Gallery of equations. 1. Introduction
|
|
- Herbert Richardson
- 5 years ago
- Views:
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
10 Trading strategies involving options
10 Trading strategies involving options It will not do to leave a live dragon out of your plans if you live near one. J.R.R. Tolkien Overview Strategies involving a single option and a stock Spreads 2
More informationFinancial Markets and Products
Financial Markets and Products 1. Which of the following types of traders never take position in the derivative instruments? a) Speculators b) Hedgers c) Arbitrageurs d) None of the above 2. Which of the
More informationFinancial Markets and Products
Financial Markets and Products 1. Eric sold a call option on a stock trading at $40 and having a strike of $35 for $7. What is the profit of the Eric from the transaction if at expiry the stock is trading
More informationFinancial Markets & Risk
Financial Markets & Risk Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA259 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com Session 3 Derivatives Binomial
More informationAnswers to Selected Problems
Answers to Selected Problems Problem 1.11. he farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offset the loss on the sale
More informationSwaps 7.1 MECHANICS OF INTEREST RATE SWAPS LIBOR
7C H A P T E R Swaps The first swap contracts were negotiated in the early 1980s. Since then the market has seen phenomenal growth. Swaps now occupy a position of central importance in derivatives markets.
More informationChapter 1 Introduction. Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull
Chapter 1 Introduction 1 What is a Derivative? A derivative is an instrument whose value depends on, or is derived from, the value of another asset. Examples: futures, forwards, swaps, options, exotics
More informationDerivatives Analysis & Valuation (Futures)
6.1 Derivatives Analysis & Valuation (Futures) LOS 1 : Introduction Study Session 6 Define Forward Contract, Future Contract. Forward Contract, In Forward Contract one party agrees to buy, and the counterparty
More informationAnswers to Selected Problems
Answers to Selected Problems Problem 1.11. he farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offset the loss on the sale
More informationDerivative Instruments
Derivative Instruments Paris Dauphine University - Master I.E.F. (272) Autumn 2016 Jérôme MATHIS jerome.mathis@dauphine.fr (object: IEF272) http://jerome.mathis.free.fr/ief272 Slides on book: John C. Hull,
More informationDerivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage.
Derivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage. Question 2 What is the difference between entering into a long forward contract when the forward
More informationFNCE4830 Investment Banking Seminar
FNCE4830 Investment Banking Seminar Introduction on Derivatives What is a Derivative? A derivative is an instrument whose value depends on, or is derived from, the value of another asset. Examples: Futures
More informationMathematics of Financial Derivatives
Mathematics of Financial Derivatives Lecture 11 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics Table of contents 1. Mechanics of interest rate swaps (continued)
More informationMeasuring Interest Rates. Interest Rates Chapter 4. Continuous Compounding (Page 77) Types of Rates
Interest Rates Chapter 4 Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement The difference between quarterly and annual compounding is analogous to
More informationForwards, Futures, Options and Swaps
Forwards, Futures, Options and Swaps A derivative asset is any asset whose payoff, price or value depends on the payoff, price or value of another asset. The underlying or primitive asset may be almost
More informationFNCE4830 Investment Banking Seminar
FNCE4830 Investment Banking Seminar Introduction on Derivatives What is a Derivative? A derivative is an instrument whose value depends on, or is derived from, the value of another asset. Examples: Futures
More informationLecture 7: Trading Strategies Involve Options ( ) 11.2 Strategies Involving A Single Option and A Stock
11.2 Strategies Involving A Single Option and A Stock In Figure 11.1a, the portfolio consists of a long position in a stock plus a short position in a European call option à writing a covered call o The
More informationFinancial Derivatives Section 1
Financial Derivatives Section 1 Forwards & Futures Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of Piraeus)
More informationInterest Rate Markets
Interest Rate Markets 5. Chapter 5 5. Types of Rates Treasury rates LIBOR rates Repo rates 5.3 Zero Rates A zero rate (or spot rate) for maturity T is the rate of interest earned on an investment with
More informationFINM2002 NOTES INTRODUCTION FUTURES'AND'FORWARDS'PAYOFFS' FORWARDS'VS.'FUTURES'
FINM2002 NOTES INTRODUCTION Uses of derivatives: o Hedge risks o Speculate! Take a view on the future direction of the market o Lock in an arbitrage profit o Change the nature of a liability Eg. swap o
More information2. Futures and Forward Markets 2.1. Institutions
2. Futures and Forward Markets 2.1. Institutions 1. (Hull 2.3) Suppose that you enter into a short futures contract to sell July silver for $5.20 per ounce on the New York Commodity Exchange. The size
More informationMathematics of Financial Derivatives. Zero-coupon rates and bond pricing. Lecture 9. Zero-coupons. Notes. Notes
Mathematics of Financial Derivatives Lecture 9 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics Zero-coupon rates and bond pricing Zero-coupons Definition:
More informationMathematics of Financial Derivatives
Mathematics of Financial Derivatives Lecture 9 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics Table of contents 1. Zero-coupon rates and bond pricing 2.
