Tài chính doanh nghiệp - Chapter 19: Options and contingent claims

Chapter 19Options and Contingent ClaimsLearning ObjectivesUnderstand the major types and characteristics of options and distinguish between options and futures.Identify and explain the factors that affect option prices.Understand and apply basic option pricing theorems, including put–call parity.Learning Objectives (cont.)Understand the binomial model and Black–Scholes model of option pricing and calculate option prices using these models.Explain the characteristics and uses of foreign currency options and options on futures.Define a contingent claims and explain the option-like features of several contingent claims.Options and Option MarketsAn ‘option’ is the right to force a transaction to occur at some future time on terms and conditions decided now.It is a contract which gives the purchaser the right, but not the obligation, to assume a long (buy) or short (sell) position in the relevant underlying financial instrument or future at a predetermined exercise (strike) price, at a time in the future.Options and Option Markets (cont.) In return for this right the purchaser pays a the option price to the seller (‘writer’) of the option.Unlike FRAs and futures contracts, options allow the benefits of favourable price movements and provide protection against unfavourable price movements.Exchange Traded Options (ETO)An option that is traded on an exchange in a standardised denomination and follows fixed maturities, with predetermined standard terms.These options do not need to be held to expiry, they can be easily sold on the appropriate derivatives exchange before expiry.Option TerminologyCall optionPurchaser has the right but not the obligation to buy an asset at the specified exercise price. Purchaser’s risk is limited to the price paid. Writer’s profit is limited to the price received and has unlimited upside risk should prices rise.Option Terminology (cont.) Put optionGives the purchaser the right, but not the obligation, to sell a specified asset at a specified exercise price. The purchaser’s risk is limited to the price paid. The writer of the put option has limited risk (up to the exercise price of the put option) should prices fall, and profit is limited to the price received.Option Terminology (cont.) Exercise price (strike price)The price at which a particular option can be exercised.American-type optionsBuyer may exercise option at any time up to (and including) the expiry date.European-type optionsBuyer may only exercise option on the expiry date (and not before).Option Terminology (cont.)In the moneyCall option whose exercise price is below, or a put option whose exercise price is above, the current price of the asset on which the option is written.Out-of-the-moneyCall option whose exercise price is above, or a put option whose exercise price below, the current price of the asset on which the option is written.How Options are Created and TradedOptions can be created by the company whose shares underlie the option contract.May be issued to raise capital for the company.May be issued to employees of the company to form part of a compensation package.Options can also be created by parties who may have no association with the company.For example, two share market observers may enter into a private option contract on the shares of BHP.Option Contracts and Futures ContractsBoth involve delivery of some underlying asset at a future date and at a predetermined price.However, a futures contract requires delivery, whereas an option buyer chooses whether or not delivery will occur.Option Contracts and Futures Contracts (cont.)Payment of the futures price is not required until the expiry date, but when an option contract is created, the buyer must immediately pay the option price to the writer. If the option is exercised, there is a further transaction where the exercise price is paid.Payoff Structure for CallsSet of future cash flows, example is for a call with an exercise price of $14.Table 19.1Payoff Structure for Calls (Bought) Figure 19.1Payoff Structure for PutsTable 19.2Set of future cash flows, example is for a bought put option with an exercise price of $14.50.Payoff Structure for Puts (Bought) Figure 19.2Option PricingAt expiry a call is worth, and should therefore be priced at: At expiry a put is worth, and should therefore be priced at: Components of Option PriceIntrinsic valueProfit that could be made if option was exercised immediately.Time valueValue of an option in excess of its intrinsic value.Time value reflects the value of the ability to defer final payment for the asset in the case of a call.Value due to the chance that at some point in the option’s life the intrinsic value may become positive.Option Pricing Option price depends on:Underlying physical price Strike or exercise priceThe risk-free interest rateTime to maturityVolatility of underlying assetExpected dividendsFactors Affecting Call OptionsThe current share priceThe higher the current share price, the greater is the probability that the share price will increase above the exercise price and, therefore, the higher the call price becomes, other things being equal.The exercise priceThe higher the exercise price, the lower is the probability that the share price will increase above the exercise price and, therefore, the lower the call price becomes, other things being equal.Factors Affecting Call Options (cont.)The term to expiryA longer-term option dominates a short-term option, because there is more time for the share price to increase above the exercise price.The length of the term to expiry also affects the value of deferral of payment of