INSTITUTIONAL FINANCELecture 08: Dynamic Arbitrage to Replicate Non-Linear1PayoffsOriginally prepared by Ufuk Ince, Ekaterina Emm, supplementalmaterial courtesy of Wei Xiong, Princeton UniversityBINOMIAL OPTION PRICINGBINOMIAL OPTION PRICING´ Consider a European call option maturing at time T with strike K: C=max(SK 0) no cash flows inwith strike K: CT=max(ST-K,0), no cash flows in between´ Is there a way to statically replicate this payoff?« Not using just the stock and risk-free bond – required stock position changes for each period until maturity (as we willposition changes for each period until maturity (as we will see)« Need to dynamically hedge – compare with static hedge such as hedging a forward or hedge using putcall parityas hedging a forward, or hedge using put-call parity´ Replication strategy depends on specified random process of stock price – need to know how price evolves over time. Binomial (Cox-Rubinstein-Ross) model is canonicalASSUMPTIONSASSUMPTIONS´ 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 byR=erhdSR e« No arbitrage: u > R > d« Note: simplistic model, but as we will see, with enough periods begins to look more realisticperiods begins to look more realisticA ONE-PERIOD BINOMIAL TREEA ONE-PERIOD BINOMIAL TREE´ Example of a single-period model« S=50, u = 2, d= 0.5, R=1.2510050«What is value of a European call option with K=50?2550«What is value of a European call option with K=50?« Option payoff: max(ST-K,0)50C?« Use replication to price0C = ?ppSINGLE-PERIOD REPLICATIONSINGLEPERIOD REPLICATION´ Consider a long position of ∆ in the stock and B dll i b ddollars 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 option payoff in down state (Cu=50,Cd=0 here)´ Replicating strategy must match payoffs:Cu=∆uS+RBC=∆dS+RBCd=∆dS+RBSINGLE-PERIOD REPLICATIONSINGLE-PERIOD REPLICATION´ Solving these equations yields:)(dCCduSCCdu−−=Δ´In previous example ∆=2/3 and B=-13 33 so the)( 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 optionto hedge risk of option« To hedge a long position in call, short ∆ shares of stockRISK-NEUTRAL PROBABILITIESRISK-NEUTRAL 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⎥⎦⎤⎢⎣⎡−−+−−=duCduRuCdudRR1´Define p = (R-d)/(u-d), note that 1-p = (u-R)/(u-d), so[]duCppCRC )1(1−+=´ Interpretation of p: probability the stock goes to uS in world where everyone is risk-neutralRISK-NEUTRAL PROBABILITIESRISKNEUTRAL PROBABILITIES´ Note that p is the probability that would justify the tt k i Si iktl 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 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.construct Qmeasure and price the option.THE BINOMIAL FORMULA IN A GRAPHTHE BINOMIAL FORMULA IN A GRAPHTWO-PERIOD BINOMIAL TREETWOPERIOD BINOMIAL TREE´ Concatenation of single-period trees:u2SuSSudSdSd2STWO-PERIOD BINOMIAL TREETWOPERIOD BINOMIAL TREE´ Example: S=50, u=2, d=0.5, R=1.252001002005025505012 5´ Option payoff:12.5150CuC0Cd0TWO-PERIOD BINOMIAL TREETWOPERIOD 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 before:5.05.025.025.1=−−=−−=dudRp´Three-step procedure:[]60)1(1=−+=uduuuCppCRCThreestep procedure:« 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..TWO-PERIOD BINOMIAL TREETWOPERIOD 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- ∆SCudSynthetic option requiresdynamic hedgingd(uddd)()Bd=Cd- ∆dSCdd´Synthetic option requires dynamic hedging« Must change the portfolio as stock price movesCddARBITRAGING A MISPRICED OPTIONARBITRAGING A MISPRICED OPTION´ Consider a 3-period tree with S=80, K=80, u=1.5, d05R11d=0.5, R=1.1´ Implies p = (R-d)/(u-d) = 0.6´Can dynamically replicate this option using 3period´Can dynamically replicate this option using 3-period binomial tree. Turns out that the cost is $34.08´ If the call is selling for $36, how to arbitrage?« Sell the real call« Buy the synthetic call´What do you get up front?´What do you get up front?« C-∆S+B = 36 – 34.08 = 1.92ARBITRAGING A MISPRICED OPTIONARBITRAGING A MISPRICED OPTION´ Suppose that one period goes by (2 periods from i ti ) d S 120 If l itiexpiration), and now S=120. If you close your position, what do you get in the following scenarios?« Call price equals “theoretical value”, $60.50.pq« Call price is less than 60.50« Call price is more than 60.50´Answer:´Answer:« Closing the position yields zero if call equals theoretical« If call price is less than 60.50, closing position yields more than zero since it is cheaper to buy back call.« If call price is more than 60.50, closing out position yields a loss! What do you do? (Rebalance and wait.)TOWARDS BLACK-SCHOLESTOWARDS BLACKSCHOLES´ Black-Scholes can be viewed as the limit of a binomial tree where the number of periods n goes to infinity´ Take parameters:´ Where:nTnTeudeu///1,σσ−===« n = number of periods in tree« T = time to expiration (e.g., measured in years)«σ = standard deviation of continuously compounded return«σ = standard deviation of continuously compounded return´ Also takenrTR/nrTeR/=TOWARDS BLACK-SCHOLESTOWARDS BLACKSCHOLES´ General binomial formula for a