Derivatives CFA Level II PDF

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Derivatives CFA Level II...

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SS14: Derivatives Pricing & Valuation of Forward Commitments Long Forward Position  Party to the forward contract that agree to buy the asset Short Forward Position  Party to the forward that agree to sell / deliver the asset Forward Contract: - No money changes hands at inception (unlike futures contracts: initial deposit is made called the margin as a guarantee of performance) - The party to the forward contract with the Negative value will owe money to the other side - Value is 0 at inception. Price that makes the contract a 0-value investment at BGN The No-Arbitrage Principle: - Transactions costs are 0 - There are No restrictions on short sales or on the use of short sale proceeds - Borrowing & lending can be done in unlimited amounts at the Rf The price of a Forward contract is not the price to purchase the contract because the parties to a forward contract typically pay nothing to enter into the contract at its inception. It refers to the price of the underlying asset under the terms of the forward. The price is the value of both the long & the short positions zero at contract initiation. Forward Price = Price that prevents profitable riskless arbitrage in frictionless markets FP = S0 x (1 + RF)T Or: S0 = FP / (1 + RF)T

FP = Forward Price S0 = Spot Price at inception of the contract Rf = Annual Risk-free rate T = Forward contract term in years

If the price of a Forward is Overpriced: Borrow money -> Buy the spot asset -> Short the asset in the forward market If the price of a Forward is Underpriced: Borrow asset -> Short Spot asset -> Lend Money -> Buy forward 3 variations used in the CFA: For LIBOR based contracts (FRA, Swap, Floors…): 360 days per year & simple interest & multiply “r” by days / 360 For Equity Bonds, currencies & stock options: 365 days & periodic compound interest & Raise (1+r) to an exponent of days / 365 Equity Indexes: 365 days per year & continuous compounding & exponent of “r” x days/365

Equity Forward Contracts with Discrete Dividends: In order to price a contract with dividend, we must either adjust the spot price for the Present value of the expected dividend (PVD) over the life of the contract or adjust the Forward price for the future value of the dividends (FVD) over the life of the contract. The No-arbitrage price of an Equity forward price in either case is: - FP = (S0 – PVD) x (1 + Rf)T - FP = (S0 x (1 + Rf)T ) - FVD

To calculate the value of the long position in a Forward contract on a dividend-paying stock, we make the adjustment for the PV of the remaining expected discrete dividends at time t: Vt (long position) = (St – PVDt) – (FP / (1 + Rf)T-t ) Contango  When the Futures prices exceeds the Spot price Over the life of a SWAP, the price of the SWAP DOES NOT Change The Theorical Price of a Forward Contract is the No-Arbitrage Price At Expiration, Spot Price – Futures Prices = 0  Spot = Futures

Equity Forward Contracts with Continuous Dividends We have to make the calculation as if the dividends are paid continuously (rather than discrete times) at the dividend yield rate on the index. FP (on a equity index) = S0 x e(Rf-dc)xT Rf = Continuously compounded Rf & Dc = Continuously compounded dividend Plain Vanilla receive-fixed interest rate swap = Long position in a bond coupled with the issuance of a floating rate note

Forward & Futures on Fixed Income securities: FP (on a fixed income security) = (S 0 – PVC) x (1 + Rf)T Or: S0 x (1 + Rf)T – FVC

Interest rate Swaps can be replicated with a series of PUT & CALL positions with expiration dates on the payment dates of the Swap

It is the same method as for a dividend paying stock, we simply substituting the PV of the expected coupon payment (PVC) over the life of the contract. The value of the Forward contract prior to expiration is as follow: Vt (long position) = (St – PVCt) – (FP / (1 + Rf)T-t ) WARNING: Difference between the pricing & the valuation of Forward: Pricing  Determining the appropriate forward price or rate when initiating the forward contract Valuation  Determining the appropriate value of the forward typically after it has been initiated!

