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Investment Analysis and Portfolio ManagementFirst Canadian EditionBy Reilly, Brown, Hedges, Chang14

Copyright 2010 by Nelson Education Ltd. 14-*

An Overview of Forward and Futures TradingHedging with Forwards and FuturesValuation of Forward and FuturesFinancial FuturesAn Overview of Option Markets and ContractsThe Fundamental Option ValuationSwapsOption-Like Securities

Chapter 14 Derivatives: Analysis and Valuation


Forward contracts are negotiated directly between two parties in the OTC marketsIndividually designed to meet specific needsSubject to default riskFutures contracts are bought through brokers on an exchangeNo direct interaction between the two partiesExchange clearinghouse oversees delivery and settles daily gains and lossesCustomers post initial margin accountAn Overview of Forward & Futures Trading


An Overview of Forward & Futures Trading


An Overview of Forward & Futures Trading


An Overview of Forward & Futures TradingFutures Contract MechanicsFutures exchange requires each customer to post an initial margin account in the form of cash or government securities when the contract is originatedThe margin account is marked to market at the end of each trading day according to that days price movementsForward contracts may not require either counterparty to post collateralAll outstanding contract positions are adjusted to the settlement price set by the exchange after trading ends


An Overview of Forward & Futures TradingWith commodity futures, it usually is the case that delivery can take place any time during the month at the discretion of the short position

FuturesForwardsDesign FlexibilityStandardizedCan be customizedCredit RiskClearinghouse riskCounterparty riskLiquidity RiskDepends on tradingNegotiated risk


Hedging with Forwards & FuturesCreate a position that will offset the price risk of another more fundamental holdingShort hedge: Holding a short forward position against the long position in the commodity Long hedge: Supplements a short commodity holding with a long forward positionThe basic premise behind any hedge is that as the price of the underlying commodity changes, so too will the price of a forward contract based on that commodity


Hedging with Forwards & Futures: Defining the BasisBasis is spot price minus the forward price for a contract maturing at date T:

Bt,T = St - Ft,T

where St the Date t spot priceFt,T the Date t forward price for a contract maturing at Date TInitial basis, B0,T, is always knownMaturity basis, BT,T, is always zero. That is, forward and spot prices converge as the contract expiresCover basis: Bt, T


Hedging with Forwards & Futures: Understanding Basis RiskThe terminal value of the combined position is defined as the cover basis minus the initial basis Bt, T B0, T = (St - Ft,T ) - (S0 F0,T )

Basis RiskInvestors terminal value is directly related to Bt, T Bt, T depends on the future spot and forward pricesIf St and Ft,T are not correlated perfectly, Bt, T will change and cause the basis risk


Hedging with Forwards & Futures: Understanding Basis RiskHedging ExposureIt is to the correlation between future changes in the spot and forward contract prices Perfect correlation with customized contracts


Valuation of Forwards & Futures: The Cost to Carry ModelThe Cost of CarryCommissions paid for storing the commodity, PC0,TCost of financing the initial purchase, i0,TCash flows received between Dates 0 and T, D0,T


Valuation of Forwards & Futures: Contango & BackwardatedContango MarketWhen F0,T > S0Normally with high storage costs and no dividendsBackwardation MarketWhen F0,T < S0Normally with no storage costs and pays dividendsPremium for owning the commodity Convenience yieldCan results from small supply at date 0 relative to what is expected at date T (after the crop harvest)


Financial FuturesInterest Rate FuturesInterest rate forwards and futures were among the first derivatives to specify a financial security as the underlying assetForward rate agreementsInterest rate swapsBasic Types Long-term interest rate futuresShort-term interest rate futuresStock index futuresCurrency forwards and futures


Financial FuturesInterest rate futures available at the Montreal ExchangeBAX (Three-Month Canadian Bankers Acceptance FuturesOBX (Options on Three-Month Canadian Bankers Acceptance Futures)CGB (Ten-Year Government of Canada Bond Future)


