Probability Density of the CIR Model

Vanyolos, Andras; Cho, Maxx; Glasgow, Scott

doi:10.2139/ssrn.2508699

Cited by 2 publications

(3 citation statements)

References 13 publications

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“…It has a unique strong solution. This equation is not explicitly solvable, but its transition density is explicitly known (Vanyolos et al, 2014) and its moments and distributions have been intensively studied.…”

Section: Scenario (I): Strong Selection Arbitrary Sample Sizementioning

confidence: 99%

Latent mutations in the ancestries of alleles under selection

Fan¹,

Wakeley

2023

Preprint

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We consider a single genetic locus with two alleles A_1 and A_2 in a large haploid population. The locus is subject to selection and two-way, or recurrent, mutation. Assuming the allele frequencies follow a Wright-Fisher diffusion and have reached stationarity, we describe the asymptotic behaviors of the conditional gene genealogy and the latent mutations of a sample with known allele counts, when the count n_1 of allele A_1 is fixed, and when either or both the sample size n and the selection strength |β| tend to infinity. Our study extends previous work under neutrality to the case of non-neutral rare alleles, asserting that when selection is not too strong relative to the sample size, even if it is strongly positive or strongly negative in the usual sense (β → -∞ or β → +∞), the number of latent mutations of the n_1 copies of allele A_1 follows the same distribution as the number of alleles in the Ewens sampling formula. On the other hand, very strong positive selection relative to the sample size leads to neutral gene genealogies with a single ancient latent mutation. We also demonstrate robustness of our asymptotic results against changing population sizes, when one of |β| or n is large.

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Section: Scenario (I): Strong Selection Arbitrary Sample Sizementioning

confidence: 99%

Latent mutations in the ancestries of alleles under selection

Fan¹,

Wakeley

2023

Preprint

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“…T ] and z > 0, where y → f (v − t, r t , x, y) is the probability density function of xZ rt v−t (see [3], [13] ) given by…”

Section: The Arbitrage-free Price and The Rational Exercise Boundarymentioning

confidence: 99%

“…and δ(x − y) is the delta function (see [13]). Thus, J(t, r r , x, v, z) > 0 for all (t, r t , x) ∈ D.…”

Section: The Arbitrage-free Price and The Rational Exercise Boundarymentioning

confidence: 99%

British Call Option on Stocks under Stochastic Interest Rate

Falcasantos

Sumalpong²

2022

Eur. J. Pure Appl. Math.

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The closed form expression for the price of the British put and call options have longbeen established where both interest rate and volatility are assumed to be constant. In reality,these assumptions do not fully reflect the variable nature of the financial markets. In this paper, wederived a closed form expression for the arbitrage-free price of the British call option by assumingstochastic interest rate which follows the Cox-Ingersoll-Ross model and constant volatility as where the first term is the arbitrage-free price of the European call option under stochastic interestrate and the second term is the early-exercise premmium. We have also shown that the pricefunction of the British call option satisfies the partial differential equation given by Moreover, we have shown that the contract drift satisfies μc < rt+ρσ1σ2√rtλ(0, t+u) for u ∈ [0, τ ] and t ∈ [0, T].

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Probability Density of the CIR Model

Cited by 2 publications

References 13 publications

Latent mutations in the ancestries of alleles under selection

Latent mutations in the ancestries of alleles under selection

British Call Option on Stocks under Stochastic Interest Rate

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