Bond Futures Contract are often allowing the short an option to deliver any of several bonds, which will satisfy the delivery terms of the contract. This is called a delivery option & is valuable to the short. Each bond is given a Conversion Factor that is used to adjust the long’s payment at delivery so the more valuable bonds receive a larger payment. They are multipliers for the futures price at settlement. The long pays the future price at expiration multiplied by the Conversion factor. At settlement, the buyer pays the clean price (bond price) + an accrued interest or the full price: Full price = Clean Price + Accrued Interest Accrued Interest = (days since last coupon payment / Days between coupon payments) x Coupon Amount The Future price can then be calculated as: FP = (Full price x (1 + Rf)T-t - AIt – FVC) where AIt = accrued interest at maturity of the Futures The Quoted Futures Price adjusts this price based on the Conversion Factor (CF) as: QFD = FP / CF = (full price x (1 + Rf)T-t - AIt – FVC) (1 / CF)

Carry Arbitrage  When F0 (T) > FV (S0) ( Quand Future > Spot) Reverse Carry Arbitrage  When F0 (T) < FV (S0)

Libor-based Loans & Forward Rate Agreements: Eurodollar Deposit  Deposits in large banks outside the US denominated in $ LIBOR  Lending rate on $ denominated loans between banks on a 360-days year basis. LIBOR is an Add-on rate (like a yield quote on short-term certificate of deposit) & is used as a reference rate for floating US $-denominated loans worldwide.

EURIBOR  Euro lending rate for the European Central Bank, established in Frankfurt Long Position in a Forward Rate Agreement (FRA)  Party that would borrow the money. If the floating rate at contract expiration is > at the rate specified in the agreement, the long position in the contract can be viewed as the right to borrow at below market rates & the long will receive a payment. Notation: 2x3 FRA = expires in 2 months & the underlying asset is settled in 3 months. The price of a FRA is always quoted as a Percentage! Pricing FRAs:

Valuing an FRA at Maturity: If the interest rates increase, the long will profit as the contract has fixed a borrowing rate below the now-current market rate.

Valuing an FRA before Maturity: To value an FRA before the settlement date, we need to know the number of days that have passed since the initiation of the contract. 110 = 4 x 30 – days that have passed  120 – 10 20 = 30 - 10

Pricing Currency Forward Contracts: 365-days basis The price & value of a currency forward contract is an application of Covered interest parity which is based on the assumption that you should make the same amount when you lend at the Rf in your country as you would if you bought one unit of foreign currency at the current

spot rate invested it at the foreign Rf & entered into a forward contract to exchange the proceeds of the investment at maturity for the home currency at the forward rate. Covered interest rate parity gives us the No-arbitrage Forward price of a unit of foreign currency in terms of the home currency for a currency forward contract of length T in years: FT (currency Forward contract) = S0 x (1+ RPC)T / (1+ RBC)T (Pc = Price Currency & BC = Base C)

Valuing Currency Forward Contracts after Initiation Vt = (FPt – FP) x Contract Size / (1+ RPC)(T-t)

Vt = Value to the party long the base currency FP = Forward price at inception of the contract FPt = Forward price at time t of contract maturing at T RPC = Interest rate of price currency

Futures Contracts: Similar to the Forward contracts except that they trade on Organized Exchange with a clearing house. Clearing houses guarantee that traders in the future market will honour their obligations acting as the opposite side of each position. The exchange requires both sides of the trade to post margin & settle their accounts on a daily basis. The margin in the future markets is a performance guarantee. Margin-to-Market  Process of adjusting the Margin balance in a futures account each day for the change in the value of the contract from the previous trading day based on the settlement price. Mark-to-Market  Payment or receipt of funds necessary to adjust for the gains or losses. Like Forward Contracts, futures contracts have No value at Contract initiation Unlike Forward Contracts, futures contracts do not accumulate value changes over the term of the contract. The value of futures contracts is adjusted on a daily basis, the day’s gains & losses in contract value is always 0. Value of Future Contracts = Current Futures Price – Previous Mark-to-Market Price If the futures price increases, the value of the long position increases. The value is set back to 0 by the mark-to-market at the end of the mark-to-market period. Interest Rate Swap  One party agrees to pay floating & receive fixed. At initiation, the fixed rate is selected so that the PV of the floating-rate payment is equal to the PV of the fixed-rate payments, which means that the SWAP value is 0 to both parties. Determining the swap rate = price the swap.

