Presentation on theme: "DERIVATIVES: ANALYSIS AND VALUATION"— Presentation transcript:
1 DERIVATIVES: ANALYSIS AND VALUATION Chapter 12DERIVATIVES: ANALYSIS AND VALUATION
2 Chapter 12 Questions How are spot and futures prices related? What is basis risk?What is program trading and stock index arbitrage? How can futures be used to hedge or speculate on changes in yield curve spreads and credit quality spreads?Why would investors want to invest in an option on a futures contract?
3 Chapter 12 Questions What factors influence the price of an option? How does one use the Black-Scholes option-pricing model?Why are the terms delta,theta, vega, rho, and gamma important to option investors?How do option-like features affect the price of bonds?
4 Futures Valuation Issues Cost of Carry ModelSuppose that you needed some commodity in three months. You have at least the following two options:Purchase the commodity now at the current spot market price (S0) and “carry” the commodity for 3 monthsBuy a futures contract for delivery of the commodity in 3 months for the current futures price (F0,3)
5 Futures Valuation Issues Cost of Carry ModelThe futures prices and spot prices must be related to one another in order for there to be no arbitrage opportunities for investors.If the carrying cost only amounts to forgone interest at a risk-free rate (rf) for T time periods, then the following relationship must hold:F0,T = S0 (1+rf)T
6 Futures Valuation Issues Cost of Carry Model Example: Suppose that you can buy gold in the spot market for $300. The monthly risk-free is .25%. You need the gold in three months.What should be the current futures price?F0,T = 300 ( )3 =What if the futures price is $305?You have a risk-less profit opportunity. Buy gold at $300, sell futures at $305. In three months, delivery the gold, pay the known interest, pocket the difference.
7 Futures Valuations Issues Similar futures-spot price relationships can be derived when there are “market imperfections” involved with carrying the commodity or financial assetIncorporating storage and insurance costs as a percentage of contract value (SI):F0,T = S0 (1+rf +SI)TIncorporating ownership benefits lost with a futures position, especially dividends(d):F0,T = S0 (1+rf +SI -d)T
8 Futures Valuation Issues BasisBasis is the difference between the spot and futures prices.For a contract expiring at time T, the basis at time t is:Bt,T = St – Ft,TOver time, the spot and futures prices converge, and basis becomes zero at expirationBetween time t and expiration, basis can change as the difference between spot and futures prices vary (known as basis risk)
9 Advanced Applications of Financial Futures Stock Index ArbitrageAn example of a program trading strategy designed to take advantage of temporarily “mis-pricing” of securitiesMonitor the parity condition:F0,T = S0 (1+rf +-d)TIf it does not hold, construct a risk-free position to take advantage of the situation.
10 Advanced Applications of Financial Futures T-Bond/T-Note Futures Spread“Note over bond” (NOB) spreadStrategies based on speculating the changing slope of the yield curve
11 Options on Futures Also known as Futures Options Options on Stock Index FuturesGives the owner the right to buy (call) or sell (put) a stock futures contractOptions on Treasury Bond FuturesGives the owner the right to buy (call) or sell (put) a Treasury bond futures contract
12 Options on Futures Why would they be attractive? If exercised, it would seem to have been better to simply buy a futures contract instead (no option premium to pay)One primary advantage can be found when looking at all the potential price movementsFutures contracts used for hedging offset portfolio value changes; thus, advantageous price movements for a portfolio are offset by the futures positionOptions give the right (but not the obligation) to purchase the futures contract; thus, favorable price movements will be offset only by the option premium rather than by a corresponding loss on the futures position
13 Valuation of Options Factors influencing the value of a call option: Stock price (+)For a given exercise price, the higher the stock price, the greater the intrinsic value of the option (or at least the closer to being in-the-money)Exercise price (-)The lower the price at which you can buy, the more valueTime to expiration (+)The longer the time to expiration, the more likely the option will be valuable
14 Valuation of Options Factors influencing the value of a call option: Interest rate (+)Options involve less money to invest, lower opportunity costsVolatility of underlying stock price (+)The greater the volatility of the underlying stock, the more likely that the option position will be valuable
15 Valuation of Options Factors influencing the value of a put option: The same listed, but different directions for several items.Stock price (-)Exercise price (+)Time to expiration (+)Interest rate (-)Volatility of underlying stock price (+)
16 Black-Scholes Option Pricing Model Model for determining the value of American call optionsThis work warranted the awarding of the 1997 Nobel Prize in Economics!