Stock Index FuturesIntended to provide general hedges against stock market movements and can be applied to a portfolio or individual stocksHedging an individual stock with an index isolates the unsystematic portion of that securitys riskCan only be settled in cash, similar to the Eurodollar (i.e., LIBOR) contractStock Index Arbitrage: Use the stock index futures to convert a stock portfolio into synthetic riskless positionsProminent in program trading


Overview of Options Markets & ContractsOption Market ConventionsOption contracts have been traded for centuriesCustomized options traded on OTC marketIn April 1973, standardized options began trading on the Chicago Board Option ExchangeContracts offered by the CBOE are standardized in terms of the underlying common stock, the number of shares covered, the delivery dates, and the range of available exercise pricesOptions Clearing Corporation (OCC) acts as guarantor of each CBOE-traded options


Overview of Options Markets & ContractsPrice Quotations for Exchange-Traded OptionsEquity OptionsCBOE, AMEX, PHLX, PSETypical contract for 100 sharesRequire secondary transaction if exercisedTime premium affects pricingStock Index OptionsFirst traded on the CBOE in 1983Index options can only be settled in cashIndex puts are particularly useful in portfolio insurance applications


Overview of Options Markets & ContractsOptions on Futures Contracts Options on futures contracts have only been exchange-traded since 1982Give the right, but not the obligation, to enter into a futures contract on an underlying security or commodity at a later date at a predetermined priceLeverage is the primary attraction of this derivative


Fundamentals of Option ValuationRisk reduction tools when used as a hedgeForecasting the volatility of future asset pricesdirection and magnitudeHedge ratio is based on the range of possible option outcomes related to the range of possible stock outcomesRisk-free hedge buys one share of stock and sells call options to neutralize riskHedge portfolio should grow at the risk-free rate


Fundamentals of Option ValuationThe Basic ApproachAssume the WYZ stock price as the followingAssume the risk-free rate is 8%Want the price of a call option (C0) with X = $52.50

Stock price nowPrice in one year$50$65$40


Fundamentals of Option ValuationStep 1: Estimate the number of call optionsCalculate the options payoffs for each possible future stock priceIf stock goes to $65, option pays off $12.50If stock goes to $40, option pays off $0Determine the composition of the hedge portfolioIt contains one share of stock and h call optionsIf stock goes up, portfolio will pay:$65 + (h)($12.50)If stock goes down, portfolio will pay:$40 + (h)($0)


Fundamentals of Option ValuationStep 1 (Continued)To determine the composition of the hedge portfolio, find the number of options that equates the payoffs $65 + $12.50h = $40 + $0hImplies h = -2Hedge portfolio is long one share of stock and short two call optionsValue of hedge portfolio today:$50 - 2.00(C0)


Fundamentals of Option ValuationStep 2: Determine the PV of the portfolioWe know the hedge portfolio will pay $40 in one year with certaintyThus the value of that portfolio right now is40/(1+RFR)T Step 3: Compute the price of a call optionCondition of no risk-free arbitrage$50 - 2.00(C0) = 40/(1+RFR)T When T=1 and RFR=8%, solve for C0C0=$6.48


The Binomial Option Pricing FormulaIn the jth state in any sub-period, the value of the option can be calculated byFundamentals of Option Valuation where

andr = one plus the risk-free rate over the sub-period


Fundamentals of Option ValuationAt Date 0, the binomial option pricing formula can be expressed as follows:

m is the smallest integer number of up moves guaranteeing that the option will be in the money at expiration


Fundamentals of Option ValuationThe hedge ratio for any state j becomes

()()jujdjjCCSduh--=


The Black-Scholes Valuation ModelBinomial model is discrete method for valuing options because it allows security price changes to occur in distinct upward or downward movementsPrices can change continuously throughout time Advantage of Black-Scholes approach is relatively simple, closed-form equation capable of valuing options accurately under a wide array of circumstances


Assuming the continuously compounded risk-free rate and the stocks variance (i.e., 2) remain constant until the expiration date T, Black and Scholes used the riskless hedge intuition to derive the following formula for valuing a call option on a non-dividend-paying stock:C0 = SN(d1) X(e-(RFR)T)N(d2)The Black-Scholes Valuation Model


where:C0 = market value of call optionS = current market price of underlying stockX = exercise price of call optione-(RFR)T = discount function for continuously compounded variablesN(d1) = cumulative density function of d1 defined as

