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- 1. UNIVERSITY OF SOUTH AFRICA
- 2. College of Economic andManagement SciencesDepartment of Finance & Risk Management & Banking
- 3. INV3703: DerivativesPrimary Lecturer:Coert Erasmuserasmcf@unisa.ac.zaAJH 5-100012 429 4949
- 4. INV3703: DerivativesSecondary Lecturer:Godfrey Marozva, CFAmarozg@unisa.ac.zaAJH 5-99012 429 4977
- 5. General information• Exam is 2 hours• Formula sheet will be provided.• Formulate the LOS into questions to testyour knowledge of the Subject Unit• Examination includes both theory andcalculations• Mark composition:Questions PercentagesMCQ 15 x 2 marks=30 60Long questions 2 x 10 marks=20 40Total 50 marks 100%
- 6. ReferencesThese slides were created byMs E Botha andMr G MarozvaBoth are employed at Unisa and areacademic staff within the DepartmentFinance, Risk management andBanking
- 7. References (Continued)The prescribed textbook for this moduleis:Analysis of Derivatives for the CFAprogram ~ By Don Chance (2003)The prescribed textbook was used tocreate the following slides.
- 8. INV3703INVESTMENTS: DERIVATIVESCHAPTER 1FORWARD MARKETS ANDCONTRACTS
- 9. • Read• Basic concepts and terminologyWhat is a:• Forward contract• Futures contract• Swap• Option– Call option– Put optionForwardcommitmentHolder has a rightbut does not imposean obligation
- 10. Chapter 1 – Introduction cont…Forward Contracts Contingent ClaimNo premium paid at Premium Paid atinception inceptionQuestion : What is the advantage of a contingentclaims over forward commitments ?Answer: Permit gain while protectingagainst losses……Why is it so?
- 11. Chapter 1 – Introduction cont…If a risk free rate of interest is 7% and aninvestor enters into a transaction that hasno risk, what would be the rate of returnthe investor should earn in the absence ofthe riskA. 0%B. between 0% and 7%C. 7%D. Less than 7%
- 12. Chapter 1 – Introduction cont…The spot price of Gold is R930 per ounceand the risk free-rate of interest is 5% perannum. Calculate the equilibrium 6-monthforward price per ounce of gold.930×(1+(0.05/2))=R953.25Why divide by 2… (6-months i.e. half a year)
- 13. INV3703INVESTMENTS: DERIVATIVESCHAPTER 2FORWARD MARKETS ANDCONTRACTS
- 14. DefinitionA forward contract is anagreement between two parties inwhich one party, the buyer, agrees tobuy from the other party, the seller,an underlying asset at a future date ata price established today. The contractis customised and each party issubject to the possibility that the otherparty will default.
- 15. ForwardsEquity forwardsBond/Fixed-income forwardsInterest rate forwards (FRAs)Currency forwards
- 16. Forwards FuturesOver the counter Futures exchangePrivate PublicCustomized StandardizedDefault risk Default freeNot marked to market Marked to marketHeld until expiration Offset possibleNot liquid LiquidUnregulated Regulated
- 17. Differentiate between the positions held by the longand short parties to a forward contractLF (Long Forward) LALong party SA Short party SF• Party that agrees to buy the asset has a long forward position• Party that agrees to sell the asset has a short forwardposition
- 18. Pricing and valuation of forward contractsAre pricing and valuation not the same thing?• The price is agreed on the initiationdate (Forward price or forward rate)i.e. pricing means to determining theforward price or forward rate.• Valuation, however, means todetermine the amount of money thatone would need to pay or wouldexpect to receive to engage in thetransaction
