Consider a typical term in g(x,t) where can be either 2n(u ) or 2 +

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Consider a typical term in g(x,t) 

exp(  44) exp( - 2017). 202 (x - )2 221

where ξ can be either 2n(u − ℓ) or 2ℓ + 2n(u − ℓ). Show that the above term can be rewritten as

eo 221 exp -(x - - ut) . 201

Hence, show that 

= f" 8(x g(x, t) dx  exp n=- - - 2u(u - l) l - - 2n(u l)  [x("0 3014 12)-N( 2 300) N - exp (2n(u - l) 02 2[l

Use the above result to derive the price formula of the European double knockout call option coLU [see Sect. 4.1.3].

We generalize the barriers to become exponential functions in time. Suppose the upper and lower barriers are set to be Ueδ1t and Leδ2t ,t ∈ [0,T ]. Here, δ1 and δ2 are constant parameters and the barriers do not intersect over [0,T ]. Show that the price formula of the European double knock-out call option can be expressed as (Kunitomo and Ikeda, 1992)

CLU= = S where n=- {(F)"(5) - (US) - - Xe-T d = d3 Ln+1 3-2 Uns [N(d3) - N(d4)] Un 141-2 L {( {()^ (5) **

f1= H3 143 = 2[r 82 n(81-82)] 02 2[r 82 + n(8182)] 02 - - +1, 81 - 82 y 22n- +1, F = UeT

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