initial forward price F 0 and spot price S 0 for an investment asset that pays no dividend are related by F 0 S 0 ct, where... F 0 S 0 ct (.28) Proposition.28. The initial forward price F 0 and spot price S 0 for a consumption asset that pays no dividend are related by F 0 S 0 cyt, where... F 0 S 0 cyt (.29)

16 6 Ian Buckley Delivery options Futures prices and expected future spot prices 6. Interest Rate Futures Day count conventions Interest earned between two dates Number of days between dates Interest earned in reference period Number of days in reference period (.30) In US Treasury Bonds: Corporate Bonds: 30/360 Money Market Instruments: Actual/Actual (in period) Actual/360 *"&& 2-3 4", " 4"5$ 2( 3 %+$4 '$ )# 0$45#6#, ) * "&& 2-3 4", " 4"5$ 2( 3 %+$4 '$ )# 0$45#6 #, ) Money market Instruments Actual/360 Quoted using a discount rate

17 CMFM03 Financial Markets 7 P Y n (.3) Quotations for Treasury bonds Since last coupon Cash price Quoted price Accrued interest (.32) 64#&&."&&&&& ""- 5"&#&"#78"' 4#&&. 2$ #6! (# 9+#: ) #$; 3 <$/+( / / ) 23 #$+ / / /6 Treasury bond futures Conversion factors Assume interest rate for all maturities is 6% per annum (semi annual compounding) round to nearest 3 mos Cash price received by party with short position Cash price received by party with short position Most recent settlement price Conversion factor Accrued interest (.33) Cheapest to deliver bond Given that the cost to purchase a bond is: Quoted bond price Accrued interest ' #$# While the cash price received is:

18 8 Ian Buckley Quoted futures price Conversion factor Accrued interest ' # A cheapest to deliver bond can be found where the difference is a minimum: Quoted bond price Quoted futures price Conversion factor (.34) Determining the futures price F 0 S 0 I rt (.35), 30 "#- "%9 #.& 9"&- 7 "#& ( (7 "#- Coupon 60 Coupon Now 48 0 Maturityfut Coupon 305 2$ ) ' = 6" + $+ $ > 2+ $ 0 #,?.+/+ ',, ) ) 6" + $ ))) Eurodollar futures Eurodollar dollar deposited in bank outside United States Eurodollar dollar deposited in bank outside United States Forward vs futures interest rates Forward rate futures rate 2 2 t t 2 (.36)

19 CMFM03 Financial Markets 9 Extending the LIBOR Zero Curve F i R it i R i T i T i T i (.37) R i F it i T i R i T i T i (.38) Duration-based hedging strategies Duration Matching Duration-based hedge ratio Number of contracts required to hedge against an uncertain y is N P D P F C D F (.39) Hedging portfolios of assets and liabilities 7. Swaps Definition.29. A swap is an agreement to exchange cash flows at specified future times according to certain specified rules. Mechanics of interest rate swaps Definition.30. A plain vanilla interest rate swap is an agreement in which a company agrees to pay cash flows equal to interest at a predetermined fixed rate in return for interest at a floating rate, on a notional principal, for a period of time. An Example of a Plain Vanilla Interest Rate Swap 5.0% LIBOR Figure.3: Interest rate swap between Microsoft and Intel Table.6. Cash flows (millions of dollars) to Microsoft, in a $00 million, 3-year interest rate swap, when a fixed rate of 5% is paid and LIBOR is received. The net cash flow is the difference. (Ignore day count issues.)

20 20 Ian Buckley Date 6-month LIBOR (%) Floating received Fixed paid Net cash flow Mar. 5, Sept. 5, Mar. 5, Sept. 5, Mar. 5, Sept. 5, Mar. 5, Typical Uses of an Interest Rate Swap Cashflow to transform Liability Asset 5.0% 5.2% LIBOR LIBOR 0.2% 5.0% LIBOR LIBOR 0.% 4.7% Figure.4: Use of an interest rate swap to transform a liability or an asset, without a financial intermediary. Cashflow to transform Liability 4.985% 5.2% LIBOR 5.05% LIBOR LIBOR 0.% Asset LIBOR 0.2% 4.985% LIBOR 5.05% LIBOR 4.7% Figure.5: Use of an interest rate swap to transform a liability or an asset, with a financial intermediary. Day count issues LIBOR (e.g. 6-mo in Table 7.) is a money market rate, hence quoted on actual/360 basis Fixed rate cash flows actual/365 30/360