- 19. Pricing and valuation of forward contracts cont…F(0,T)- The forward contract price initiated attime 0 and expiring at time TVo(0,T) – the value of a forward contractinitiated at time 0 and expiring at time TVt(0,T) – the value of a forward contractat the point in time during the life of acontract such as tVT(0,T)- Value at expiration
- 20. Pricing and valuation of forward contracts cont…( )= +TT 0F S 1 r
- 21. If Vo ≠ 0 arbitrage would the prevailPricing and valuation of forward contracts cont…The forward price that eliminates arbitrage:( )= −+T0 0 TFV S1 r( )= +TT 0F S 1 r
- 22. By definition an asset’svalue is the present valueof future cash flows thus,Pricing and valuation of forward contracts cont…( )−= −+Tt t T tFV S1 r0SotStTFTRemaining period (T-t)
- 23. When dividends are paid continuouslyPricing and valuation of forward contracts cont…( ) ( )−+= − += ∑ T tiTT 0Di(1 r)F S PV(D) 1 rPV(D)−∂= ×= +l lcr ttT 0cF Sr Ln(1 r)
- 24. FRAs0Todaygm-day Eurodollar deposith(Expiration)h + mh - is the day on which the FRA expiresm – is the number of days on the particularEurodollar deposit i.e. The underlying asseth + m is the number of days until the maturitydate of the Eurodollar instrument on which theFRA rate is basedg – is the date during the life of the FRA at whichwe want to determine the value of the FRA
- 25. FRAs Pricing and Valuation( )( )+++ −− += − + += −+ + −+ −h m0(h m) 360 3600,h,m mh0(h) 360m0,h,m 360g h m gh gg 360g 3601 L ( )FRA 11 L ( )1 FRA ( )1V (0,h,m)1 L (h m g)( )1 L (h g)
- 26. Calculate and interpret the payment atexpiration of a FRA and identify each ofthe component terms
- 27. • ESKOM P/L is expecting to receive a cash inflow ofR20, 000,000.00 in 90 days. Short term interestrates are expected to fall during the next 90 days. Inorder to hedge against this risk, the company decidesto use an FRA that expires in 90 days and is basedon 90day LIBOR. The FRA is quoted at 6%. Atexpiration LIBOR is 5%. Indicate whether thecompany should take a long or short position tohedge interest rate risk. Using the appropriateterminology, identify the type of FRA used here.Calculate the gain or loss to ESKOM P/L as aconsequence of entering the FRA.
- 28. • Identify the characteristics ofcurrency forwards• Exchange of currencies• Exchange rate specified• Manage foreign exchange risk– Domestic risk-free rate– Foreign risk free rate– Interest rate parity (IRP)– Covered interest arbitrage
- 29. Determine the price of a forwardcontract• Initial or delivery priceFT =S0(1+r)^T =K• Forward price during periodFT =St(1+r)^(T-t)
- 30. Determine the value of a forwardcontract at initiation, during thelife of the contract, and atexpirationalternatively
- 31. Calculate the price and value of aforward contract on a currency• Price – currency forwardDiscrete interestContinuous interest
- 32. Value – currency forwardDiscrete interest
- 33. • Covered interest arbitrage= ++ SFrfrdTT)1()1(
- 34. INV3703INVESTMENTS: DERIVATIVESCHAPTER 3FUTURES MARKETS AND CONTRACTS
- 35. DefinitionA forward contract is an agreement between twoparties in which one party, the buyer, agrees to buyfrom the other party, the seller, an underlying asset at afuture date at a price established today. The contract iscustomized and each party is subject to the possibilitythat the other party will default.A futures contract is a variation of a forwardcontract that has essentially the same basic definition,but some clearly distinguishable additional features,the most important being that it is not a private andcustomized transaction. It is a public, standardizedtransaction that takes place on a futures exchange.
