Electronic Companion Option Pricing under a Mixed-exponential Jump Diffusion Model

Proof. Without loss of generality, we assume that pu > 0, qd > 0, pi 6= 0 for i = 1, · · · ,m, and qj 6= 0 for j = 1, · · · , n. First of all, it is easily seen that G(x) − α has the same roots as (G(x)−α) mi=1(x−ηi) ∏n j=1(x+ θj), which is a polynomial with order m+n+2. This implies that for any α ∈ R, the function G(x) = α has at most (m + n + 2) real roots. From now on we shall show that for sufficiently large α > 0, the function has exactly (m + n + 2) real roots, among which m + 1 are positive and n + 1 are negative. Due to the symmetry, we will focus only on arguing that for sufficiently large α > 0, the function has (m + 1) positive roots. Note that there exist m positive singularities η1, · · · , ηm for the function G(x), which divide the positive real line into (m + 1) disjoint intervals: (0, η1), (η1, η2), · · · , (ηm−1, ηm), (ηm,+∞). Since G(η1−) = +∞ (because p1 > 0) and G(0) − α = −α, we know that for any α > 0, G(x) = α has at least one real root on the interval (0, η1). We plan to show there exist m real roots on the other m intervals for sufficiently large α > 0. For convenience, we define pm+1 := +∞, ηm+1 = +∞, and two sets S+ and S− as follows S := {i ∈ {1, · · · ,m} : pi > 0 and pi+1 < 0}, S− := {i ∈ {1, · · · ,m} : pi < 0 and pi+1 > 0}. Noting that p1 > 0 and pm+1 = +∞ > 0, we can easily see that the number of elements in S+ and that in S− are identical. Moreover, if the number of elements is k > 0 and if we assume