21 CMFM03 Financial Markets 2 The comparative-advantage argument Table.7. Borrowing rates for two corporations Company Fixed Floating AAACorp 4.0% LIBOR + 0.3% BBBCorp 5.2% LIBOR +.0% Financial intermediary Agreement No 4% 3.95% LIBOR LIBOR % Yes 4% 3.93% LIBOR 3.97% LIBOR LIBOR % Figure.6: Illustration of comparative advantage agreement for two corporations, without and with a financial intermediary. The nature of swap rates Six-month LIBOR is short-term AA borrowing rate Definition.3. Swap rates are the fixed rates at which financial institutions offer interest rate swap contracts to their clients. Definition.32. The swap rate is that value of the fixed rate that makes the value of the swap zero at inception. Determining the LIBOR/swap zero rates Overview of argument Consider a new swap with fixed rate = swap rate Add principals on both sides on final payment date swap exchange of fixed rate and floating rate bonds Value of bonds/swaps Floating-rate rate bond par. Swap zero. fixed-rate bond worth par. swap rates define par yield bonds; used to bootstrap the LIBOR (or LIBOR/swap) zero curve

22 22 Ian Buckley Swap rates from bond prices Proposition.33. The n-period swap rates S n can be expressed in terms of prices of zero coupon bonds P 0n by the relationship S n P 0n n i P 0i S n P 0n n i P 0i (.40) Proof Consider 4-year swap S 4 S 4 S 4 S Figure.7: Diagram of cashflows for fixed leg of a 4-year swap agreement Pay to get etc... 4 coupons of swap rate initial investment back at 4 Bond prices from swap rates S 4 P 0 S 4 P 02 S 4 P 03 S 4 P 04 Proposition.34. The prices of zero coupon bonds P 0n can be expressed in terms of the n-period swap rate S n by the relationship P 0n S n r0 n r k0 S nk n P 0n S n r0 r k0 S nk (.4) Proof The swap rate is the level of the fixed rates such that the swap has zero value at inception Zero value occurs when floating rate bond equals fixed rate bond At inception, value of floating rate bond is unity (up to a common factor of the principal) S P 0 S 2 P 0 S 2 P 02 S 3 P 0 S 3 P 02 S 3 P 03 n S n i P 0i P 0n Solve iteratively in terms of P 0i

23 CMFM03 Financial Markets 23 P 0 P 02 P 03 P 0n S S S 2 P 0 S S 2 S 2 S 2 S S 2 S S 2 2 S S 2 S 2 S S 2 2 S S 2 S 2 S S 2 S 2S 2 S 3 S 3 S n r0 n r k0 S 2 S 2 S 3 S nk S S 2 S 3 FRNs are worth par after a coupon payment Consider a 3-yr investment paying a LIBOR coupon, annually L 0 L 2 L Figure.8: Diagram of cashflows for 3-yr investment paying a LIBOR coupon, annually With a compounding frequency of m, the LIBOR rate and price of a zero coupon ba bond are related: P ab L ab b a (.42) For our simple case, b a L ab P ab Argument Each LIBOR coupon payment is discounted back to the previous payment date using a LIBOR discount rate L 0 L 0 L 2 L L 2 L 23 Figure.9: Diagram of cashflows for 3-yr investment with LIBOR coupons expressed in terms of bond prices Value of 3rd coupon at time 2 L 23 L 23 Value of 3rd and 2nd coupons at time previous step L 2 Repeat iteratively down to t 0 L 2 Value of 3rd, 2nd & st coupons at time 0

24 24 Ian Buckley previous step L 0 L 0 If coupon has just been paid, floating rate note is worth par For an alternative proof see e.g. Cuthbertson & Nitzsche (200) Financial Engineering: Derivatives and Risk Management Valuation of interest rate swaps Swap value V swap B fix B fl (.43) Fixed rate bond n B fix c i r it i r nt n i n c i P 0i P 0n i (.44) Value of floating rate bond Table.8. Expressions for the value of a floating rate bond Time, relative to coupon payment Immediately after Immediately before Time t before Value of bond k k r t Example value swap as pair of bonds and as portfolio of FRAs : ' ;<=6> 2- "&& "# ;<=6>$8?""&-"&-?""- ';<=6> "&#-

25 CMFM03 Financial Markets 25 A / ' +( +@ + /)) / / Table.9. Table accompanying exercise Time B fix cash flow B fl cash flow Discount factor PV B fix cash flow PV B fl cash flow Total Valuation in Terms of FRAs : >* *8 * $# 6#A! *+ $# 6$+ #+!;*%$ A< % * % % % 6 ;<! $ '#+% 6 Table.0. Table accompanying exercise