- 36. Forwards FuturesOver the counter Futures exchangePrivate PublicCustomized StandardizedDefault risk Default freeNot marked to market Marked to marketHeld until expiration Offset possibleNot liquid LiquidUnregulated RegulatedIdentify the primary characteristics of futurescontracts and distinguish between futuresand forwards
- 37. Describe how a futures contract can beterminated at or prior to expiration byclose out, delivery, equivalent cashsettlement, orexchange for physicalsClose out (prior to expiration)– Opposite (offsetting) transactionDelivery– Close out before expiration or take delivery– Short delivers underlying to long (certain date andlocation)Cash settlement– No need to close out (leave position open)– Marked to market (final gain/loss)Exchange for physicals– Counterparties arrange alternative delivery
- 38. Stock index futuresUnderlying – individual share or share indexS&P500 Stock Index futureFutures price quoted in same way as index levelContract price = futures price x multiplier ($250)Cash settledSouth AfricaALSI, INDI, FINI etc. (R10 x Index level)ALSI contract – (10 x 29150) R291,500 per one futuresALMI (Index level ÷ 10)Value of one traded ALMI contract is 29150 ÷ 10 =R2,915
- 39. Explain why the futures price must converge tothe spot price at expirationTo prevent arbitrage: ft(T) = STSpot price (S) current price for immediatedeliveryFutures price (f) current price for future deliveryAt expiration f becomes the current price forimmediate delivery (S)If : fT (T) < STBuy contract; take delivery of underlying and pay lowerfutures priceIf : fT (T) > STSell contract; buy underlying, deliver and receivehigher futures price
- 40. Determine the value of a futurescontract• Value before m-t-m = gain/loss accumulated since last m-t-m• Gain/loss captured through the m-t-m process• Contract re-priced at current market price and value = zerott-1 T0 j
- 41. Summary: Value of a futures contract• Like forwards, futures have no value atinitiation• Unlike forwards, futures do not accumulateany value• Value always zero after adjusting for day’sgain or loss (m-t-m)• Value different from zero only during m-t-mintervals• Futures value = current price – price at lastm-t-m time• Futures price increase -> value of longposition increases• Value set back to zero by the end-of-daymark to market
- 42. Contrast forward and futures prices• Futures – settle daily and essentially free of default risk• Forwards – settle at expiration and subject to default riskInterest rates positively correlated with futures prices• Long positions prefer futures to forwards – futures priceshigher• Gains generated when rates increase – invest at higher rates• Incur losses when rates decrease – cover losses at lower rates• For example: Gold futuresInterest rates negatively correlated with futures prices• Prefer not to mark to market – forward prices higher• For example: Interest rate futures• Correlation low/close to zero – difference between F and fsmall• Assumption: Forward and futures prices are the same• Ignore effects of marking a futures contract to market
- 43. No-arbitrage futures pricesCash-and-carry arbitrage:Reverse C&C arbitrage:Today:Sell futures contractBorrow moneyBuy underlyingAt expiration:Collect loan plus interestPay futures price and takedeliveryAt expiration:Deliver asset and receivefutures priceRepay loan plus interestToday:•Buy futures contract•Sell/short underlying•Invest proceeds
- 44. Identify the different types of monetaryand non-monetary benefits and costsassociated with holding the underlying,and how they affect the futures priceCosts associated with storing or holding the underlying▪ Increase the no-arbitrage futures priceFinancial assets▪ No storage costs▪ Monetary benefit to holding asset− Earn dividends, coupons or interest− Decrease the no-arbitrage futures price▪ Non-monetary benefit to holding asset− Asset in short supply− Convenience yield (decrease price)
- 45. • No cost or benefit to holding the asset• Net cost or benefit to holding an asset+ FV CB• Cost-of-carry model▪ CB -> cost minus benefit (negative or positive value)▪ Costs exceed benefits (net cost) – future value added▪ Benefits exceed costs (net benefit) – FV subtracted• Financial assets▪ High(er) cash flows -> lower futures
- 46. • Forward price• Futures price
- 47. Contrast backwardation andcontango• Backwardation– Futures price below the spot price– Significant benefit to holding asset– Net benefit (negative cost of carry)•Contango– Futures price above spot price– Little/no benefit to holding asset– Net cost (positive cost of carry)
- 48. Example:Calculate the no-arbitrage futures price of a1.2 year futures contract on a 7% T-bondwith exactly 10 years to maturity and a priceof $1,040. The annual risk-free rate is 5%.Assume the cheapest to deliver bond has aconversion factor of 1.13.Treasury bond futures
- 49. Answer:The semi-annual coupon is $35. Abondholder will receive two couponsduring the contract term – i.e., apayment 0.5 years and 1 year from now.
