Return Elasticity of Drift

This would represent the mean reversion factor for the process

E.g.

ΔS(t) = ρ ΔS(t – 1) + ε

ΔS(t) – ΔS(t – 1) = ρ ΔS(t – 1) – ΔS(t – 1) + ε

ΔΔS(t) = (ρ – 1) ΔS(t – 1) + ε


As the sapling interval becomes smaller

ρ ≈ 1 – c Δt (Assuming an exponentially decaying function proportional to the sampling interval)

Then,

ΔΔS(t) = – c ΔS(t - 1) Δt + ε

Best regards, Suminda Sirinath Salpitikorala Dharmasena

Serial Correlation

As I think, this would represent the serial correlation for the process

E.g.

S(t) = ρ S(t – 1) + ε

S(t) – S(t – 1) = ρ S(t – 1) – S(t – 1) + ε

ΔS(t) = (ρ – 1) S(t – 1) + ε


As the sapling interval becomes smaller

ρ ≈ 1 – c Δt (Assuming an exponentially decaying function proportional to the sampling interval)

S(t – 1) -> S(t)

Then,

ΔS(t) = – c S(t) Δt + ε


Similarly,

ΔS(t) = (ρ – 1) S(t – 1) + μ S(t – 1) Δt + ε

Becomes

ΔS(t) = (μ – c) S(t) Δt + ε

Best regards, Suminda Sirinath Salpitikorala Dharmasena

P.S. I am not a mathematician so please help me if there is a problem with my thinking.

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

RE: Ideal Asset Pricing Model to Value Derivatives

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

Correction to Previous Post Equation

(S(t) – S(t – 1) / S(t)) (1 – ξ exp(-Δt ^ λ)) = 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)

Ideal Asset Pricing Model to Value Derivatives

Generally accepted theoretical consensus is that asset pricing is a random walk.




But for all practical purposes this might not be true. A practical scheme must capture:

• Mean reversion trends

• Serial correlation

• Price elasticity of variance

• Stochastic volatility

• Return correction component

• Jumps



And the effects of the above change with continuously. Therefore, the calibration will also need to be dynamic.



Modeling all this in a mathematically proven model is difficult, but still computer simulations can be done using soft techniques which profile the needed parameters given the situation. The parameters estimation can be done based on situational set for each parameter. The ideal method to do this is subjected to further investigation.



The type of process which needs representing in discretized for is:



(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)



Best regards, Suminda Sirinath Salpitikorala Dharmasena

Annualizing Retunes deriving the Variance produce different results from calculating the Variance and Annualizing it

Annualizing returns and variance seam to have an odd effect. Annualize the returns and calculate the variance produces different results than calculating the variance and annualizing.