Consider a down-and-in American call C di (S, ; X,B), where the down-and-in trigger clause entitles

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Consider a down-and-in American call Cdi (S, τ ; X,B), where the down-and-in trigger clause entitles the holder to receive an American call option with strike price X when the asset price S falls below the threshold level B. The underlying asset pays dividend yield q and let r denote the riskless interest rate. Let C(S,τ ; X) and c(S,τ ; X) denote the price function of the American call and European call with strike price X, respectively. Show that when B ≤ max (X, r/q X), we have

Cdi (S, T; X, B) 2(r-q) 1- B2 where cdi (S, τ ; X,B) is the price function of the European down-and-in call counterpart. Find the corresponding form of the price function Cdi (S, τ ; X,B) when (i) B ≥ S(∞) and (ii) S(0+) ∗(∞), where S(τ) is the optimal exercise boundary of the American non-barrier call C(S,τ ; X) (Dai and Kwok, 2004). 

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