Binomial Tree for Two Assets A two-asset binomial tree with (x, y)-coordinates representing the assets, and time-coordinate t, is assumed to develop as follows: Each node with position (x, y) may develop for t → t + Δt with equal probabilities 0.25 to the four positions (xu, yA), (xu, yB), (xd, yC), (xd, yD) (∗) for constants u,

d, A, B, C, D.

a) Show that the tree is recombining for AD = BC. Hint: Sketch the possible values in a (x, y)-plane. Following [Rub94b], a tree is defined for interest rate r, asset parameters σ1, σ2, correlation ρ, and dividend rates δ1, δ2, by μi := r − δi − σ2 i /2 for i = 1, 2 u := exp(μ1Δt + σ1 √ Δt) d := exp(μ1Δt − σ1 √ Δt) A := exp(μ2Δt + σ2 √ Δt[ρ + 1 − ρ2]) B := exp(μ2Δt + σ2 √ Δt[ρ − 1 − ρ2]) C := exp(μ2Δt − σ2 √ Δt[ρ − 1 − ρ2]) D := exp(μ2Δt − σ2 √ Δt[ρ + 1 − ρ2]) For initial prices x0 := S0 1 , y0 := S0 2 , and time level tν := νΔt, the S1- components of the grid according to (∗) distribute in the same way as for the one-dimensional tree, xν i := S0 1ui dν−i for i = 0,...,ν.

b) Show that the second (S2-)components belonging to xν i are yν i,j := S0 2 exp(μ2νΔt) exp  σ2 √ Δt  ρ(2i − ν) + 1 − ρ2(2j − ν)  . for j = 0,...,ν. Hints: For ν → ν + 1, u corresponds to i → i + 1, and d corresponds to i → i; C exp(2σ2 √ Δt) = B.

c) Set up a computer program that implements this binomial method. Analogously as in Section 1.4 work in a backward recursion for ν = M,..., 0. For each time level tν set up the (x, y)-grid with the above rules and Δt = T /M. For tM = T fix V by the payoff Ψ, and use for ν