26 26 Ian Buckley Time Fixed cash flow Float cash flow Net Discount factor PV net cash flow Total Agrees At inception, and later, FRAs do not have zero value PVs cash flows for T i, T i are: floating R i T i R i T i (convert cc forward rate to discrete and discount) fixed sm m R i T i Valuation formula Value of a vanilla payer (fixed-for-floating) interest-rate swap, with swap rate s m, with discrete compounding frequency m T i T i for i,, n: V swap s m R n T n sm n m i R i T i (.45) Currency swaps Comparative Advantage Arguments for Currency Swaps USD 5.0% USD 5% AUD.9% USD 6.3% AUD 3.0% AUD 3.0%

27 CMFM03 Financial Markets 27 Valuation of currency swaps Credit risk Other types of swaps 8. Mechanics of Options Markets Types of options Option positions Underlying assets Specification of stock options Newspaper quotes Trading Commissions Margins The options clearing corporation Regulation Taxation Warrants, executive stock options, and convertibles Over-the-counter markets 9. Properties of Stock Options Factors affecting option prices

28 28 Ian Buckley Assumptions and notation Upper and lower bounds for option prices Upper bounds Call options Proposition.35. The stock price is an upper bound to the price of an American or European call option. c S 0, C S 0 (.46) Put options Proposition.36. The strike price is an upper bound to the price of an American call option. The discounted strike price is an upper bound to the price of a European call option. p K, P K (.47) Lower bounds for European calls on non-dividend paying stocks Proposition.37. The expression maxs 0 K, 0 is a lower bound to the price of a European call option. maxs 0 K, 0 c (.48) Example 0 & Instrument Holding Value at 0 Value at T Stock S 0 S T Bank/bond K K K Call c maxs T K, 0 Total c S 0 K maxs T K, 0 S T K maxk S T, 0! 0 Proof of proposition Using the strategy from the table Value at T of strategy is greater than or equal to zero.

29 CMFM03 Financial Markets 29 must be true at earlier times also (why?) c S 0 K! 0 Also c! 0 Combining: c! maxs 0 K, 0 Lower bounds for European puts on non-dividend paying stocks Proposition.38. The expression maxk S 0, 0 is a lower bound to the price of a European put option. maxk S 0, 0 p (.49) Proof Seek lower bound, so show if put price lower than bound arb opp. Cheap put long put Table.. Strategy to find lower bound for a European put price Instrument Holding Value at 0 Value at T Stock S 0 S T Bank/bond K K K Put p maxks T,0 Total pks 0 maxks T,0 KS T maxs T K,0!0 Value of strategy at T is greater than or equal to zero. true at earlier times p K S 0! 0 Also p! 0 Combining: p! maxk S 0, 0 Put-call parity European Proposition.39. The prices of European call and put options are related by c K p S 0 c K p S 0 (.50) Proof Table.2. Strategy to establish put-call parity for European options

30 30 Ian Buckley Instrument Holding Value at 0 Value at T Call c maxs T K,0 Put p maxks T,0 Stock S 0 S T Bank/bond K K K Total cps 0 K 0 American options Proposition.40. Bounds on the prices of American call and put options are given by S 0 K C P S 0 K S 0 K C P S 0 K (.5) Proof Upper bound puts more No dividend case: American calls equally below.) P! p c K S 0 C K S 0 C P S 0 K Lower bound than puts valuable European. (See to calls Consider value immediately after entering strategy, given that value of American options (call or put) is always greater or equal to value of immediate exercise " is time at which it is optimal to early exercise the put: " T K K rt" K K # K r " Table.3. Strategy to establish put-call parity lower bound for C P for American options

31 CMFM03 Financial Markets 3 Instrument Holding Value at 0 Value at " Value at T, no early excs Call Cc!S " K lower bound for Euro call S T K Bank/bond K K K # K r" K # K rt Subtotal CK!maxS ",K K # K Put P KS " value when we exercise maxs T,K KK # KS T Stock S 0 S " S T Subtotal PS 0 maxs ",K maxs T,K Total CPS 0 K >0 payoff greater than from a bull spread 0 C P S 0 K! 0 S 0 K C P Early exercise: calls on a non-dividend paying stock Proposition.4. It is never optimal to early exercise an American call option on a non-dividend paying stock. Plausibility argument Formal argument S 0 K Consider Lower bound on c c r 0 S 0 K C American at least as valuableas European Condition for early exercise S 0 K C Cannot both be true, so early exercise can never be optimal C Early exercise: puts on a non-dividend paying stock Early exercise of American puts can be optimal Consider extreme case 0S 0 K