the exercise price, this is linked to the risk-free rate’s effect, mentioned below.Therefore, the longer the term to expiry, the greater the call price, other things being equal.Factors Affecting Call Options (cont.)The volatility of the shareHigher share price volatility increases the chance of both large increases and large decreases in the share price. However, the asymmetric features of options mean that the holder of a call gains more from the increased chance of a large increase in the share price than is lost from the increased chance of a large decrease.Higher volatility increases the price of a call, other things being equal.Factors Affecting Call Options (cont.)The risk-free interest rateThe buyer of a call option can defer paying for the shares.Because interest rates are positive, money has a time value, so the right to defer payment is valuable.The higher the interest rate, the more valuable is this right.Therefore, the higher the risk-free interest rate, the higher the price of a call, other things being equal.Factors Affecting Call Options (cont.)Expected dividendsIf a company pays a dividend to its ordinary shareholders, the share price will fall on the ex-dividend date.Therefore, a call on a share that will go ex-dividend before the expiry of the call, is worth less than if the share either never pays dividends or, if it does pay dividends, will not reach the next ex-dividend date until after the call has expired.Factors Affecting Call Options (cont.)Examples of factors affecting call option prices. Factors Affecting Call Options (cont.)Other things being equal, call prices should be higher (lower):The higher (lower) the current share price.The lower (higher) the exercise price.The longer (shorter) the term to expiry.The more (less) volatile the underlying share.The higher (lower) the risk-free interest rate.The lower (higher) the expected dividend to be paid following an ex-dividend date that occurs during the term of the call.Basic Features of Put Option PricingOther things being equal, put prices should be higher (lower):The lower (higher) the current share priceThe higher (lower) the exercise price.The longer (shorter) the term to expiry (for American puts).The more (less) volatile the underlying share.The lower (higher) the risk-free interest rate.The higher (lower) the expected dividend to be paid following an ex-dividend date that occurs during the term of the put.Basic Features of Put Option Pricing (cont)Examples of factors affecting put option prices.Put–Call ParityFor European options on shares that do not pay dividends, there is an equilibrium relationship between the prices of puts and calls that:Are written on the same underlying shareAre traded simultaneouslyHave the same exercise price, and Have the same term to expiryPut–Call Parity (cont.) The put–call parity relationship can be written as: Put–Call Parity (cont.) The put–call parity relationship is based on the idea of arbitrage between a portfolio that includes a put option and another portfolio that includes a call option.These two portfolios are constructed such that the payoff structure is identical for all possible share price movements.The first, portfolio A, comprises a call option and a zero coupon bond that will pay out the exercise price, X, at maturity of bond and option.Put–Call Parity (cont.) The second, portfolio B, comprises a put option plus the share over which the options are held.The payoffs to these portfolios are the same for all possible share price movements, thus the values of the portfolios must be equal, giving rise to put–call parity. Put–Call Parity (cont.)While there is no simple equation linking the values of American puts and calls, the following upper and lower bounds have been established: The Minimum Value of CallsThe minimum value of a European call on a non-dividend paying share: In the absence of dividends, American call options should not be exercised before expiry.Supposing that S > X and the holder of the option decided to dispose of the call, the payoff from selling the call [S – X / (1 + r )] will always be greater than the payoff from exercising the call [S – X ]. The Minimum Value of Calls (cont.)The minimum value equation can be verified through a no arbitrage condition.Consider portfolio A from put call parity and portfolio B’ which comprises only a share. The Minimum Value of Calls (cont.)The payoffs to these two portfolios leads to the following condition on the values of the portfolios:Which gives us part of the minimum value of a call.This minimum value is bounded below by zero because of the limited liability associated with call options — that is, an option is a right, not an obligation.The Minimum Value of PutsThe minimum value of a European put is: Unlike calls, it can be rational to exercise an American put before expiry.In some circumstances, the benefit of receiving an early cash flow from early exercise will outweigh the cost of forfeiting some of the option’s time value.Binomial Option PricingBinomial option pricing was developed by Cox, Ross and Rubinstein (1979).Assumption that after each time period, the price of the underlying asset can only take one of two values.Risk-neutrality: enables us to price options regardless of the risk preferences of traders in the market:Situation in which investors are indifferent to risk; assets are therefore priced such that they are expected to yield the risk-free interest rate.Binomial Option Pricing (cont.)Multi-period binomial option pricingStage 1: Building up a lattice of share prices.Stage 2: Calculating the option payoffs at expiry from the expiry share prices.Stage 3: Calculating option prices by