N(d2) = cumulative density function of d2 defined as

The Black-Scholes Valuation Model


Properties of the ModelOptions value is a function of five variablesCurrent security priceExercise priceTime to expirationRisk-free rateSecurity price volatilityFunctionally, the Black-Scholes model holds that C = f (S, X, T, RFR, )The Black-Scholes Valuation Model


The Black-Scholes Valuation Model


Estimating VolatilityThe standard deviation of returns to the underlying asset can be estimated in two waysTraditional mean and standard deviation of a series of price relativesEstimate implied volatility from Black-Scholes formulaIf we know the current price of the option (call it C*) and the four other variables, we can calculate the level of No simple closed-form solution exists for performing this calculation; it must be done by trial and errorThe Black-Scholes Valuation Model


Problems With Black-Scholes ValuationStock prices do not change continuouslyArbitrageable differences between option values and prices (due to brokerage fees, bid-ask spreads, and inflexible position sizes)Risk-free rate and volatility levels do not remain constant until the expiration dateEmpirical studies showed that the Black-Scholes model overvalued out-of-the-money call options and undervalued in-the-money contractsAny violation of the assumptions upon which the Black-Scholes model is based could lead to a misevaluation of the option contractThe Black-Scholes Valuation Model


SwapsSwapsThey are agreements to exchange a series of cash flows on periodic settlement dates over a certain time period (e.g., quarterly payments over two years). The length of a swap is termed the tenor of the swap that ends on termination date.Forward-Based Interest Rate ContractsForward Rate Agreement (FRA)Interest Rate Swaps


Swaps


Swaps


Extensions of SwapsEquity Index-Linked SwapsEquivalent to portfolios of forward contracts calling for the exchange of cash flows based on two different investment rates: A variable-debt rate (e.g., three-month LIBOR)Return to an equity index (e.g., Standard & Poors 500)Payment is based on: Total return, or Percentage index change for settlement period plus a fixed spread adjustment


Extensions of SwapsCredit-Related SwapsThese swaps are designed to help investors manage their credit risk exposuresOne of the newest swap contracting extensions introduced in the late 1990sCredit-related swaps have grown in popularity, exceeding $45 trillion in notional value by mid-2007 Types: Total Return SwapCredit Default Swap (CDS)Collateralized Debt Obligations (CDOs)


Option-Like SecuritiesWarrantsEquity call option issued directly by company whose stock serves as the underlying assetKey feature that distinguishes it from an ordinary call option is that, if exercised, the company will create new shares of stock to give to the warrant holderThus, exercise of a warrant will increase total number of outstanding shares, which reduces the value of each individual share. Because of this dilutive effect, the warrant is not as valuable as an otherwise comparable option contract.


Option-Like SecuritiesWarrants ValuationExpiration Date Value, WT

where:N = the current number of outstanding sharesNW = the shares created if the warrants are exercisedVT = the value of the firm before the warrants are exercisedX = the exercise price


Option-Like SecuritiesWarrants Valuation

orwhere:CT = the expiration date value of a regular call option with otherwise identical terms as the warrant


Option-Like SecuritiesConvertible BondsA convertible security gives its owner the right, but not the obligation, to convert the existing investment into another formTypically, the original security is either a bond or a share of preferred stock, which can be exchanged into common stock according to a predetermined formulaA hybrid securityBond or preferred stock holdingA call option that allows for the conversion


Option-Like SecuritiesConvertible BondsConversion ratio number of shares of common stock for which a convertible security may be exchangedConversion parity price price at which common stock can be obtained by surrendering the convertible instrument at par value


Option-Like SecuritiesCallable BondProvides the issuer with an option to call the bond under certain conditions and pay it off with funds from a new issue sold at a lower yieldBond with an embedded option (Chapter 12)


Option Like Securities

Copyright 2010 Nelson

*Copyright 2010 Nelson























**Copyright 2010 Nelson

**Copyright 2010 Nelson



















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