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The CRR Market Model 7: The CRR Market Model Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2016 7: The CRR Market Model Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein Market Model 2 The CRR Call Option Pricing Formula 3 Call and Put Options of American Style 4 Dynamic Programming Approach to American Claims 5 Examples: American Call and Put Options 7: The CRR Market Model PART 1 THE COX-ROSS-RUBINSTEIN MARKET MODEL 7: The CRR Market Model Introduction The Cox-Ross-Rubinstein market model (CRR model) is an example of a multi-period market model of the stock price. At each point in time, the stock price is assumed to either go ‘up’ by a fixed factor u or go ‘down’ by a fixed factor d. Only three parameters are needed to specify the binomial asset pricing model: u > d > 0 and r > 1. − Note that we do not postulate that d < 1 < u. The real-world probability of an ‘up’ mouvement is assumed to be the same 0 < p < 1 for each period and is assumed to be independent of all previous stock price mouvements. 7: The CRR Market Model Bernoulli Processes Definition (Bernoulli Process) A process X = (Xt )1 t T on a probability space (Ω, , P) is called ≤ ≤ F the Bernoulli process with parameter 0 < p < 1 if the random variables X1, X2,..., XT are independent and have the following common probability distribution P(Xt = 1) = 1 P(Xt =0)= p. − Definition (Bernoulli Counting Process) The Bernoulli counting process N = (Nt )0 t T is defined by setting N = 0 and, for every t = 1,..., T and≤ ≤ω Ω, 0 ∈ Nt (ω) := X (ω)+ X (ω)+ + Xt (ω). 1 2 · · · The process N is a special case of an additive random walk. 7: The CRR Market Model Stock Price Definition (Stock Price) The stock price process in the CRR model is defined via an initial value S > 0 and, for 1 t T and all ω Ω, 0 ≤ ≤ ∈ Nt (ω) t Nt (ω) St (ω) := S0u d − . The underlying Bernoulli process X governs the ‘up’ and ‘down’ mouvements of the stock. The stock price moves up at time t if Xt (ω) = 1 and moves down if Xt (ω) = 0. The dynamics of the stock price can be seen as an example of a multiplicative random walk. The Bernoulli counting process N counts the up mouvements. Before and including time t, the stock price moves up Nt times and down t Nt times. − 7: The CRR Market Model Distribution of the Stock Price For each t = 1, 2,..., T , the random variable Nt has the binomial distribution with parameters p and t. Specifically, for every t = 1,..., T and k = 0,..., t we have that t k t k P(Nt = k)= p (1 p) − . k − Hence the probability distribution of the stock price St at time t is given by k t k t k t k P(St = S u d − )= p (1 p) − 0 k − for k = 0, 1,..., t. 7: The CRR Market Model Stock Price Lattice 0.8 S u4 0 0.6 ... S u3 0 0.4 S u2 S u3d 0 0 0.2 S u S u2d 0 0 0 ... S S ud S u2d2 0 0 0 −0.2 S d S ud2 0 0 −0.4 3 S d2 S ud 0 0 −0.6 ... S d3 0 −0.8 S d4 0 −1 Figure: Stock Price Lattice in the CRR Model 7: The CRR Market Model Risk-Neutral Probability Measure Proposition (7.1) Assume that d < 1+ r < u. Then a probability measure P on (Ω, T ) is a risk-neutral probability measure for the CRR model F e = (B, S) with parameters p, u, d, r and time horizon T if and onlyM if: 1 X1, X2, X3,..., XT are independent under the probability measure P, 2 P 0 < p :=e (Xt = 1) < 1 for all t = 1,..., T , 3 pu + (1 p)d = (1+ r), e −e where X is the Bernoulli process governing the stock price S. e e 7: The CRR Market Model Risk-Neutral Probability Measure Proposition (7.2) If d < 1+ r < u then the CRR market model = (B, S) is arbitrage-free and complete. M Since the CRR model is complete, the unique arbitrage price of any European contingent claim can be computed using the risk-neutral valuation formula E X πt (X )= Bt P t BT F e We will apply this formula to the call option on the stock. 