RE: Ideal Asset Pricing Model to Value Derivatives

Typos corrected!

This would work for small Δt


(S(t) – S(t – 1) / S(t)) (1 – ξ Δt) = ά (μ* – μ) [S(t) ^ (Θ1 / 2 – 1)]Δt + (ρ – 1) S(t – 1) / S(t) + [S(t) ^ (Θ2 / 2 – 1)] σ1 Δ W + ή (κ + σ2 ΔB) / S(t)



0 <= ρ <= 1 (ρ = 1 and ξ = 0 and Θ1 = 2 then diffusion process)



The LHS can be augmented as:

(S(t) – S(t – 1) / S(t)) (1 – ξ exp(–Δt))



If in the discreatised version the time between sampling becomes considerable.



This also can be further augmented to capture return elasticity of drift.



(S(t) – S(t – 1) / S(t)) + ξ exp(–Δt) ((S(t - 1) – S(t – 2) / S(t - 1)) - (S(t) – S(t – 1) / S(t)))



i.e.,



(ΔS(t) / S(t)) + ξ exp(–Δt) ((ΔS(t - 1) / S(t - 1)) - (ΔS(t) / S(t)))



Best regards, Suminda Sirinath Salpitikorala Dharmasena