(剩下五天了, 還沒把東西全消化, 更不用說背重點了, 今次悲哀了)

今天做練習時發現了有矛盾的地方, 想各位看看我有沒有logical error
Financial Theory and Corporate Policy 關於capital structure那一課
其中有一部份是用option pricing model 來計算beta 的
書derivation:
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Ito's lemma: dC = (delta)dS +0.5 (gamma) (dS)^2 +(theta)dt
(dS)^2 proprotional to dt, let dt->0
dC= (delta) dS
dC/C = (delta)/C dS=(delta) (S/C) (dS/S)
return of call = rc
return of stock = rs
return of market = rm
rc = N(d1)(S/C) rs
but Beta(call) = Cov(rc, rm)/Var(rm) = Cov (N(d1)(S/C) rs, rm)/Var(rm) = N(d1)(S/C) Cov(rs, rm)/Var(rm) = N(d1)(S/C) Beta(stock)
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問題:
i) by rc = N(d1)(S/C) rs, rc = N(d1)(S/C) (rf + Beta(S) [rm-rf] )
ii) by Beta(call) = N(d1)(S/C) Beta(stock), rc' = rf + Beta(C) [rm-rf] = rf +  N(d1)(S/C) Beta(stock) [rm-rf]
then rc' - rc = rf (1 - N(d1)(S/C)) =/= 0 -> contradiction

其實我認為 let dt->0 是有問題的, 不過書這樣寫我也沒有辦法
有人能來解答嗎?