7: The CRR Market Model PART 2 THE CRR CALL OPTION PRICING FORMULA 7: The CRR Market Model CRR Call Option Pricing Formula Proposition (7.3) The arbitrage price at time t = 0 of the European call option + CT = (ST K) in the binomial market model = (B, S) is given by the− CRR call pricing formula M T T T k T k K T k T k C = S pˆ (1 pˆ) − p (1 p) − 0 0 k − − (1 + r)T k − Xk=kˆ Xk=kˆ e e where 1+ r d pu p = − , pˆ = u d 1+ r − e and kˆ is the smalleste integer k such that u K k log > log T . d S0d 7: The CRR Market Model Proof of Proposition 7.3 Proof of Proposition 7.3. + The price at time t = 0 of the claim X = CT = (ST K) can be − computed using the risk-neutral valuation under P 1 e C = E (CT ) . 0 (1 + r)T P e In view of Proposition 7.1, we obtain T 1 T k T k k T k C = p (1 p) − max 0, S u d − K . 0 (1 + r)T k − 0 − Xk=0 e e We note that k k T k u K S0u d − K > 0 > T − ⇔ d S0d u K k log > log T ⇔ d S0d 7: The CRR Market Model Proof of Proposition 7.3 Proof of Proposition 7.3 (Continued). We define kˆ = kˆ(S0, T ) as the smallest integer k such that the last inequality is satisfied. If there are less than kˆ upward mouvements there is no chance that the option will expire worthless. Therefore, we obtain T 1 T k T k k T k C = p (1 p) − S u d − K 0 (1 + r)T k − 0 − Xk=kˆ T e e S0 T k T k k T k = p (1 p) − u d − (1 + r)T k − Xk=kˆ T e e K T k T k p (1 p) − − (1 + r)T k − Xk=kˆ e e 7: The CRR Market Model Proof of Proposition 7.3 Proof of Proposition 7.3 (Continued). Consequently, T k T k T pu (1 p)d − C = S − 0 0 k 1+ r 1+ r Xk=kˆ e e T K T k T k p (1 p) − − (1 + r)T k − Xk=kˆ e e and thus T T T k T k K T k T k C = S pˆ (1 pˆ) − p (1 p) − 0 0 k − − (1 + r)T k − Xk=kˆ Xk=kˆ e e where we denotep ˆ = pu . 1+e r 7: The CRR Market Model Martingale Approach (MATH3975) pu 1+r d Check that 0 < pˆ = 1+r < 1 whenever 0 < p = u −d < 1. e − P Let be the probability measure obtained bye setting p = p in Proposition 7.1. Then the process B is a martingale under P. b S b For t = 0, the price of the call satisfies b C = S P(D) KB(0, T ) P(D) 0 0 − b e where D = ω Ω : ST (ω) > K . { ∈ } Recall that 1 + Ct = Bt E B− (ST K) t P T − | F e Using the abstract Bayes formula, one can show that Ct = St P(D t ) KB(t, T ) P(D t ). | F − | F b e 7: The CRR Market Model Put-Call Parity It is possible to derive explicit pricing formula for the call option at any date t = 0, 1,..., T . Since CT PT = ST K, we see that the following put-call − − parity holds at any date t = 0, 1,..., T (T t) Ct Pt = St K(1 + r)− − = St KB(t, T ) − − − where (T t) B(t, T )=(1+ r)− − is the price at time t of zero-coupon bond maturing at T . Using Proposition 7.3 and the put-call parity, one can derive an explicit pricing formula for the European put option with + the payoff PT = (K ST ) . This is left as an exercise. − 7: The CRR Market Model PART 3 CALL AND PUT OPTIONS OF AMERICAN STYLE 7: The CRR Market Model American Options In contrast to a contingent claim of a European style, a claim of an American style can by exercised by its holder at any date before its expiration date T . Definition (American Call and Put Options) An American call (put) option is a contract which gives the holder the right to buy (sell) an asset at any time t T at strike price K. ≤ It the study of an American claim, we are concerned with the price process and the ‘optimal’ exercise policy by its holder. If the holder of an American option exercises it at τ [0, T ], ∈ τ is called the exercise time. 7: The CRR Market Model Stopping Times An admissible exercise time should belong to the class of stopping times.
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