More informationFinancial Derivatives Section 3
Financial Derivatives Section 3 Introduction to Option Pricing Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un.
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationDerivative Instruments
Derivative Instruments Paris Dauphine University - Master I.E.F. (272) Autumn 2016 Jérôme MATHIS jerome.mathis@dauphine.fr (object: IEF272) http://jerome.mathis.free.fr/ief272 Slides on book: John C. Hull,
More informationCHAPTER 1 Introduction to Derivative Instruments
CHAPTER 1 Introduction to Derivative Instruments In the past decades, we have witnessed the revolution in the trading of financial derivative securities in financial markets around the world. A derivative
More informationBuilding a Zero Coupon Yield Curve
Building a Zero Coupon Yield Curve Clive Bastow, CFA, CAIA ABSTRACT Create and use a zero- coupon yield curve from quoted LIBOR, Eurodollar Futures, PAR Swap and OIS rates. www.elpitcafinancial.com Risk-
More informationAFM 371 Winter 2008 Chapter 26 - Derivatives and Hedging Risk Part 2 - Interest Rate Risk Management ( )
AFM 371 Winter 2008 Chapter 26 - Derivatives and Hedging Risk Part 2 - Interest Rate Risk Management (26.4-26.7) 1 / 30 Outline Term Structure Forward Contracts on Bonds Interest Rate Futures Contracts
More informationFixed-Income Analysis. Assignment 5
FIN 684 Professor Robert B.H. Hauswald Fixed-Income Analysis Kogod School of Business, AU Assignment 5 Please be reminded that you are expected to use contemporary computer software to solve the following
More informationLecture 9. Basics on Swaps
Lecture 9 Basics on Swaps Agenda: 1. Introduction to Swaps ~ Definition: ~ Basic functions ~ Comparative advantage: 2. Swap quotes and LIBOR zero rate ~ Interest rate swap is combination of two bonds:
More informationAppendix A Financial Calculations
Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY
More informationOptions Markets: Introduction
17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value
More informationP1.T3. Financial Markets & Products. Hull, Options, Futures & Other Derivatives. Trading Strategies Involving Options
P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Trading Strategies Involving Options Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Trading Strategies Involving
More informationSection 1: Advanced Derivatives
Section 1: Advanced Derivatives Options, Futures, and Other Derivatives (6th edition) by Hull Chapter Mechanics of Futures Markets (Sections.7-.10 only) 3 Chapter 5 Determination of Forward and Futures
More informationDerivative Instruments
Derivative Instruments Paris Dauphine University - Master I.E.F. (272) Autumn 2016 Jérôme MATHIS jerome.mathis@dauphine.fr (object: IEF272) http://jerome.mathis.free.fr/ief272 Slides on book: John C. Hull,
More informationPricing Options with Mathematical Models
Pricing Options with Mathematical Models 1. OVERVIEW Some of the content of these slides is based on material from the book Introduction to the Economics and Mathematics of Financial Markets by Jaksa Cvitanic
More informationForwards and Futures
Options, Futures and Structured Products Jos van Bommel Aalto Period 5 2017 Class 7b Course summary Forwards and Futures Forward contracts, and forward prices, quoted OTC. Futures: a standardized forward
More informationINV2601 DISCUSSION CLASS SEMESTER 2 INVESTMENTS: AN INTRODUCTION INV2601 DEPARTMENT OF FINANCE, RISK MANAGEMENT AND BANKING
INV2601 DISCUSSION CLASS SEMESTER 2 INVESTMENTS: AN INTRODUCTION INV2601 DEPARTMENT OF FINANCE, RISK MANAGEMENT AND BANKING Examination Duration of exam 2 hours. 40 multiple choice questions. Total marks
More informationUNIVERSITY OF SOUTH AFRICA
UNIVERSITY OF SOUTH AFRICA Vision Towards the African university in the service of humanity College of Economic and Management Sciences Department of Finance & Risk Management & Banking General information
More informationMathematics of Financial Derivatives
Mathematics of Financial Derivatives Lecture 8 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics Table of contents 1. The Greek letters (continued) 2. Volatility
More informationINTEREST RATE FORWARDS AND FUTURES
