stochastic.225

# stochastic.225 - 224 2 3 7 1 6 r X1 !1  = 1 r 6 1 d X2...

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Unformatted text preview: 224 2 3 7 1 6 r X1 !1  = 1 + r 6 1 + , , d X2 !z ; H  + u , 1 + r X2!z1; T 7 ; 4 u d | 1 5 u,d | V ! ;H  V ! ;T  1 + r , d u , 1 + r X T  ; 1 X0 = 1 + r u , d X1H  + u , d 1 ! T 1 !1  = X2 !1 ; H  , X2! 1;;T  ; S2 !1; H  , S2 1 X1 H  , X1T  : 0 = S H  , S T  1 1 1 1 The probabilities of the stock price paths are irrelevant, because we have a hedge which works on every path. From a practical point of view, what matters is that the paths in the model include all the possibilities. We want to ﬁnd a description of the paths in the model. They all have the property log Sk+1 , log Sk 2 = Sk+1 2 log Sk =  log u2 = log u2 : Let = log u 0. Then n,1 X k=0 log Sk+1 , log Sk 2 = 2 n: The paths of log Sk accumulate quadratic variation at rate 2 per unit time. If we change u, then we change , and the pricing and hedging formulas on the previous page will give different results. We reiterate that the probabilities are only introduced as an aid to understanding and computation. Recall: Xk+1 = k Sk+1 + 1 + rXk , k Sk : Deﬁne k Then Xk+1 =  Sk+1 + Xk ,  Sk ; k k k+1 i.e., = 1 + rk : k+1 k k Xk+1 , Xk =  Sk+1 , Sk : k k+1 k k+1 In continuous time, we will have the analogous equation k t d Xtt = t d St : ...
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