A New Logistic-Type Model for Pricing European Options

Londoño, Jaime A.; Sandoval, Javier

doi:10.2139/ssrn.2580670

Cited by 1 publication

(1 citation statement)

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“…( 2) is the stochastic Verhulst or stochastic logistic model, which are wellknown in the population dynamics and epidemiology; see, eg., (Verhulst, 1838;Bacaer, 2011;Giet et al, 2015) and references therein. In the past, several authors attempted to use this model in finance; see, eg., (Chen, 2010;Londono and Sandoval, 2015;Halperin and Feldshteyn, 2018). In our case, the stochastic Verhulst equation has the form…”

Section: The Modified Bk (Verhulst) Modelmentioning

confidence: 90%

From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy

Itkin,

Lipton,

Muravey

2020

Preprint

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In this paper, we argue that some of the most popular short-term interest models have to be revisited and modified to reflect current market conditions better. In particular, we propose a modification of the popular Black-Karasinski model, which is widely used by practitioners for modeling interest rates, credit, and commodities. Our adjustment gives rise to the stochastic Verhulst model, which is well-known in the population dynamics and epidemiology as a logistic model. We demonstrate that the Verhulst model's dynamics are well suited to the current economic environment and the Fed's actions. Besides, we derive new integral equations for the zero-coupon bond prices for both the BK and Verhulst models. For the BK model for small maturities up to 2 years, we solve the corresponding integral equation by using the reduced differential transform method. For the Verhulst integral equation, under some mild assumptions, we find the closed-form solution. Numerical examples show that computationally our approach is significantly more efficient than the standard finite difference method.1. The diffusion rt is recurrent if and only if q ≤ 0, where q = 1 2 − κ(t) θ(t) σ 2 (t) . 2. If q < 0, the diffusion rt converges in law towards the unique stationary Gamma probability distribution γ −2q, σ 2 (t)/(2κ(t) t→∞ . 3. If q > 0, the diffusion goes a.s. to zero when time goes to infinity.Proof. The proof can be obtained by applying the Itô's lemma to Eq. ( 5) and using Eq. ( 6). The second part follows from Proposition 3.3 in (Giet et al., 2015). It is interesting to note, that the condition q < 1/2 is precisely the Feller condition for the famous CIR model, (Andersen and Piterbarg, 2010).

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Section: The Modified Bk (Verhulst) Modelmentioning

confidence: 90%