17 Black-Scholes Option Pricing Formula P0 = PS[N(d1)] - X[e-rt][N(d2)]where:P0 = market value of call optionPS = current market price of underlying stockN(d1) = cumulative density function of d1 as defined laterX = exercise price of call optionr = current annualized market interest rate for prime commercial papert = time remaining before expiration (in years)N(d2) = cumulative density function of d2 as defined later
18 Black-Scholes Option Pricing Formula P0 = PS[N(d1)] - X[e-rt][N(d2)]The cumulative density functions are defined as:Where:ln(PS/X) = natural logarithm of (Ps/X)S = standard deviation of annual rate of return on underlying stock
19 Using the Black-Scholes Formula Besides mathematical values, there are five inputs needed to use this model:Current stock price (Ps)Exercise price (X)Market interest rate (r)Time to expiration (t)Standard deviation of annual returns (s)Of these, only the last in not observableAlso, using the put/call parity, we can value put options as well after calculating call value
20 Option Valuation Terminology DeltaThe sensitivity of an option’s price to the price of the underlying securityPositive for calls, negative for putsThetaMeasures how the option premium changes as expiration approaches
21 Option Valuation Terminology VegaThe sensitivity of the option premium to the price volatility (s) of the underlying securityRhoMeasures the sensitivity of the option premium to changes in interest ratesGammaMeasures the sensitivity of delta to changes in the underlying security price
22 Option-like Securities Several types of securities contain embedded options:Callable and Putable BondsWarrantsConvertible Securities
23 Callable and Putable Bonds Callable Bonds contain a “call provision”The issuer has the option of buying the bonds back at the call (exercise) price rather than having to wait until maturityAttractive option for issuers if interest rates fall, since they can purchase back old bonds and refinance (refunding) with new, lower interest bondsTypically will trade at no more than the call price, since call becomes likely at that point
24 Callable and Putable Bonds Putable Bonds contain a “put provision”Investors may resell the bonds back to the issuer prior to maturity at the put (exercise) price, often par valuePuts can generally be exercised only when designated events take place
25 WarrantsWarrant is an option to buy a stated number of shares of common stock at a specified price at any time during the life of the warrantSimilar to a call option, but usually with a much longer lifeIssued by the company whose stock the warrant is for
26 Intrinsic Value = (Stock Price – Exercise Price) x Number of Share WarrantsIntrinsic value is the difference between the market price of the common stock and the warrant exercise priceIntrinsic Value = (Stock Price – Exercise Price) x Number of ShareSpeculative value is the value of the warrant above its intrinsic valueLike other options, the value is higher than intrinsic value, except at maturity
27 Convertible Securities Allows the holder to convert one type of security into a stipulated amount of another type (usually common stock) at the investor’s discretionWith convertible securities, value depends both on the value of the original asset and the value if conversion takes placeValue cannot fall below the greater of the two values
28 Convertible Securities Convertible BondsAdvantages to issuing firmsLower interest rate on debtDebt represents potential common stockAdvantages to investorsUpside potential of common stockDownside protection of a bond
29 Convertible Securities Convertible bondsConversion ratio = number of shares obtained if convertedConversion price = Face Value/Number of sharesValuation of convertible bondsCombination value of stock and bondTwo step process to determine minimum value
30 Convertible Securities Convertible BondsValue of a convertible as a bondDetermine the bond’s value as if it had no conversion featureThis is the convertible’s investment value or floor valueValue of a convertible as stockCompute the value of the common stock received on conversionThis is the conversion value
31 Convertible Securities Convertible BondsMinimum Value = Max (Bond Value, Conversion Value)Like other options, including embedded options, they typically only sell at their minimum, intrinsic value only at maturity.Conversion Premium = (Market Price – Minimum Value)/Minimum Value
32 Convertible Securities Convertible BondsConversion Parity Price = Market Price/Conversion RatioAn risk-free profit opportunity would exist if the price of the convertible below this price, since immediate conversion of the bond and then selling the stock would yield a profitPaybackHow long it takes the higher-interest income from the convertible bond (compared to the stock dividend) to make up for the conversion premium
33 Convertible Securities Convertible Preferred StockCombination of preferred stock and common stockCommon characteristics:Cumulative but not participating dividendsNo sinking fund or purchase fundFixed conversion rateWaiting period not required before conversionConversion privilege does not expireUsually issued in connection with mergers
34 Convertible Securities Convertible Preferred StockValue as preferred stockValue as common stock, given the conversion rateParity relationships imply that the value has to be higher than the maximum of the two values