INTEREST RATE FORWARDS AND FUTURES FORWARD RATES The forward rate is the future zero rate implied by today s term structure of interest rates BAHATTIN BUYUKSAHIN, CELSO BRUNETTI 1 0 /4/2009 2 IMPLIED FORWARD
More informationGlobal Securities & Investment Management Target Audience: Objectives:
Global Securities & Investment Management Target Audience: This course is focused at those who are seeking to acquire an overview of Finance, more specifically a foundation in capital markets, products,
More informationFutures and Forward Contracts
Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 Forward contracts Forward contracts and their payoffs Valuing forward contracts 2 Futures contracts Futures contracts and their prices
More informationB. Combinations. 1. Synthetic Call (Put-Call Parity). 2. Writing a Covered Call. 3. Straddle, Strangle. 4. Spreads (Bull, Bear, Butterfly).
1 EG, Ch. 22; Options I. Overview. A. Definitions. 1. Option - contract in entitling holder to buy/sell a certain asset at or before a certain time at a specified price. Gives holder the right, but not
More informationMarket, exchange over the counter, standardised ( amt, maturity), OTC private, specifically tailored)
Lecture 1 Page 1 Lecture 2 Page 5 Lecture 3 Page 10 Lecture 4 Page 15 Lecture 5 Page 22 Lecture 6 Page 26 Lecture 7 Page 29 Lecture 8 Page 30 Lecture 9 Page 36 Lecture 10 Page 40 #1 - DS FUNDAMENTALS (
More informationDerivatives: part I 1
Derivatives: part I 1 Derivatives Derivatives are financial products whose value depends on the value of underlying variables. The main use of derivatives is to reduce risk for one party. Thediverse range
More informationGlossary of Swap Terminology
Glossary of Swap Terminology Arbitrage: The opportunity to exploit price differentials on tv~otherwise identical sets of cash flows. In arbitrage-free financial markets, any two transactions with the same
More informationVanilla interest rate options
Vanilla interest rate options Marco Marchioro derivati2@marchioro.org October 26, 2011 Vanilla interest rate options 1 Summary Probability evolution at information arrival Brownian motion and option pricing
More informationSWAPS. Types and Valuation SWAPS
SWAPS Types and Valuation SWAPS Definition A swap is a contract between two parties to deliver one sum of money against another sum of money at periodic intervals. Obviously, the sums exchanged should
More informationEurocurrency Contracts. Eurocurrency Futures
Eurocurrency Contracts Futures Contracts, FRAs, & Options Eurocurrency Futures Eurocurrency time deposit Euro-zzz: The currency of denomination of the zzz instrument is not the official currency of the
More informationFIN 684 Fixed-Income Analysis Swaps
FIN 684 Fixed-Income Analysis Swaps Professor Robert B.H. Hauswald Kogod School of Business, AU Swap Fundamentals In a swap, two counterparties agree to a contractual arrangement wherein they agree to
More informationMBF1243 Derivatives. L7: Swaps
MBF1243 Derivatives L7: Swaps Nature of Swaps A swap is an agreement to exchange of payments at specified future times according to certain specified rules The agreement defines the dates when the cash
More informationSwaps. Chapter 6. Nature of Swaps. Uses of Swaps: Transforming a Liability (Figure 6.2, page 136) Typical Uses of an Interest Rate Swap
6.1 6.2 Swaps Chapter 6 Nature of Swaps A swap is an agreement to exchange cash flows at specified future times according to specified rules Example: A Plain Vanilla Interest Rate Swap The agreement on
More informationProfit settlement End of contract Daily Option writer collects premium on T+1
DERIVATIVES A derivative contract is a financial instrument whose payoff structure is derived from the value of the underlying asset. A forward contract is an agreement entered today under which one party
More informationFixed-Income Analysis. Solutions 5
FIN 684 Professor Robert B.H. Hauswald Fixed-Income Analysis Kogod School of Business, AU Solutions 5 1. Forward Rate Curve. (a) Discount factors and discount yield curve: in fact, P t = 100 1 = 100 =
More informationCHAPTER 14 SWAPS. To examine the reasons for undertaking plain vanilla, interest rate and currency swaps.
