Constant Elasticity of Variance (
            <scp>CEV</scp>
            ) Diffusion Model

Linetsky, Vadim; Mendoza, Rafael

doi:10.1002/9780470061602.eqf08015

Cited by 18 publications

(16 citation statements)

References 19 publications

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“…In particular, we explicitly provide the closed-form expressions for the first four terms. Numerical experiments suggest that the corresponding expansion formula up to the third order performs very well for a broad range of diffusion models, including not only those satisfying these regularity conditions such as the BSM and the Brennan-Schwartz process, but also those violating them, e.g., the CIR model (see [17]) and the general CEV models (see, e.g., [16], [45] and [47]).…”

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confidence: 99%

Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models

Cai

Shi

2014

Mathematics of OR

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In this paper we propose a closed-form asymptotic expansion approach to pricing discretely monitored Asian options in general one-dimensional diffusion models. Our expansion is a small-time expansion because the expansion parameter is selected to be the square root of the length of monitoring interval. This expansion method is distinguished from many other pricing-oriented expansion algorithms in the literature due to two appealing features. First, we illustrate that it is possible to explicitly calculate not only the first several expansion terms but also any general expansion term in a systematic way. Second, the convergence of the expansion is proved rigorously under some regularity conditions. Numerical experiments suggest that the closed-form expansion formula with only a few terms (e.g., four terms up to the third order) is accurate, fast, and easy to implement for a broad range of diffusion models, even including those violating the regularity conditions.

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confidence: 99%

Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models

Cai

Shi

2014

Mathematics of OR

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“…Essentially, for processes governed by equations of the form dXt=f(Xt,t)dt+g(Xt-,t-)dBt, the integral ∫ g ( X t − , t − ) dB t can acquire non-zero expectation if the function g grows too quickly. A classic example is the CEV model of quantitative finance [31], which is of the form f ( X t , t ) = X t and g(Xt,t)=Xtγ for γ > 1. However, one can use the fact that the exponential version of the LES model, i.e.…”

Section: Appendixmentioning

confidence: 99%

Stationary moments, diffusion limits, and extinction times for logistic growth with random catastrophes

Schlomann

2018

Journal of Theoretical Biology

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A central problem in population ecology is understanding the consequences of stochastic fluctuations. Analytically tractable models with Gaussian driving noise have led to important, general insights, but they fail to capture rare, catastrophic events, which are increasingly observed at scales ranging from global fisheries to intestinal microbiota. Due to mathematical challenges, growth processes with random catastrophes are less well characterized and it remains unclear how their consequences differ from those of Gaussian processes. In the face of a changing climate and predicted increases in ecological catastrophes, as well as increased interest in harnessing microbes for therapeutics, these processes have never been more relevant. To better understand them, I revisit here a differential equation model of logistic growth coupled to density-independent catastrophes that arrive as a Poisson process, and derive new analytic results that reveal its statistical structure. First, I derive exact expressions for the model's stationary moments, revealing a single effective catastrophe parameter that largely controls low order statistics. Then, I use weak convergence theorems to construct its Gaussian analog in a limit of frequent, small catastrophes, keeping the stationary population mean constant for normalization. Numerically computing statistics along this limit shows how they transform as the dynamics shifts from catastrophes to diffusions, enabling quantitative comparisons. For example, the mean time to extinction increases monotonically by orders of magnitude, demonstrating significantly higher extinction risk under catastrophes than under diffusions. Together, these results provide insight into a wide range of stochastic dynamical systems important for ecology and conservation.

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“…The CEV diffusion of Cox () is a non‐negative diffusion process with the instantaneous volatility

σ (x) = a x^{β},

where

β < 0

is the volatility elasticity parameter and

a > 0

is the volatility scale parameter (see Schroder ; Davydov and Linetsky , ; Linetsky and Mendoza ), and

k (x) \equiv 0

in equation . The negative specification of the volatility elasticity parameter is consistent with the leverage effect (volatility increases when the stock price falls) and results in the implied volatility skew in stock options in this model.…”

Section: Examples Of Financial Applicationsmentioning

confidence: 99%

Multivariate Subordination of Markov Processes With Financial Applications

Mendoza-Arriaga

Linetsky²

2014

Mathematical Finance

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This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes X i with killing and an independent d-dimensional time change T , we construct a new process by time, changing each of the Markov processes X i with a coordinate . This construction provides a rich modeling architecture for building multivariate models in finance with time-and state-dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit-equity models and correlated commodity models.

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Constant Elasticity of Variance ( CEV ) Diffusion Model

Abstract: The constant elasticity of variance (CEV) model is a one-dimensional diffusion process that solves a stochastic differential equation (SDE) dS t = µS t dt + aS β+1 t

Cited by 18 publications

References 19 publications

Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models

Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models

Stationary moments, diffusion limits, and extinction times for logistic growth with random catastrophes

Multivariate Subordination of Markov Processes With Financial Applications

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