- 50. Stock futuresExample:Calculate the no-arbitrage price for a 120-day future on a stock currently priced at $30and expected to pay a $0.40 dividend in 15days and in 105 days. The annual risk-freerate is 5%.
- 51. Stock index futuresExample:The current level of the Nasdaq Index is1,780. The continuous dividend is 1.1%and the continuously compounded risk-free rate is 3.7%. Calculate the no-arbitrage futures price of an 87-dayfutures contract on this index.
- 52. Currency futuresExample:The risk-free rates are 5% in U.S. Dollars($) and 6.5% in British pounds (£). Thecurrent spot exchange rate is $1.7301/£.Calculate the no-arbitrage $ price of a 6-month futures contract.
- 53. Currency futuresAnswer:
- 54. INV3703INVESTMENTS: DERIVATIVESCHAPTER 4OPTION MARKETS AND CONTRACTS
- 55. • Identify the basic elements and characteristics of option contracts• Call options grant the holder (long position) theopportunity to buy the underlying security at a pricebelow the current market price, provided that themarket price exceeds the call strike before or atexpiration (specified contingency).• Put options grant the holder (long position) theopportunity to sell the underlying security at a priceabove the current market price, provided that the putstrike exceeds the market price before or at expiration(specified contingency).• The option seller (short position) in both instancesreceives a payment (premium) compellingperformance at the discretion of the holder.
- 56. Option shapesLong Call (LC)Short Call (SC)Long Put (LP)Short Put (SP)Right to buy Right to sellObligation to sell Obligation to buy57
- 57. Call option Put option+ At the money + At the moneyOut of money In the moneyIn the moneyOut of money0 0- Share price =X - Share price =X
- 58. Identify the different varieties of options in termsof the types of instruments underlying them• Financial optionsEquity options (individual or stock index), bondoptions, interest rate options, currency options• Options on futuresCall options – long position in futures uponexercisePut options – short position in futures uponexercise• Commodity optionsRight to either buy or sell a fixed quantity ofphysicalasset at a (fixed) strike price
- 59. JSE Equity Options: Why Trade Options?• Time to decide• Protect the value of your shares (put)• Leverage: small initial requirement can lead to high exposureand profit potential.• Option holder’s losses limited to premium• Take advantage of unique opportunities:• Profit in volatile markets (Straddle and Strangle)• Can be combined with shares already owned to protect the valueof your shares at very little cost (Fence)• Limiting both profit and loss (Collar)• Flexibility: Tailor-made to your needs
- 60. Notation and variablesVariable Notation State Call option valuePut optionvalueSpot price S Increase Increase DecreaseStrike price X Higher Lower HigherVolatility σ Higher Higher HigherTime tomaturityt Longer Higher UncertainInterest rates r Higher Higher LowerCall option C or c max(0; S – X)Put option P or p max(0; X – S)61
- 61. Explain putcall parity for European options and relate arbitrage and the construction of synthetic instrumentsFiduciary call – buying a call and investing PV(Payoff is X (otm) or X + (S-X) = S (itm)Protective put – buying a put and holding assetPayoff is S (otm) or (X-S) + S = X (itm)Therefore:When call is itm put is otm > payoff is S , -When call is otm, put is itm -> payoff is XS+ p = c+ PV(X) S =c +PV(X) -p P= c +PV(X)- S C=S+ p -PV(X) PV(X)= S+ p -c
- 62. Put call parity arbitrageS + p = c+ PV(X)c - p = S-PV(X)c - p >S-PV(X)Sell call; buy put; buy spot; borrow PV(X)c - p < S-PV(X)Buy call; sell put; sell spot; invest PV(X)