calculating expected values and then discounting at the risk-free interest rate.Binomial Option Pricing (cont.)Example 19.3We wish to value a three-month call option with an exercise price of $10.25.The current share price is $10 and the risk-free interest rate is 1.5% per month.Three time periods of 1 month each.It is assumed that at the end of each month the share price can move to only one of two values.Binomial Option Pricing (cont.)Example 19.3 (cont.)Stage 1: The lattice of share pricesThe objective is to lay out all the future share prices that can arise, given our assumptions.In this example, it is assumed that each month the share price can rise by 4% or fall by 3.846% (a rise in one month will be exactly offset if there is a fall in the following month, and vice versa).If the share price increases in the first month: $10.00 × 1.04, then the price will be $10.40.Binomial Option Pricing (cont.)Example 19.3 (cont.)Stage 2: Option payoffs at expiryAs we know the expiry share prices from Stage 1, it is a simple matter to calculate the matching call option payoffs.If after three rises in price the share is $11.2486, then the payoff is $0.9986 because the exercise price is $10.25.Binomial Option Pricing (cont.)Example 19.3 (cont.)Stage 3: DiscountingWe first need to find the probabilities of a rising and falling share price.Using node B as an example: This equation solves to give p = 0.6814 and (1 – p) = 0.3186.Binomial Option Pricing (cont.)Example 19.3 (cont.)We can now work back through the lattice from expiry to the present, at each node calculating the present value of the expected payoff.For example, at node D, the call’s price is:Working back through the lattice to today (node A) gives the call’s price as $0.3658 or about 37 cents.Binomial Option Pricing (cont.)Figure 19.3 Binomial Option Pricing (cont.)The previous example is a realistic treatment of binomial option pricing in all but two respects:The number of time periods was set at only three, whereas it should be set at 30 or more to gain an accurate answer.No attempt to justify the choice of ‘up’ and ‘down’ factors of 4% and 3.846% were made. In practice, these factors are selected very carefully, using some estimate of share price volatility during the life of the option.Applying the Binomial Approach to Other Option ProblemsBinomial model is easy to adapt to other option pricing problems such as valuing put options and to incorporate dividends.Once the lattice of shares is laid out, a put option can be priced as easily as a call option.The American feature is easily incorporated just by checking, at each node, whether the calculated option price is less than the payoff from immediate exercise.Dividend-paying shares: simply reduce share price at the ex-dividend date to reflect the payment of the dividend.Black–Scholes Model of Call Option PricingBlack and Scholes presented a model that determines the price of a call as a function of five variables: The current price of the underlying share.The exercise price of the call.The call’s term to expiry.The volatility of the share (as measured by the variance of the distribution of returns on the share).The risk-free interest rate.Black–Scholes Model of Call Option Pricing (cont.)Assumptions:Constant risk-free interest rate at which investors can borrow and lend unlimited amounts.Share returns follow random walk in continuous time with a variance proportional to the square of the share price. The variance rate is a known constant.No transaction costs or taxes.Short selling is allowed, with no restrictions or penalties.There are no dividends or rights issues.The call is of the European type.Black–Scholes Model of Call Option Pricing (cont.)The Black–Scholes call option pricing formula is given by: N(d) indicates the cumulative standard normal density function with upper integral limit d. T is the term to expiry.Black–Scholes Model of Call Option Pricing (cont.) Example 19.4:Current share price: S = $17.60Exercise price: X = $16.00Term to expiry: T = 3 months = 0.25 yrsVolatility (variance): = 0.09 p.a.Risk-free interest rate: r  = 0.1 p.a. continuously compoundingBlack–Scholes Model of Call Option Pricing (cont.)Example 19.4: Calculate d1 and d2:Black–Scholes Model of Call Option Pricing (cont.)Example 19.4:N (0.877) from normal density function is 0.8098N (0.727) from normal density function is 0.7664The discounting factorThe Black–Scholes call price is, therefore, approximately $2.29.Black–Scholes Model of Call Option Pricing (cont.)The Black–Scholes call option pricing formula is specified for continuous compounding.It can be modified to assume discrete compounding: In this case R is the observed effective risk-free interest rate for the term of the option.Black–Scholes Model of Put Option Pricing (cont.)The Black–Scholes put pricing model for European puts on shares that do not pay dividends:This exploits the put-call parity formula.The Black–Scholes formula has been combined with put–call parity to arrive at the above formula for pricing a put.Black–Scholes Model: Brief AssessmentNo assumption concerning the attitude of investors towards risk is required or implied.The variables that are required are for the most part observable, and have reliable data available.Measuring the share’s volatility is, however, subject to error (the big unknown).Empirical evidence suggests that the Black–Scholes model can price exchange-traded call options accurately, provided that a dividend-adjusted model is used. Finance in Action: Hedging the WeatherCan view an option as an insurance contract.On possible type of option can protect the purchaser from adverse weather.Example of golf course