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Chapter 11OptionsRoad MapPart A Introduction to ﬁnance.Part B Valuation of assets, given discount rates.Part C Determination of risk-adjusted discount rate.Part D Introduction to derivatives.• Forwards and futures.• Options.• Real options.Main Issues• Introduction to Options• Use of Options• Properties of Option Prices• Valuation Models of OptionsChapter 11 Options 11-11 Introduction to Options1.1 DeﬁnitionsOption types:Call: Gives owner the right to purchase an as-set (the underlying asset) for a given price(exercise price) on or before a given date(expiration date).Put: Gives owner the right to sell an asset for agiven price on or before the expiration date.Exercise styles:European: Gives owner the right to exercise theoption only on the expiration date.American: Gives owner the right to exercise theoption on or before the expiration date.Key elements in deﬁning an option:• Underlying asset and its price S• Exercise price (strike price) K• Expiration date (maturity date) T (today is 0)• European or American.Fall 2006cJ. Wang 15.401 Lecture Notes11-2 Options Chapter 111.2 Option PayoﬀThe payoﬀ of an option on the expiration date is determined bythe price of the underlying asset.Example. Consider a European call option on IBM with exerciseprice $100. This gives the owner (buyer) of the option the right(not the obligation) to buy one share of IBM at $100 on theexpiration date. Depending on the share price of IBM on theexpiration date, the option owner’s payoﬀ looks as follows:IBM Price Action Payoﬀ... Not Exercise 080 Not Exercise 090 Not Exercise 0100 Not Exercise 0110 Exercise 10120 Exercise 20130 Exercise 30... Exercise ST− 100Note:• The payoﬀ of an option is never negative.• Sometimes, it is positive.• Actual payoﬀ depends on the price of the underlying asset.15.401 Lecture NotescJ. Wang Fall 2006Chapter 11 Options 11-3 Payoﬀs of calls and puts can be described by plotting theirpayoﬀs at expiration as function of the price of the underlyingasset:-6AssetpricePayoﬀ ofbuyinga call100100-6AssetpricePayoﬀ ofbuyinga [email protected]@@@@@@-?AssetpricePayoﬀ ofsellinga [email protected]@@@@@@-?AssetpricePayoﬀ ofsellinga put100-100Fall 2006cJ. Wang 15.401 Lecture Notes11-4 Options Chapter 11The net payoﬀ from an option must includes its cost.Example. A European call on IBM shares with an exercise priceof $100 and maturity of three months is trading at $5. The3-month interest rate, not annualized, is 0.5%. What is the priceof IBM that makes the call break-even?At maturity, the call’s net payoﬀ is as follows:IBM Price Action Payoﬀ Net payoﬀ...Not Exercise 0 - 5.02580 Not Exercise 0 - 5.02590 Not Exercise 0 - 5.025100 Not Exercise 0 - 5.025110 Exercise 10 4.975120 Exercise 20 14.975130 Exercise 30 24.975...Exercise ST− 100 ST− 100 − 5.25-6AssetpricePayoﬀ100100-5.0250aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaThe break even point is given by:Net payoﬀ= ST− 100 − (5)(1 + 0.005) = 0orST= $105.025.15.401 Lecture NotescJ. Wang Fall 2006Chapter 11 Options 11-5Using the payoﬀ diagrams, we can also examine the payoﬀ of aportfolio consisting of options as well as other assets.Example. Consider the following portfolio (a straddle): buy onecall and one put (with the same exercise price). Its payoﬀ is:-6AssetpricePayoﬀ ofa [email protected]@@@@@@@Example. The underlying asset and the bond (with face value$100) have the following payoﬀ diagram:-6AssetpricePayoﬀ ofasset100100-6AssetpricePayoﬀ ofbond100100Fall 2006cJ. Wang 15.401 Lecture Notes11-6 Options Chapter 111.3 Corporate Securities as OptionsExample. Consider two ﬁrms, A and B, with identical assets butdiﬀerent capital structures (in market value terms).Balance sheet of A Balance sheet of BAsset $30 $0 Bond Asset $30 $25 Bond30 Equity 5 Equity$30 $30 $30 $30• Firm B’s bond has a face value of $50. Thus default is likely.• Consider the value of stock A, stock B, and a call on theunderlying asset of ﬁrm B with an exercise price $50:Asset Value of Value of Value ofValue Stock A Stock B Call$20 20 0 040 40 0 050 50 0 060 60 10 1080 80 30 30100 100 50 50• Stock B gives exactly the same payoﬀ as a call option writtenon its asset.• Thus B’s common stocks really are call options.15.401 Lecture NotescJ. Wang Fall 2006Chapter 11 Options 11-7Indeed, many corporate securities can be viewed as options:Common Stock: A call option on the assets of the ﬁrmwith the exercise price being its bond’sredemption value.Bond: A portfolio combining the ﬁrm’s assetsand a short position in the call with exer-cise price equal bond redemption value.Warrant: Call options on the stock issued by theﬁrm.Convertible bond: A portfolio combining straight bonds anda call option on the ﬁrm’s stock with theexercise price related to the conversionratio.Callable bond: A portfolio combining straight bonds anda call written on the bonds.Fall 2006cJ. Wang 15.401 Lecture Notes11-8 Options Chapter 112 Use of OptionsHedging Downside while Keeping Upside.The put option allows one to hedge the downside risk of an asset.-6AssetAssetpricePayoﬀ ofasset & put100100@@@@@@@-6Net PayoﬀAssetpricePayoﬀ ofasset + put100100Speculating on Changes in PricesBuying puts (calls) is a convenient way of speculating on decreases(increases) in the price of the underlying asset. Options requireonly a small initial investment.-6AssetpricePayoﬀ ofa call100100-6AssetpricePayoﬀ ofa [email protected]@@@@@@15.401 Lecture NotescJ. Wang Fall 2006Chapter 11 Options 11-93 Properties of OptionsFor convenience, we refer to the underlying asset as stock. Itcould also be a bond, foreign currency or some other asset.Notation:S: Price of stock nowST: Price of stock at TB: Price of discount bond with face value$1 and maturityT (clearly, B ≤ 1)C: Price of a European call with strike priceK and maturity TP: Price of a European put with strike priceK and maturity Tc: Price of an American call with strikepriceK and maturity Tp: Price of an American put with strikepriceK and maturity T .Fall 2006cJ. Wang 15.401 Lecture Notes11-10 Options Chapter 11Price BoundsFirst consider European options on a non-dividend paying stock.1.C ≥ 0.2.C ≤ S — The payoﬀ of stock dominates that of

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