- 63. Determine the minimum and maximum values ofEuropean optionsThe lower bound for any option is zero (otm option)Upper bound Lower boundCall options c ≤ S c ≥ S – PV(X)Put options p ≤ PV(X) p ≥ PV(X) - S
- 64. Chapter 5 - Swaps• Equity SWAPs• Interest rate SWAPs• Equity and interest rate SWAPs65
- 65. Equity SwapsConsider an equity swap in which the assetmanager receives the return of the Russel2000 Index in return for paying the return onthe DJIA. At the inception of the equity swap,the Russel 2000 is at 520.12 and the DJIA is at9867.33. Calculate the market value of theswap a few months later when the Russel 2000is at 554.29 and the DJIA is at 9975.54. Thenotional principal of the swap is $15 million.= =554.29Russel 1.0657520.12= =9,975.54DJIA 1.01109,867.33( )= − =pay _DJIAV $15,000,000 1.0657 1.0110 $820,500
- 66. Interest rate swaps• Determining the swap rate = pricing ofswap• As rates change over time, the PV of floatingpayments will either exceed or be less thanthe PV of fixed payments– Difference = value of swap• Market value = difference between bonds– Fixed bond minus floating bond– Domestic bond minus foreign bond• PV (receive) minus PV (pay)
- 67. Interest rate swaps....Consider a two-year interest rate swap withsemi-annual payments. Assume a notionalprincipal of $50 million.Calculate the semi-annual fixed paymentand the annualized fixed rate on the swap ifthe current term structure of LIBOR interestrates is as follows:L0(180) = 0.0688L0(360) = 0.0700L0(540) = 0.0715L0(720) = 0.0723
- 68. ( )( )= =+01B 180 0.96671 0.0688 180 360( )( )= =+01B 360 0.93461 0.0700 360 360( )( )= =+01B 540 0.90311 0.0715 540 360( )( )= =+01B 720 0.87371 0.0723 720 360( )−= =+ + +1 0.8737FS 0,4,180 0.03430.9667 0.9346 0.9031 0.8737= × =Fixed payment 0.0343 $50,000,000 $1,715,000( )= =Annualized fixed rate 3.43% 360 180 6.86%
- 69. Calculate the market value of the swap 120days later from the point of view of the partypaying the floating rate and receiving thefixed rate, and from the point of view of theparty paying the fixed rate and receiving thefloating rate if the term structure 120 dayslater is as follows:• L120(60) = 0.0620• L120(240) = 0.0631• L120(420) = 0.0649• L120(600)= 0.0687
- 70. ( )( )= =+1201B 180 0.98981 0.0620 60 360( )( )= =+1201B 360 0.95961 0.0631 240 360( )( )= =+1201B 540 0.92961 0.0649 420 360( )( )= =+1201B 720 0.89731 0.0687 600 360( ) ( )= + + + +=Fixed 0.0343 0.9898 0.9596 0.9296 0.8973 1 0.89731.0268( ) ( ) = + = st1 Floating payment 1 0.0688 180 360 1.0344
- 71. Ans cont...= × =Float 1.0344 0.9898 1.0239( )= − =pay _ floatV $50,000,000 1.0268 1.0239 $145,000( )= − = −pay _ fixedV $50,000,000 1.0239 1.0268 $145,000Discounted with the 60 day present value factor of 0.9898:
- 72. Equity and interest rate swapAssume an asset manager enters into a one-year equity swap in which he will receive thereturn on the Nasdaq 100 Index in returnfor paying a floating interest rate. The swapcalls for quarterly payments. The Nasdaq100 is at 1651.72 at the beginning of theswap. Ninety days later, the rate L90(90) is0.0665. Calculate the market value of theswap 100 days from the beginning of theswap if the Nasdaq 100 is at 1695.27, thenotional principal of the swap is $50million, and the term structure is:L100(80) = 0.0654L100(170) = 0.0558L100(260) = 0.0507
- 73. ( )( )= =+1001B 80 0.98571 0.0654 80 360( ) ( ) = + = Next floating payment 1 0.0665 90 360 1.0168= × =Float 1.0168 0.9857 1.0023= =1,695.27Equity 1.02641,651.72( )= − =pay _ floatV $50,000,000 1.0264 1.0023 $1,205,000Discounted with the 80 day present value factor of 0.9857:
- 74. BEST OF LUCK IN YOUREXAMS

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