Boyle (1988) proposed the following three-jump process for the approximation of the asset price process over one

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Boyle (1988) proposed the following three-jump process for the approximation of the asset price process over one period: 

Nature of jump up horizontal down Probability P1 P2 P3 Asset price us S ds

where S is the current asset price. The middle jump ratio m is chosen to be 1. There are five parameters in Boyle’s trinomial model: u,d and the probability values. The governing equations for the parameters can be obtained by: 

(i) Setting the sum of probabilities to be 1; and 

P1+ P2 + P3 = 1,

(ii) Equating the first two moments of the approximating discrete distribution and the corresponding continuous lognormal distribution 

Pu + P2 +p3d = erat = R Pu+ P2 + P3d - (pu + P2 + p3d) = er^t (eot - 1).

The last equation can be simplified as 

Pu + P2 + P3d = erAt eo At

The remaining two conditions can be chosen freely. They were chosen by Boyle (1988) to be 

and u= edo At ud = 1  is a free parameter.

By solving the five equations together, show that 

P1 (WR)u - (R  1) (u - 1)(u - 1) P3 = (WR)u (R-1)u (u - 1)(u - 1)

where W = R2eσ2Δt. Also show that Boyle’s trinomial model reduces to the Cox–Ross–Rubinstein binomial scheme when λ = 1.

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