32 32 Ian Buckley Effect of dividends Lower bounds for calls and puts Table.4. Strategy to find lower bound for a European put price Instrument Holding Value at 0 Value at T Stock S 0 S T Bank/bond K D KD K Call c maxs T K,0 Total cs 0 KD maxs T K,0 S T K maxks T,0!0 Therefore: c! S 0 D K (.52) Similarly p! D K S 0 (.53) Early exercise Now sometimes optimal to early exercise a call Put call parity Equality for European options c D K p S 0 (.54) Bounds for American options S 0 D K C P S 0 K (.55) 0. Trading Strategies Involving Options Strategies involving a single option and a stock Spreads! 2 options of same type (all calls, or all puts) Combinations mixture of calls and puts Other payoffs

33 CMFM03 Financial Markets 33 Strategies involving a single option and a stock Spreads Types of spreads Bull Bear Box Butterfly Calendar Diagonal Bull spreads Bull Spread = long call at K + short call at K 2, K K 2 Limits upside and downside hope stock 3 types to do with moneyness of calls, in order of aggressiveness Both in One in, one out Both out Alternatively, long put at K, short put at K 2 Profit Profit S T K K 2 S T K K 2 Figure.: Bull spread using calls (left) and puts (right) Payoffs in each interval Table.5. Payoff from a bull spread created using calls Interval Long call Short call Total K 2 S T S T K S T K 2 K 2 K K S T K 2 S T K 0 S T K S T K Example

34 34 Ian Buckley * 9 $& $ 9 $ " (@A (@ 9 $&$A (@A (@ 9 //6$ (# 2 + #/ / /? '?+ ' ' B Bear spreads Anticipation prices Bear Spread = short call at K + long call at K 2, K K 2 Profit Profit K K 2 S T K K 2 K K 2 S T Figure.2: Bear spread using calls (left) and puts (right) Payoffs in each interval Table.6. Payoff from a bear spread created using puts Interval Long put Short put Total K 2 S T K S T K 2 K 2 S T 0 K 2 S T S T K K 2 S T K S T K 2 K Box spreads Combination of spreads: bull call

35 CMFM03 Financial Markets 35 bear put Valuation European box spread worth PV of difference between strikes American not so Payoffs in each interval Table.7. Payoff from a box spread Interval Bull call Bear put Total K 2 S T K 2 K 0 K 2 K K S T K 2 S T K K 2 S T K 2 K S T K 0 K 2 K K 2 K Butterfly spreads Butterfly = long call at K + short two calls at K 2 + long one call at K 3 Buy low and high, sell intermediate strike Bet on stock price staying put Small outlay required Profit Profit K K 2 K 3 S T K K 2 K 3 S T Figure.3: Butterfly spread using calls (left) and puts (right) Payoffs in each interval Take K 2 K K 3 2 Table.8. Payoff from a butterfly spread Interval Long call Long call 2 Short calls Total S T K K S T K 2 S T K 0 0 S T K K 2 S T K 3 S T K 0 2S T K 2 K 3 S T K 3 S T S T K S T K 3 2S T K 2 0 Calendar spreads

36 36 Ian Buckley Combinations Both calls and puts on same stock Types of combinations Straddle Strips and straps Strangles Straddle Investor expects move, but unsure of direction Also bottom straddle, or straddle purchase Cf. top straddle, or straddle write is reverse Profit K S T Figure.5: Straddle Payoffs in each interval Table.9. Payoff from a straddle Interval Call Put Total S T K 0 KS T KS T KS T S T K 0 S T K Strips and straps Strip long one call and two puts; bullish, but more bearish Strap long two calls and one put; bearish, but more bullish

37 CMFM03 Financial Markets 37 Profit Profit K S T K S T Strip Strap Figure.6: Strip (left) and strap (right) Strangles Also bottom vertical combination Buy call and put with different strikes Bet on move, unsure of direction cf. straddle Distance between strikes increases downside risk distance stock moves until profit Cf. top vertical combination is sale of strangle, has unlimited loss Profit K K 2 S T Figure.7: Strangle Payoffs in each interval Table.20. Payoff from a strangle Interval Call Put Total S T K 0 K S T K S T K S T K K 2 S T S T K 2 0 S T K 2