The swap will have a Positive value for the fixed-rate payer & the swap contract value to the other party will be Negative, a liability to the pay-floating party. SWAP = Series of OFF-Market FRAs. SWAP Rate  Set the initial value of the SWAP to 0 The price of an interest rate SWAP is the Fixed Rate of Interest! Computing the SWAP Fixed Rate: It is derived from the LIBOR 1) Calculate the Discount Factors (Z) as: Z = 1 / (1 + (LIBOR x Days/360)) 2) The Periodic swap fixed rate (SFR) can then be calculated as: SFR = (1 – Last Discount Factor) / Sum of Discount Factors Fixed Rate Receiver  Profit when Short rates fall & losses when short rate rise = Long interest-rate puts & Short calls Currency Swap Receiver Long position in a foreign bond + issuance of a $-denominated floating rate note Fixed Rate Payer  Issuing a fixed-rate bond & buying a floating rate bond

Calculating the Market Value of an Interest Rate Swap: After the initiation of an interest swap, the swap will take on a positive or negative value as interest rates change. The party that is Fixed-rate payer benefits if rates increase because they are paying the older fixed rate & receiving the newer floating rate. Value to the payer =  Z x (SFRnew – SFR old) x (Days / 360) x Notional principal ( Z = sum of discount factors associated with the remaining settlement periods)

Currency Swaps: Determining the Fixed Rate & Foreign Notional Principal

The value of a Currency swap is calculated the same way as an interest rate swap. The value to any party is the PV of the cash flows they expect to receive minus the PV of the CF they are obligated to pay.

Equity Swaps: To price an N-period pay fixed equity swap, we can use the same formula as for a plain vanilla swap: SFR (periodic) = (1 – Last discount factor) / Sum of Discount factors To value this swap after time has passed, determine the value of the equity or index portfolio & the value of the fixed-rate payments.

A swap of returns on 2 different stocks can be viewed as buying one stock & shorting an equal value of a different stock. There is no “pricing” at swap initiation & we can value the swap at any point in time by taking the difference in returns multiplied by the notional principal.

Valuation of Contingent Claims: Binomial Model  Based on the idea that over the next period, the value of an asset will change. We need to know the beginning asset value the size of the 2 possible changes & the probability of each of these changes occurring. 2 Options: -

Increased by 33,3 % Decreased by 25 %

 U = (1 + Rf – D) / (U – D) (Avec U = 1.333 & D = 0,75)

We can Calculate the value of an option on the stock by: - Calculating the payoff of the option at maturity in both the up-move & down-moves - Calculating the expected value of the option in 1 year as the probability-weighted average of the payoffs in each state - Discounting the expected value back to today at the Rf

Put-Call Parity  The value of a Fiduciary Call (Long call + Zero-coupon bond) = Protective Put (Long stock + Long Put) Put-Call Parity  S0 + P0 = C0 + PV (X) Deductible  Loss an insured is willing to accept = Stock Price – Strike Price

Two-Period Binomial Model:

American-Style Options  Allow for exercise Any point until maturity of the option. Deepin-the-money Put options could benefit from early exercise. When an investor exercises an option early, it captures only the intrinsic value of the option & forgoes the time value.

For a Deep-in-the-money put options, the upside is limited because the stock price cannot fall below 0. Early Exercise of “In-the-Money” American option on Forward is sometimes worthwhile but never is for Options on Futures American Option CANNOT be valued by the “Expectation Approach” because of the possibility of early exercise!

Arbitrage with a One-period Binomial Model: The Fractional share of stock needed in the Arbitrage Trade is calculated as: Hedge Ratio = (C + - C -) / (S + - S -). (Avec S = Strike Price)

Interest Rate Options: It has a Positive Payoff when the Reference Rate is > than the Exercise Rate: Call Payoff = Notional Principal x (Max (0, Reference rate – Exercise rate)) Interest rate call options increase in value when rates increase.

An interest rate put option has a Positive Payoff when the Reference rate is < than Exercise rate. Put Payoff = Notional principal x ((Max (0, Exercise rate – Reference rate)) Interest rate put option values increase in value when rates decrease. Black-Scholes Merton Valuation Model  Values options in Continuous times, but is based on the No-Arbitrage Condition we used in valuing options in discrete time with a binomial model. To derived the BSM model, an “instantaneously” riskless portfolio is used to solve for the option price. The Assumptions of the Model are: -

The underlying asset price follow a Geometric Brownian motion process. The return on the underlying asset follows a Lognormal Distribution. The logarithmic continuously compounded return is Normally distributed. The Rf is constant & known. Borrowing & lending are both at the Rf The Volatility of the returns on the underlying asset is Constant & know. The price of the underlying changes smoothly Markets are “Frictionless”: no taxes, no transactions costs & no restrictions on short sales. No Arbitrage opportunities in the market place. The Continuously Compounded yield on the underlying asset is Constant The options are European Options.

