Chapter 11 Options - its.caltech.edu

1 Chapter 11 OptionsRoad MapPart AIntroduction to BValuation of assets, given discount CDetermination of risk-adjusted discount DIntroduction to derivatives. Forwards and futures. Options . Real Issues Introduction to Options Use of Options Properties of Option Prices Valuation Models of OptionsChapter 11 Options11-11 Introduction to 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 styles:European: Gives owner the right to exercise theoption only on the expiration : Gives owner the right to exercise theoption on or before the expiration elements in defining an option: Underlying asset and its priceS Exercise price (strike price)K Expiration date (maturity date)T(today is 0) European or 2006c J.

2 Lecture Notes11-2 OptionsChapter Option PayoffThe payoff of an option on the expiration date is determined bythe price of the underlying 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 Exercise080 Not Exercise090 Not Exercise0100 Not 100 Note: The payoff of an option is never negative. Sometimes, it is positive. Actual payoff depends on the price of the underlying Lecture Notesc J. WangFall 2006 Chapter 11 Options11-3 Payoffs of calls and puts can be described by plotting theirpayoffs at expiration as function of the price of the underlyingasset: AssetpricePayoff ofbuyinga call100100 AssetpricePayoff ofbuyinga put100100 AssetpricePayoff ofsellinga call100-100 AssetpricePayoff ofsellinga put100-100 Fall 2006c J.

3 Lecture Notes11-4 OptionsChapter 11 The net payoff from an option must includes its 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 What is the priceof IBM that makes the call break-even?At maturity, the call s net payoff is as follows:IBM PriceActionPayoffNet Exercise0- Exercise0- Exercise0- Exercise0- 100ST 100 The break even point is given by:Net payoff=ST 100 (5)(1 + ) = 0orST= $ Lecture Notesc J. WangFall 2006 Chapter 11 Options11-5 Using the payoff diagrams, we can also examine the payoff of aportfolio consisting of Options as well as other the following portfolio (a straddle): buy onecall and one put (with the same exercise price). Its payoff is: AssetpricePayoff ofa straddle100100 underlying asset and the bond (with face value$100) have the following payoff diagram: AssetpricePayoff ofasset100100 AssetpricePayoff ofbond100100 Fall 2006c J.

4 Lecture Notes11-6 OptionsChapter Corporate Securities as two firms, A and B, with identical assets butdifferent capital structures (in market value terms).Balance sheet of ABalance sheet of BAsset $30$0 BondAsset $30$25 Bond30 Equity5 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:AssetValue ofValue ofValue ofValueStock AStock BCall$2020004040005050006060101080803030 1001005050 Stock B gives exactly the same payoff as a call option writtenon its asset. Thus B s common stocks really are call Lecture Notesc J. WangFall 2006 Chapter 11 Options11-7 Indeed, 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 :A portfolio combining the firm s assetsand a short position in the call with exer-cise price equal bond redemption :Call Options on the stock issued by bond: A portfolio combining straight bonds anda call option on the firm s stock with theexercise price related to the bond:A portfolio combining straight bonds anda call written on the 2006c J.

5 Lecture Notes11-8 OptionsChapter 112 Use of OptionsHedging Downside while Keeping put option allows one to hedge the downside risk of an asset. AssetAssetpricePayoff ofasset & put100100 Net PayoffAssetpricePayoff ofasset + put100100 Speculating 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. AssetpricePayoff ofa call100100 AssetpricePayoff ofa put100100 Lecture Notesc J. WangFall 2006 Chapter 11 Options11-93 Properties of OptionsFor convenience, we refer to the underlying asset as stock. Itcould also be a bond, foreign currency or some other :S: Price of stock nowST: Price of stock atTB: Price of discount bond with face value$1 and maturityT(clearly,B 1)C: Price of a European call with strike priceKand maturityTP: Price of a European put with strike priceKand maturityTc:Price of an American call with strikepriceKand maturityTp:Price of an American put with strikepriceKand 2006c J.

