# 3b - multistep binomial model - secondary market

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Multi Period Binomial Model - Secondary Market Now we are going to add one contingent claim (derivative) to the multi period binomial Market. In this section, we start by adding a European contingent claim Stock-Bond-Contingent Claim Market Consider a financial market in which, in addition to stock and bond, there is one European contingent claim that can be traded. ® (: Price of the contingent claim at time t = 0 e T:expiration time of the contingent claim e X: The final value of the contingent claim at the expiratointimet =T ¢ Ingeneral, X can depend on the stock price during the whole interval t = 0,1,2,..,T (and not only the final stock price St) X = F(So, 51, 52, ,ST) Example 1. If X is a European call option with expiration time T and strike price K: T . X = (S-r - KB P S?C—OQ' G' )( Qv\\a ée?tué\ an S.'. Example 2. If X is a European put option with expiration time T and strike price K: X: (,k' S'T \* 3?'—0&' G' )( Qw\a éd»]mé\ an S-.\. Example 3. We can potentially consider a European contingent claim with expiration time T=4 whose final value is defined by X = Max(S3,S4) Example 4. Consider a binomial model with S = 4,d = 0.5,u = 3,r = 0.5. Let X be a European contingent claim with expiration time T=2. For the following cases, find all possible final values of X. .k (a) X is a European call option with strike price K=5. X: KSJ__":)\ X __w\fiX(Sq,'al 4") (b) X = Max (Sg, S1, S2) o PRI, SRV KT RV SRUIe! (R § NS Pt ¥ s L3N - \ \ Sy - w88z by (hn)L () __Mx(h,n,L)_ n SQ=L(' S 14y 6 ' RN 17 SXD) = £ 1ea) s\ (Ue) =mere( §, L8 = \s.-."%-'L A - e = _ \ . - Lx;'— \ Xs(\:S)-'Fo {(uj\a"'"'""'"l"\fq'
Replication strategy in multi period binomial model The idea is similar to the single period case: Definition. Let X be the final value of European contingent claim with expiration time T. A replicationg strategy for X is a trading strategy @™ = {(a], f1), (@3, 53), ..., (ar, B)} in the pimary market such that Vr(@®) =X [for all possible outcomes] How to find replicating strategies? Idea: Move backwards in the graph. Use what we know from single period. Example. Consider a binomial model with T=2, §; = 100, u=1.5, d=0.5, r=0.2 , and let X be the European put option with expiration time T=2 and strike price K=90. Find a replicating strategy for X, and find the manufactring price that replicating strategy. Ros ) .12 Byt X= (10-8, Y' (0'5_. Q,_\ T, \ s, - 75 , X=5 x\u ?\'-7. SL" Jz)_ ) X: 'q \$,:5€ / o~ . (""\- ?1\1'7. \ V\z". 57,:1;: X'-é')' Tl Y, WSe) s Ty =Y
S,.150 / SL= 215, Vy=o S\ \L o0 \ S, : 45 V,-15 . X Vo =4, x o-1H o(';, = = -0, 115 -15 @.b& ) 0.\ x 25 (SLZ 1.4 < X Vi (f) = & %'*' B, oy« LUL P = (5 zr{o(x* ¥ LGk ?; = b5 EY oA - _\ , T - 6L1.5