owner buying an option against more than 50 rainy days during summer.This option pays out as the number of rainy days goes above 50.Weather risk management association claims up to 12 000 such contracts signed worldwide in 12 months to March 2003.Options on Foreign CurrencyWhat is an option on foreign currency?A contract that confers the right to buy (sell) an agreed quantity of foreign currency at a given exchange rate.Options on foreign currency are traded in organised markets, as well as in ‘over-the-counter’ markets and by privately arranged contracts.A company selling US$ in exchange for A$ can be said to be purchasing A$ in exchange for US$.A put option to sell US$ in exchange for A$ can be described as a call to buy A$ in exchange for US$.Pricing Options on Foreign CurrencyA call to buy foreign currency may be priced as: Options, Forwards and FuturesSimple relationship between European-type options and forward prices.Combining a put and call with identical exercise price and expiry date can replicate a forward/futures contract.If prices of such puts and calls are equal, then futures/forward price should equal exercise price of options.Options, Forwards and Futures (cont.)Low Exercise Price Option, LEPO.LEPOs are options with an exercise price of 1 cent.Option will certainly be exercised if they are written on shares worth more than 1 cent.On purchase, option price is not paid, instead traders pay margin calls.This appears like a futures contract on shares.Options on FuturesA number of futures exchanges, including the SFE, have introduced options on futures on their popular futures contracts.A call option on futures confers on the buyer of the call the right to enter into a futures contract as a buyer.A put option on futures confers on the buyer of the put the right to enter into a futures contract as a seller.Options on Futures (cont.)Uses of options on futures: Three attributes that may have prompted market participants to trade in the option.The key is the right feature of an option rather than the obligation feature of a future:Open futures positions entail very high risks for a speculator, particularly if those positions are held for a long time.Hedgers may not be certain enough of their own circumstances to justify accepting the obligations of a futures contract.The deposit/margin system is simpler for option buyers than for futures traders.Contingent ClaimsA ‘contingent claim’ is an asset whose value depends on the given value of some other asset.A call option is perhaps the simplest type of contingent claim.A large number of financial contracts are contingent claims, and this raises the possibility that such contracts might be valued using an option-pricing approach.Contingent Claims (cont.)Rights issueA shareholder is given the right to purchase new shares in the company at an issue price set by the company.The rights must be sold or taken up by a specified date.Simply a call option issued by the company.Contingent Claims (cont.)Convertible bondsA type of debt security that, in addition to paying interest, gives the investor the right to convert the security into shares of the company.Equivalent to ordinary debt plus a call option on the shares of the company.Contingent Claims (cont.)Valuation of levered shares and risky zero-coupon debt:Shareholders must make a choice which resembles the choice facing the holder of a call option.The shareholders can direct the company to repay debt on the maturity date (equivalent to the shareholders exercising a call option).The shareholders can allow the company to default (equivalent to the the shareholders allowing the call to expire unexercised).Contingent Claims (cont.)Valuation of levered shares and risky coupon-paying debt:The contingent claims approach can also be applied to the problems of valuing risky coupon-paying debt and valuing the shares of a company that has issued this type of debt.Default risk structure of interest rates:As risk of default on debt increases, required yield also increases.Can use contingent claims approach to value debt, which can be used to infer the yield and implied risk of default.Contingent Claims (cont.)Project evaluation and ‘real’ optionsThe NPV approach is based on an analogy between a proposed investment project and a bond.However, sometimes the bond analogy is not the best one available.For example, once a project has commenced, the bond analogy assumes its cash flows cannot be changed.However, in practice the investing company may intervene in the project after it has started.Contingent Claims (cont.)Project evaluation and ‘real’ options (cont.)In short, there may be ‘options’ such as abandonment available. Some real options include:The option to defer, abandon, re-open, to study or to expand.These sorts of options are written into contracts on a regular basis, which suggests they are valuable.The point is that having such options cannot reduce the value of a project or asset: at worst they have zero value.Summary Option — right to force a transaction at future date on terms decided now.Important characteristics include, call, put, American and European.Option/contingent claim value depends on the value of another asset.Put–call parity — Euro put can be replicated by share, risk-free deposit and counterpart call. Summary (cont.)Options can be priced using:Binomial option pricing modelBlack–Scholes option pricing model.Instruments such as rights, convertible bonds, shares in levered companies and default risk of interest rates can all be analysed as options.Real options, such as to abandon, delay, consider further or expand projects, have value just as the options on financial assets do.