6 Lecture Notes11-10 OptionsChapter 11 Price BoundsFirst consider European Options on a non-dividend paying S The payoff of stock dominates that of call: STPayoffKK S KB(assuming no dividends).Strategy (a): Buy a callStrategy (b): Buy a share of stock by payoff of strategy (a) dominates that of strategy (b): STPayoffK strategy (b) Lecture Notesc J. WangFall 2006 Chapter 11 Options11-11 SinceC 0,wehaveC max[S KB,0].4. Combining the above, we havemax[S KB,0] C S. Stock priceOptionpriceKB upperboundlowerboundoptionpriceFall 2006c J. Lecture Notes11-12 OptionsChapter 11 Put-Call ParityConsider the following two portfolios:1. A portfolio of a call with exercise price $100 and a bond withface value $ A portfolio of a put with exercise price $100 and a share ofthe underlying payoffs are Asset pricePayoff ofportfolio 1100100 bondcall Asset pricePayoff ofportfolio 2100100 underlying assetputTheir payoffs are identical, so must be their prices:C+K/(1 +r)T=P+ is called the put-call Lecture Notesc J.

7 WangFall 2006 Chapter 11 Options11-13 American Options and Early Exercise1. American Options are worth more than their European Without dividends, never exercise an American call early. Exercising prematurely requires paying the exercise priceearly, hence loses the time value of money. Exercising prematurely foregoes the option valuec(S, K, T)=C(S, K, T).3. Without dividends, it can be optimal to exercise an Americanput put with strike $10 on a stock with price zero. Exercise now gives $10 today Exercise later gives $10 of Dividends1. With dividends,max[S KB PV(D),0] C Dividends make early exercise more likely for American callsand less likely for American 2006c J. Lecture Notes11-14 OptionsChapter 11 Option Value and Asset VolatilityOption value increases with the volatility of underlying firms, A and B, with the same current priceof $100.

8 B has higher volatility of future prices. Consider calloptions written on A and B, respectively, with the same exerciseprice $ statebad stateProbabilityp1 pStock A12080 Stock B15050 Call on A200 Call on B500 Clearly, call on stock B should be more Lecture Notesc J. WangFall 2006 Chapter 11 Options11-154 Binomial Option Pricing ModelDeterminants of Option ValueKey factors in determining option value:1. price of underlying assetS2. strike priceK3. time to maturityT4. interest rater5. dividendsD6. volatility of underlying asset .Additional factors that can sometimes influence option value:7. expected return on the underlying asset8. additional properties of stock price movements9. investors attitude toward risk,..Fall 2006c J. Lecture Notes11-16 OptionsChapter 11 Price Process of Underlying AssetIn order to have a complete option pricing model, we need tomake additional assumptions about1.

9 Price process of the underlying asset (stock)2. Other will assume, in particular, that: Prices do not allow arbitrage. Prices are reasonable . A benchmark model Price follows a binomial t=0t= Lecture Notesc J. WangFall 2006 Chapter 11 Options11-17 One-period Bimomial of a European call on a stock. Current stock price is $50. There is one period to go. Stock price will either go up to $75 or go down to $25. There are no cash dividends. The strike price is $50. one period borrowing and lending rate is 10%.The stock and bond present two investment option s payoff at expiration is:C0250 Question:WhatisC0, the value of the option today?Fall 2006c J. Lecture Notes11-18 OptionsChapter 11 Claim: We can form a portfolio of stock and bond that givesidentical payoffs as the a portfolio(a, b)where ais the number of shares of the stock held bis the dollar amount invested in the riskless want to find(a, b)so that75a+ + is a unique solutiona= is buy half a share of stock and sell$ of bond payoff of this portfolio is identical to that of the call present value of the call must equal the current cost of this replicating portfolio which is(50)( ) = : Number of shares needed to replicate one call optionis calledhedge ratiooroption the above problem, the option delta isa:Option delta=1 Lecture Notesc J.

10 WangFall 2006 Chapter 11 Options11-19 Two-period Binomial ModelNow we generalize the above example when there are two periodsto go: period 1 and period 2. The stock price process is:S= call price follows the following process:CCuCuu= the terminal value of the call is known, and CuandCddenote the option value next period when the stockprice goesupand goesdown, derive current value of the call by working backwards: firstcompute its value next period, and then its current 2006c J. Lecture Notes11-20 OptionsChapter 11 Step with Period 1:1. Suppose the stock price goes up to $75 in period 1: Construct the replicating portfolio at node (t=1, up) + + The unique solution isa= The cost of this portfolio is( )(75) The exercise value of the option is75 50 = 25< Thus,Cu= Suppose the stock price goes down to $25 in period the above for node (t=1,down): The replicating portfolio isa=0andb=0.