1 LEARNING OBJECTIVES CHAPTER 14 SWAPS To examine the reasons for undertaking plain vanilla, interest rate and currency swaps. To demonstrate the principle of comparative advantage as the source of the
More informationChapter 2: BASICS OF FIXED INCOME SECURITIES
Chapter 2: BASICS OF FIXED INCOME SECURITIES 2.1 DISCOUNT FACTORS 2.1.1 Discount Factors across Maturities 2.1.2 Discount Factors over Time 2.1 DISCOUNT FACTORS The discount factor between two dates, t
More informationUNIVERSITY OF AGDER EXAM. Faculty of Economicsand Social Sciences. Exam code: Exam name: Date: Time: Number of pages: Number of problems: Enclosure:
UNIVERSITY OF AGDER Faculty of Economicsand Social Sciences Exam code: Exam name: Date: Time: Number of pages: Number of problems: Enclosure: Exam aids: Comments: EXAM BE-411, ORDINARY EXAM Derivatives
More informationFair Forward Price Interest Rate Parity Interest Rate Derivatives Interest Rate Swap Cross-Currency IRS. Net Present Value.
Net Present Value Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 688 0364 : LKCSB 5036 September 16, 016 Christopher Ting QF 101 Week 5 September
More informationMATH4210 Financial Mathematics ( ) Tutorial 6
MATH4210 Financial Mathematics (2015-2016) Tutorial 6 Enter the market with different strategies Strategies Involving a Single Option and a Stock Covered call Protective put Π(t) S(t) c(t) S(t) + p(t)
More informationSAMPLE SOLUTIONS FOR DERIVATIVES MARKETS
SAMPLE SOLUTIONS FOR DERIVATIVES MARKETS Question #1 If the call is at-the-money, the put option with the same cost will have a higher strike price. A purchased collar requires that the put have a lower
More informationAmortizing and Accreting Floors Vaulation
Amortizing and Accreting Floors Vaulation Alan White FinPricing http://www.finpricing.com Summary Interest Rate Amortizing and Accreting Floor Introduction The Benefits of an amortizing and accreting floor
More informationCIS March 2012 Diet. Examination Paper 2.3: Derivatives Valuation Analysis Portfolio Management Commodity Trading and Futures.
CIS March 2012 Diet Examination Paper 2.3: Derivatives Valuation Analysis Portfolio Management Commodity Trading and Futures Level 2 Derivative Valuation and Analysis (1 12) 1. A CIS student was making
More informationFinancial Market Introduction
Financial Market Introduction Alex Yang FinPricing http://www.finpricing.com Summary Financial Market Definition Financial Return Price Determination No Arbitrage and Risk Neutral Measure Fixed Income
More informationB6302 Sample Placement Exam Academic Year
Revised June 011 B630 Sample Placement Exam Academic Year 011-01 Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund
More informationPart III: Swaps. Futures, Swaps & Other Derivatives. Swaps. Previous lecture set: This lecture set -- Parts II & III. Fundamentals
Futures, Swaps & Other Derivatives Previous lecture set: Interest-Rate Derivatives FRAs T-bills futures & Euro$ Futures This lecture set -- Parts II & III Swaps Part III: Swaps Swaps Fundamentals what,
More informationMAFS601A Exotic swaps. Forward rate agreements and interest rate swaps. Asset swaps. Total return swaps. Swaptions. Credit default swaps
MAFS601A Exotic swaps Forward rate agreements and interest rate swaps Asset swaps Total return swaps Swaptions Credit default swaps Differential swaps Constant maturity swaps 1 Forward rate agreement (FRA)
More informationEXAMINATION II: Fixed Income Valuation and Analysis. Derivatives Valuation and Analysis. Portfolio Management
EXAMINATION II: Fixed Income Valuation and Analysis Derivatives Valuation and Analysis Portfolio Management Questions Final Examination March 2016 Question 1: Fixed Income Valuation and Analysis / Fixed
More informationIntroduction to FRONT ARENA. Instruments
Introduction to FRONT ARENA. Instruments Responsible teacher: Anatoliy Malyarenko August 30, 2004 Contents of the lecture. FRONT ARENA architecture. The PRIME Session Manager. Instruments. Valuation: background.
