δ j 1 (S j S j 1 ) (2.3) j=1

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1 Chapter The Binomial Model Let S be some tradable asset with prices and let S k = St k ), k = 0, 1,,....1) H = HS 0, S 1,..., S N 1, S N ).) be some option payoff with start date t 0 and end date or maturity t N. We want to replicate the option payoff.) with a suitable trading strategy in the underlying S. For notational simplicity let us assume first that we have zero interest rates r = 0. From the last chapter we know that a trading strategy holding δ k assets at the end of day t k generates the amount V N = V 0 + δ j 1 S j S j 1 ).3) Each δ k will be determined on the end of trading day t k. On such a day, the asset prices S 0, S 1,..., S k are known, but the asset prices S, S k+,..., S N are not known yet, they are lying in the future. Thus, δ k can be a function only of the known prices S 0,..., S k, δ k = δ k S 0, S 1,..., S k 1, S k ).4) Definition.1: We say that an option payoff.) can be replicated by a suitable trading strategy in the underlying if and only if there are choices of δ k of the form.4) and some initial amount V 0 such that in case of zero interest rates) HS 0, S 1,..., S N 1, S N ) = V 0 + δ j 1 S j S j 1 ).5) The initial amount V 0 which is needed to set up the replicating strategy is called the theoretical fair value of H. 11

2 1 Chapter Now let us consider the question to what extent replication of options is possible. Equation.5) can be rewritten as or HS 0, S 1,..., S N 1, S N ) = V N 1 + δ N 1 S N S N 1 ) HS 0, S 1,..., S N 1, S N ) δ N 1 S N = V N 1 δ N 1 S N 1 = some function of S 0, S 1,..., S N 1.6) That is, the right hand side of.6) is independent of S N. Let us introduce the return of the asset S from t k 1 to t k, such that Then equation.6) can be rewritten as ret k := S k S k 1 S k 1 = S k S k 1 1.7) S k = S k ret k ).8) H S 0, S 1,..., S N 1, S N ret N ) ) δ N 1 S N ret N ) = const.9) where the const in the equation above means that the left hand side of.9) has to be the same for all possible choices of ret N. Since there is only 1 free parameter in.9), namely δ N 1, we can only allow for possible choices for ret N, say, and in that case we have to have ret N {ret up, ret }.10) H S 0, S 1,..., S N 1, S N ret up ) ) δ N 1 S N ret up ) = H S 0, S 1,..., S N 1, S N ret ) ) δ N 1 S N ret ) which determines δ N 1 to δ N 1 = H S 0,..., S N 1, S N ret up ) ) H S 0,..., S N 1, S N ret ) ) S N ret up ) S N ret ).11) Thus, if we allow for possible choices of returns only as in.10), replication of option payoffs seem to be possible. This leads to the following

3 Chapter 13 Definition.: If the price process S k = St k ) of some tradable asset S has the dynamics S k = S k ret k ) with ret k {ret up, ret } k = 1,,....1) then we say that S is given by the Binomial model. Remark: Observe that in Definition. we have not introduced any probabilities for an up- or -move, that is, we have not specified a probability p up such that an up-return ret up will occur and a probability p = 1 p up for the occurence of a -return. We did that because the replicating strategy and the theoretical option fair value V 0 are actually independent of such probabilities. Nevertheless we have to remark that the definition of the Binomial model as given in the standard literature usually includes a specification of p up and p = 1 p up. Now we are in a position to formulate the following important Theorem.1: Let S be some tradable asset whose price process is given by the Binomial model.1). Let r 0 denote some constant interest rate. Then every option payoff H = HS 0,..., S N ) can be replicated. A replicating strategy is given by, for k = 0, 1,..., N 1: δ k = V S0,..., S k, S k 1 + ret up ) ) V S0,..., S k, S k 1 + ret ) ).13) S k 1 + ret up ) S k 1 + ret ) and the portfolio values V k, including the theoretical fair value V 0, can be inductively calculated through the formulae V k = 1 d k, d k, ret )V up 1 d k, d k, ret up )V.14) ret up ret V N = H where we used the abbreviations and introduced the discount factor V up, := V S0,..., S k, S k 1 + ret up, ) ).15) d k, := e rt t k ).16) Proof: For nonzero interest rates we have v N = v 0 + δ j 1 s j s j 1 )

4 14 Chapter which is equivalent to or, since v k = e rt k t 0 ) V k, v = v k + δ k s s k ) k = 0, 1,..., N 1 e rt t k ) V = V k + δ k e rt t k ) S k 1 + ret k ) S k ) Since we assume the price dynamics of the Binomial model, this equation is equivalent to ) e rt t k ) V up = V k + δ k e rt t k ) S k 1 + ret up ) S k ) e rt t k ) V = V k + δ k e rt t k ) S k 1 + ret ) S k Subtracting.18) from.17) gives.17).18) e rt t k ) V up V ) = δ k e rt t k ) S k 1 + ret up ) e rt t k ) S k 1 + ret ) ) or δ k = V up V S k 1 + ret up ) S k 1 + ret ).19) which coincides with.13). Substituting this value of δ k in equation.17) and solving for V k gives now using the abbreviation.16) for the discount factors) d k, V up δ ks k dk, 1 + ret up ) 1 ) = V k V k = d k, V up V up V dk, 1 + ret up ) 1 ) ret up ret = d k,v up ret up ret ) V up = V up = V up dk, 1 + ret ) 1 ) + V ret up ret 1 dk, 1 + ret ) ) V ret up ret and this coincides with.14). dk, 1 + ret up ) 1 ) + V dk, 1 + ret up ) 1 ) ret up ret dk, 1 + ret up ) 1 ) 1 dk, 1 + ret up ) ) Remarks: 1) If H is some then usually called european ) option which depends only on the underlying price at maturity, H = HS N ).0)

5 Chapter 15 then the δ k and the value of the replicating portfolio V k at t k depend only on the asset price S k and do not depend on earlier prices S k 1, S k,..., S 0. That is, V k = V k S k ).1) δ k = δ k S k ).) ) Assume zero interest rates such that d k, = 1. Then.14) becomes V k = ret V up + ret upv ret up ret = V up + V ret up + ret V up V.3) ret up ret If we further assume a symmetric Binomial model with ret up = q% and ret = q% we obtain the simple recursion formula V k = V up + V.4)