The formula for valuing a European option (No Dividend) using the BSM Model is:

While the BSM model formula looks complicated, its interpretations are not: - The BSM value can be thought of as the PV of the Expected Option Payoff at Expiration - Calls can be thought of as a leverage stock investment where N(D1) units of stock are purchased using e-Rt XN(D2) of borrowed funds. A portfolio replicates a put options consists of a long position in N(-D2) bonds & short position in N(-D1) stocks. - N(D2) is interpreted as the Risk-Neutral Probability that a call option will expire in the money.

Option on Dividend Paying Stocks: When dividend payments occur during the life of an option, the price of the underlying stock is reduced!  - Call & + Put

Options on Currencies: We can also use the BSM Model to value foreign exchange options. Here, the underlying is the spot exchange rate instead of a stock price. The value can be calculated as:

Analogous to the interpretations of the BSM model, an Option on futures can be as: - The value of a Call Option on futures = Value of a portfolio with a Long future position & a Short bond position  Call = Achat dans le Futur - The value of a Put Option is = to the value of a Portfolio with a Long bond & a Short futures position  Put = Vente dans le Futur - The value of a call can also be thought of as the PV of the difference between the futures prices & the exercise price Interest Rate Options  Options on forward rates or FRA. A call option on an FRA gains when rates rise & a put option on an FRA gains when rates fall. Interest rates are fixed in advance & settled in arrears (paid at maturity of the loan). FRA -> 30/360 convention & Options on FRA -> Actual / 365 conventions

Combinaison of interest rate options can be used to replicate other contracts: - A long interest rate call & a short interest rate put can be used to replicate a long FRA - A short interest rate call & a long interest rate put can be combined to replicate a short FRA position if the exercise rate = current FRA rate - A series of interest rate call options with different maturities & the same exercise price can be combined to form an interest rate cap. A floating rate loan can be hedged using a Long Interest Cap - An Interest Rate Floor is a portfolio of interest rate put options & each of these puts is known as a floorlet. Floors can be used to hedge a long position in a floating rate bond - If the exercise rate on a cap & floor is same, a long cap & short floor can be used to replicate a Payer swap  Mémo: PLS - A short cap & a long floor can replicate a receiver swap - If the exercise rate on a floor & a cap are set equal to a market swap fixed rate, the value of the cap will be equal to the value of the floor. Swaptions:

Swaption  Option that gives the holder the right to enter into an interest rate swap Payer Swaption  Right to enter into a specific swap at some date in the future at a predetermined rate as the fixed-rate payer. As interest rates increase, the right to take the pay-fixed side of a swap becomes more valuable. The holder of a payer swap would exercise it & enter into the swap if the market rate is greater than the exercise rate at expiration Receiver Swaption  Right to enter into a specific swap at some date in the future as the fixed-rate receiver at the rate specified in the Swaption. As interest rates decrease, the right to enter the receive-fixed side of a swap becomes more valuable. The holder of a receiver swap would exercise if market rates are less than the exercise rate at expiration.

Equivalencies: A Receiver Swaption can be replicated using a long receiver Swaption & a short payer Swaption with the same exercise rates. A Payer Swaption can be replicated using a long payer Swaption & short receiver Swaption with the same exercise rates. A Long callable bond can be replicated using a long-option free bond + a short receiver Swaption. There are 5 inputs to the BSM model: Asset Price, Exercise price, Asset Price Volatility, Time to expiration & the Risk Free rate. The relationship between each input & the option price is captured by sensitivity factors known as the “Greeks”: Delta -> Relationship between changes in Asset Prices & change in Option Prices. Call option deltas are positive because as the underlying asset price increases, call option value also increases. The delta of put option is negative because the put value falls as the asset price increases.

The call option intrinsic value is equal to: - 0 when the call option is out-of-the-money - Stock Price – Exercise price when the option is in-the-money

The slope of the “Prior-to-expiration” curve is the change in call price per unit change in stock price. Delta is the slope of the prior-to-expiration curve. For a put option, the put delta is close to 0 when the put is out-of-the-money. When the put is “in-the-money”, the put delta is close to -e-T. Remember that a Call Option delta is between 0 & e-T. Assuming that the underlying stock price does not change, if the call option is: - Out-of-the-money, the call delta moves closer to 0 as time passes - In-the-money, the call delta moves closer to e-T as time passes Remember that a Put Option delta i...