1 The multi period model

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1 The mlti perio moel. The moel setp In the mlti perio moel time rns in iscrete steps from t = to t = T, where T is a fixe time horizon. As before we will assme that there are two assets on the market, a bon with price process B t, an a stock with price process S t The bon price ynamics are given by B n+ = (+r)b n, B =, where r is the eterministic one perio interest rate. The stock price ynamics are given by S n+ = S n Z n, S = s, where Z,...,Z T are i.i.. ranom variables sch that P(Z n = ) = p, P(Z n = ) = p. The stock price ynamics can be illstrate with the following tree. s 2 s s s s s 2 As before, or goal is to se the moel for pricing an heging of financial erivatives. In the mlti perio moel the efinition of a financial erivative, or contingent claim, is the following. Definition A financial erivative or contingent claim is a ranom variable of the form X = φ(s T ), where the contract fnction φ is a real vale fnction. We will start by consiering a concrete example, an then we will make sre that we i the right thing afterwars.

2 Example Sppose that the parameters of the moel are r =, =.5, =.5, p =.6, an that we want to price a Eropean call option with strike price K= an exercise time T=3. This means that we are looking at a claim X sch that X = max{s T K,}. Jst like in the one perio moel it has to hol that Π(T;X) = X, or there will be arbitrage! So Π(3;X) = max{s 3,} = 9 if S 3 = 27, if S 3 = 9, if S 3 = 3, if S 3 =. The problem is to fin the price at time t < T, Π(t,X). How o we go abot oing this? Well, in the one perio moel it trne ot that one shol shol se risk netral valation. What if we se the one perio moel reslts an work or way backwars in the tree one step at a time? We shol se the martinagle probabilities which are obtaine from S() = +r EQ [S()]. or s = (q s+( q) s). +r Solving for q, an inserting or parameters we obtain q = (+r) =.5. Working or way backwars throgh the tree we get that the price of the option at time t = shol be Π(,X) = See the figre below The qestion is now if we have really one the right thing? Have we fon the arbitrage free price of the call option? Well, if we can fin a replicating portfolio h, sch that V h () =

3 then the answer is yes. To fin the replicating portfolio we start at t = an work or way forwar in the tree. To fin the replication portfolio we nee to fin h = (x,y) that solves V h () = φ(z ). This will reslt in For the first time step we have x = +r φ() φ() y = s φ() φ() φ(z ) = { 52.5 if Z =, 2.5 if Z =. Solving for the replicating portfolio we fin that h = ( 22.5,5/). The vale of the portfolio at time t = is V h () = = 27.5, which is eqal to the price of the option! So far, so goo. Now if we hol the portfolio an Z = the ol portfolio will be worth = 52.5 Solving V h (2) = φ(z ), starting from S = 2 will give s the portfolio h = ( 42.5,95/2). The vales of this new portfolio is = There is ths exactly enogh money to by the new portfolio once we have sol the ol one. This is very important, an we say that the rebalancing of the portfolio self-financing. Also note that the one perio moel is complete if the nmber of assets incling the bon is eqal to the nmber of otcomes in the sample space. For the mlti perio moel we nee intermeiary traing or the moel wol not be complete. Solving V h (2) = φ(z ), starting from S = 4 will give s the portfolio h = ( 2.5,/). The ol portfolio is now worth = 2.5 The new portfolio is worth = 2.5 The rebalancing is self-financing! We can now contine working or way forwar in the tree an will en p with the following pictre, an it is easy to check that portfoilo rebalancing is self-financing at every noe. 3

4 x=, y= 27 9 x= 42.5, y=95/2 x = 22.5, y=5/ x= 5, y=/ x= 2.5, y=/ x=, y= 2 We have therefore fon the arbirtage free price of the option at time t =, Π(,X) = 27.5, that is, since there is a self-financing portfolio replicating the payoff of the option with the same vale at time t =..2 Arbitrage an completeness Now we o the more formal analysis of the mlti perio moel. The biggest ifference is that we nee to introce the concept of self-financing portfolios an make sre that the portfolios are preictable. Here are the efinitions for the mlti perio moel. Definition 2 A portfolio strategy is a stochastic process {h t = (x t,y t ); t =,...,T} sch that h t σ{s,...,s t }. By convention h = h. We have that x t = nmber of SEK in the bank (t,t] y t = nmber of stocks yo own (t,t] The vale process V h corresponing to the portfolio strategy h is given by A portfolio strategy h is self-financing if V h t = x t (+r)+y t S t. x t (+r)y t S t = x t+ +y t+ S t t=,...,t- A portfolio strategy h is sai to be an arbitrage portfolio if h is self-financing an V h () =, ) =, > ) >. 4

5 The claim X = φ(s T ) is reachable if there exists a self-financing portfolio strategy h sch that V h T = φ(s T) with probability. If sch a portfolio h exists is sai to be a heging or replicating portfolio for the claim X. From now on we will assme that <. Proposition The moel is free of arbitrage (i.e. there exists no arbitrage portfolios) if an only if < (+r) <. The proof will be given at the en of Section.3. Jst as for the one perio moel we will have se for the martingale measre Q. Definition 3 Q is a martingale measre if. < q < (means Q P) 2. s = +r EQ [S t+ S t = s]. The expression for q will be the same as for the one perio moel. Proposition 2 The martingale probabilities are given by an q. q = (+r), Proposition 3 The mlti perio binomial moel is complete (i.e. all claims are reachable). Proof: The one perio moel + inction (see Example )..3 Pricing If X = φ(s T ) is replicate by the portfolio h then Π(t;X) = V h (t), t =,...,T. Every noe in the tree escribing the stock price evoltion is escribe by two inices (t,k) where t = time k = the nmber of p-steps We have that S t (k) = s k t k where k =,,...,t. 5

6 s 2 s s s s s 2 If we enote by V t (k) the vale of the replicating portfolio h at the noe (t,k), then V t (k) can be compte recrsively accoring to where V t (k) = V T (k) = φ(s k T k ), The replicating portfolio is given by x t (k) = y t (k) = +r {qv t(k +)+( q)v t (k)}, q = (+r). +r V t (k) V t (k +), V t (k+) V t (k). S t From the above algorithm we can obtain a risk netral valation formla. Proposition 4 The arbitrage free price at t = of the claim X is given by Π(;X) = V = (+r) T EQ [X] where Q enotes the martingale measre, or more explicitly ( ) T T Π(;X) = (+r) T q k ( q) T k φ(s k T k ) k k= Proof: To see that the secon formla is correct let Y = nmber of p-steps. Then Y Bin(T, q). We obtain X = φ(s T ) = φ(s Y T Y ) Take expectation to obtain the explicit formla. 6

7 Now we can prove Proposition, abscence of arbitrage. Proof: That abscence of arbitrage implies that < (+r) < follows from the corresponing reslt for the one perio moel. Sppose that < (+r) <, then we nee to prove abscence of arbitrage. For this prpose fix an arbitrary self-financing portfolio h sch that ) =, > ) >, h is then a potential arbitrage portfolio. From this an the risk netral valation formla we get that V h = (+r) T EQ [VT h ] >, which means that h is not an arbitrage portfolio. 7