More informationCurrency Option or FX Option Introduction and Pricing Guide
or FX Option Introduction and Pricing Guide Michael Taylor FinPricing A currency option or FX option is a contract that gives the buyer the right, but not the obligation, to buy or sell a certain currency
More informationS 0 C (30, 0.5) + P (30, 0.5) e rt 30 = PV (dividends) PV (dividends) = = $0.944.
Chapter 9 Parity and Other Option Relationships Question 9.1 This problem requires the application of put-call-parity. We have: Question 9.2 P (35, 0.5) = C (35, 0.5) e δt S 0 + e rt 35 P (35, 0.5) = $2.27
More informationMATH 425 EXERCISES G. BERKOLAIKO
MATH 425 EXERCISES G. BERKOLAIKO 1. Definitions and basic properties of options and other derivatives 1.1. Summary. Definition of European call and put options, American call and put option, forward (futures)
More informationOptions Trading Strategies
Options Trading Strategies Liuren Wu Zicklin School of Business, Baruch College Fall, 27 (Hull chapter: 1) Liuren Wu Options Trading Strategies Option Pricing, Fall, 27 1 / 18 Types of strategies Take
More informationFIXED INCOME I EXERCISES
FIXED INCOME I EXERCISES This version: 25.09.2011 Interplay between macro and financial variables 1. Read the paper: The Bond Yield Conundrum from a Macro-Finance Perspective, Glenn D. Rudebusch, Eric
More informationACI THE FINANCIAL MARKETS ASSOCIATION
ACI THE FINANCIAL MARKETS ASSOCIATION EXAMINATION FORMULAE page number INTEREST RATE..2 MONEY MARKET..... 3 FORWARD-FORWARDS & FORWARD RATE AGREEMENTS..4 FIXED INCOME.....5 FOREIGN EXCHANGE 7 OPTIONS 8
More informationAn Introduction to Derivatives and Risk Management, 7 th edition Don M. Chance and Robert Brooks. Table of Contents
An Introduction to Derivatives and Risk Management, 7 th edition Don M. Chance and Robert Brooks Table of Contents Preface Chapter 1 Introduction Derivative Markets and Instruments Options Forward Contracts
More informationFutures and Forward Markets
Futures and Forward Markets (Text reference: Chapters 19, 21.4) background hedging and speculation optimal hedge ratio forward and futures prices futures prices and expected spot prices stock index futures
More informationCreating Forward-Starting Swaps with DSFs
INTEREST RATES Creating -Starting Swaps with s JULY 23, 2013 John W. Labuszewski Managing Director Research & Product Development 312-466-7469 jlab@cmegroup.com CME Group introduced its Deliverable Swap
More informationHull, Options, Futures & Other Derivatives Exotic Options
P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Exotic Options Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Exotic Options Define and contrast exotic derivatives
More informationINV2601 SELF ASSESSMENT QUESTIONS
INV2601 SELF ASSESSMENT QUESTIONS 1. The annual holding period return of an investment that was held for four years is 5.74%. The ending value of this investment was R1 000. Calculate the beginning value
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Financial Economics
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Financial Economics June 2014 changes Questions 1-30 are from the prior version of this document. They have been edited to conform
More informationBourse de Montréal Inc. Reference Manual. Ten-year. Option on. Ten-year. Government. Government. of Canada. of Canada. Bond Futures.
