Fin 501:Asset Pricing IFin 501:Asset Pricing ILecture 10: MultiLecture 10: Multi--period Modelperiod ModelLecture 10: MultiLecture 10: Multiperiod Modelperiod ModelOptions Options –– BlackBlack--ScholesScholes--Merton modelMerton modelProf. Markus K. BrunnermeierProf. Markus K. Brunnermeier1Fin 501:Asset Pricing IFin 501:Asset Pricing IBinomialBinomialOption PricingOption PricingBinomial Binomial Option PricingOption Pricing• Consider a European call option maturing at time T ih ik K C(SK0) h fl iwith strike K: CT=max(ST‐K,0), no cash flows in betweenNt bl t tti ll li t thi ff i jt•Not able to statically replicate this payoff using just the stock and risk‐free bond•Need todynamically hedgerequired stock•Need to dynamically hedge–required stock position changes for each period until maturity–static hedge for forward using put‐call paritystatic hedge for forward, using putcall parity• Replication str ategy depends on specified random process of stock price –need to know how price p ppevolves over time. Binomial (Cox‐Ross‐Rubinstein) model is canonicalFin 501:Asset Pricing IFin 501:Asset Pricing IAssumptionsAssumptionsAssumptionsAssumptions• Assumptions:St k hi h di id d–Stock which pays no dividend– Over each period of time, stock price moves from S to either uS or dS, i.i.d. over time, so that final distribution of STis binomialbinomialuSS–Suppose length of period is h and risk‐free rate is given bydSSuppose length of period is h and riskfree rate is given byR = erh– No arbitrage: u > R > d–Note: simplistic model but as we will see with enoughNote: simplistic model, but as we will see, with enough periods begins to look more realisticFin 501:Asset Pricing IFin 501:Asset Pricing IA OneA One‐‐Period Binomial TreePeriod Binomial TreeA OneA One‐‐Period Binomial TreePeriod Binomial Tree• Example of a single‐period model–S=50, u = 2, d= 0.5, R=1.25100502550– What is value of a European call option with K=50?– Option payoff: max(ST‐K,0)50500C = ?– Use replication to price0Fin 501:Asset Pricing IFin 501:Asset Pricing ISingleSingle‐‐period replicationperiod replicationSingleSingleperiod replicationperiod replication• Consider a long position of ∆ in the stock and B dollars in bond• Payoff from portfolio:∆uS+RB=100 ∆+1.25B∆dS+RB=25 ∆+1 25B∆ S+B=50 ∆+B• Define Cuas option payoff in up state and Cdas ∆dS+RB25 ∆+1.25Budoption payoff in down state (Cu=50,Cd=0 here)• Replicating str ategy must match payoffs:Cu=∆uS+RBCd=∆dS+RBFin 501:Asset Pricing IFin 501:Asset Pricing ISingleSingle‐‐period replicationperiod replicationSingleSingle‐‐period replicationperiod replication• Solving these equations yields:CC−)(dCuCduSCCdu−−−=Δ•In previous example ∆=2/3 and B=‐13 33 so the option)( duRdCuCBud−=In previous example, ∆=2/3 and B=13.33, so the option value isC = ∆S+B = 20• Interpretation of ∆: sensitivity of call price to a change in the stock price. Equivalently, tells you how to hedge risk of optionof option– To hedge a long position in call, short ∆ shares of stockFin 501:Asset Pricing IFin 501:Asset Pricing IRiskRisk‐‐neutral probabilitiesneutral probabilitiesRiskRisk‐‐neutral probabilitiesneutral probabilities• Substituting ∆ and B from into formula for C,⎤⎡−−+−−=udduRdRduRdCuCSduSCCC1)()(•Define p = (R‐d)/(u‐d) note that 1‐p = (u‐R)/(u‐d) so⎥⎦⎤⎢⎣⎡−−+−−=duCduRuCdudRR1Define p = (Rd)/(ud), note that 1p = (uR)/(ud), so[]duCppCRC )1(1−+=• Interpretation of p: probability the stock goes to uS in world where everyone is risk‐neutralFin 501:Asset Pricing IFin 501:Asset Pricing IRiskRisk‐‐neutral probabilitiesneutral probabilitiesRiskRiskneutral probabilitiesneutral probabilities• Note that p is the probability that would justify the k i S i ikl ldcurrent stock price S in a risk‐neutral world:[]dSqquSRS −+= )1(1pdudRqR=−−=• No arbitrage requires u > R > d as claimed before•Note: didn’t need to know anything about the•Note: didn t need to know anything about the objective probability of stock going up or down (P‐measure). Just need a model of stock prices to construct Q‐measure and price the option.Fin 501:Asset Pricing IFin 501:Asset Pricing ITheThebinomialbinomialformula in a graphformula in a graphThe The binomial binomial formula in a graphformula in a graphFin 501:Asset Pricing IFin 501:Asset Pricing ITwoTwo‐‐period binomial treeperiod binomial treeTwoTwoperiod binomial treeperiod binomial tree• Concatenation of single‐period trees:u2SuSSudSdSd2SFin 501:Asset Pricing IFin 501:Asset Pricing ITwoTwo‐‐period binomial treeperiod binomial treeTwoTwoperiod binomial treeperiod binomial tree• Example: S=50, u=2, d=0.5, R=1.252001002005025505012 5• Option payoff:12.5150CuC0Cd0Fin 501:Asset Pricing IFin 501:Asset Pricing ITwoTwo‐‐period binomial treeperiod binomial treeTwoTwoperiod binomial treeperiod binomial tree• To price the option, work backwards from final period.200150•We know how to price this from before:10020050Cu1500•We know how to price this from befo re:5.05.025.025.1=−−=−−=dudRp• Three‐step procedure:[]60)1(1=−+=uduuuCppCRCp p– 1. Compute risk‐neutral probability, p– 2. Plug into formula for C at each node to for prices, going backwards from the final node.– 3. Plug into formula for ∆ and B at each node for replicating strategy, going backwards from the final node..Fin 501:Asset Pricing IFin 501:Asset Pricing ITwoTwo‐‐period binomial treeperiod binomial treeTwoTwoperiod binomial treeperiod binomial tree• General formulas for two‐period tree:•p=(Rd)/(ud)Cuu•p=(R‐d)/(u‐d)Cu=[pCuu+(1‐p)Cud]/R∆=(CC)/(u2SudS)uu∆u=(Cuu‐Cud)/(u2S‐udS)Bu=Cu‐∆uSC=[pCu+(1‐p)Cd]/RCCd=[pCud+(1‐p)Cdd]/R∆d=(Cud‐Cdd)/(udS‐d2S)=[p2Cuu+2p(1‐p)Cud+(1‐p)2Cud]/R∆=(Cu‐Cd)/(uS‐dS)B=C‐∆SCud•Synthetic option requiresdynamic hedgingd(uddd)()Bd=Cd‐∆dSCdd•Synthetic option requires dynamic hedging– Must change the portfolio as stock price movesddFin 501:Asset Pricing IFin 501:Asset Pricing IArbitraging a mispriced optionArbitraging a mispriced optionArbitraging a mispriced optionArbitraging a mispriced option• Consider a 3‐period tree with S=80, K=80, u=1.5, d=0.5, R=1.1• Implies p = (R‐d)/(u‐d) = 0.6• Can dynamically replicate this option using 3‐period binomial tree. Turns out