CGB Ten-year Government of Canada Bond Futures OGB Option on Ten-year Government of Canada Bond Futures Reference Manual Bourse de Montréal Inc. www.boursedemontreal.com Bourse de Montréal Inc. Sales and
More informationNATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION Investment Instruments: Theory and Computation
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS SEMESTER 2 EXAMINATION 2012-2013 Investment Instruments: Theory and Computation April/May 2013 Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES
More informationFinancial Mathematics Principles
1 Financial Mathematics Principles 1.1 Financial Derivatives and Derivatives Markets A financial derivative is a special type of financial contract whose value and payouts depend on the performance of
More informationDepartment of Mathematics. Mathematics of Financial Derivatives
Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, 2009 2pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5 1. (a) Suppose 0 < E 1 < E 3 and E 2
More informationFinancial Derivatives
Derivatives in ALM Financial Derivatives Swaps Hedge Contracts Forward Rate Agreements Futures Options Caps, Floors and Collars Swaps Agreement between two counterparties to exchange the cash flows. Cash
More informationForward Rate Agreement (FRA) Product and Valuation
Forward Rate Agreement (FRA) Product and Valuation Alan White FinPricing http://www.finpricing.com Summary Forward Rate Agreement (FRA) Introduction The Use of FRA FRA Payoff Valuation Practical Guide
More informationderivatives Derivatives Basics
Basis = Current Cash Price - Futures Price Spot-Future Parity: F 0,t = S 0 (1+C) Futures - Futures Parity: F 0,d = F 0,t (1+C) Implied Repo Rate: C = (F 0,t / S 0 ) - 1 Futures Pricing for Stock Indices:
More informationPortfolio Management
Portfolio Management 010-011 1. Consider the following prices (calculated under the assumption of absence of arbitrage) corresponding to three sets of options on the Dow Jones index. Each point of the
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More information= e S u S(0) From the other component of the call s replicating portfolio, we get. = e 0.015
Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin In-Term Exam II Extra problems Instructor: Milica Čudina Notes: This is a closed book and closed notes exam.
More informationFINA 1082 Financial Management
FINA 1082 Financial Management Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA257 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com 1 Lecture 13 Derivatives
More information22 Swaps: Applications. Answers to Questions and Problems
22 Swaps: Applications Answers to Questions and Problems 1. At present, you observe the following rates: FRA 0,1 5.25 percent and FRA 1,2 5.70 percent, where the subscripts refer to years. You also observe
More informationInterest Rate Forwards and Swaps
Interest Rate Forwards and Swaps 1 Outline PART ONE Chapter 1: interest rate forward contracts and their pricing and mechanics 2 Outline PART TWO Chapter 2: basic and customized swaps and their pricing
More informationRisk Management and Hedging Strategies. CFO BestPractice Conference September 13, 2011
Risk Management and Hedging Strategies CFO BestPractice Conference September 13, 2011 Introduction Why is Risk Management Important? (FX) Clients seek to maximise income and minimise costs. Reducing foreign
More informationMBF1243 Derivatives Prepared by Dr Khairul Anuar
MBF1243 Derivatives Prepared by Dr Khairul Anuar L3 Determination of Forward and Futures Prices www.mba638.wordpress.com Consumption vs Investment Assets When considering forward and futures contracts,
More informationCallable Bonds & Swaptions
Callable Bonds & Swaptions 1 Outline PART ONE Chapter 1: callable debt securities generally; intuitive approach to pricing embedded call Chapter 2: payer and receiver swaptions; intuitive pricing approach
More informationDERIVATIVES AND RISK MANAGEMENT
A IS 1! foi- 331 DERIVATIVES AND RISK MANAGEMENT RAJIV SRIVASTAVA Professor Indian Institute of Foreign Trade New Delhi QXJFORD UNIVERSITY PRKSS CONTENTS Foreword Preface 1. Derivatives An Introduction
More informationThe Convexity Bias in Eurodollar Futures
SEPTEMBER 16, 1994 The Convexity Bias in Eurodollar Futures research note note Research Department 150 S. WACKER DRIVE 15TH FLOOR CHICAGO, IL 60606 (312) 984-4345 CHICAGO Global Headquarters (312) 441-4200
More information