Bubbles and crashes in a Black–Scholes model with delay

Appleby, John A. D.; Riedle, Markus; Swords, Catherine

doi:10.1007/s00780-012-0181-4

Cited by 8 publications

(16 citation statements)

References 22 publications

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“…Our results suggest that large shocks to the price transmit quickly to the system in that case, despite the fact that the traders may use a lot of information about the past of the system. Models of this type include [1,2,4,9].…”

Section: Time Indexing In the Volterra Equationmentioning

confidence: 99%

On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

Appleby¹,

Patterson²

2016

Electron. J. Qual. Theory Differ. Equ.

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In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state-dependence in the summation is of smaller than linear order for large absolute values of the state. When the perturbation term is unbounded, it is elementary to show that solutions are also. The main results of the paper are mostly of the following form: the solution has an additional unboundedness property U if and only if the perturbation has property U . Examples of property U include monotone growth, monotone growth with fluctuation, fluctuation on R without growth, existence of time averages. We also study the connection between the times at which the perturbation and solution reach their running maximum, and the connection between the size of signed and unsigned running maxima of the solution and forcing term.

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Section: Time Indexing In the Volterra Equationmentioning

confidence: 99%

On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

Appleby¹,

Patterson²

2016

Electron. J. Qual. Theory Differ. Equ.

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“…The MT model is one of several published models that account for memory in the underlying asset [6][7][8][9][10][11][12][13]. Within this set of models, there are two reasons why the MT model is the simplest: (i) the model requires the estimation of only three parameters; and (ii) the MT model is fundamentally a discrete-time model, allowing us to bypass the technical difficulties of estimation and filtering for stochastic delay/functional differential equations.…”

mentioning

confidence: 99%

Large-Scale Empirical Tests of the Markov Tree Model

Bhat

Kumar²

2015

IJFS

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The Markov Tree model is a discrete-time option pricing model that accounts for short-term memory of the underlying asset. In this work, we compare the empirical performance of the Markov Tree model against that of the Black-Scholes model and Heston's stochastic volatility model. Leveraging a total of five years of individual equity and index option data, and using three new methods for fitting the Markov Tree model, we find that the Markov Tree model makes smaller out-of-sample hedging errors than competing models. This comparison includes versions of Markov Tree and Black-Scholes models in which volatilities are strike-and maturity-dependent. Visualizing the errors over time, we find that the Markov Tree model yields more accurate and less risky single instrument hedges than Heston's stochastic volatility model. A statistical resampling method indicates that the Markov Tree model's superior hedging performance is due to its robustness with respect to noise in option data.

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“…This is the case especially in nancial applications. A good example for this the following, which was presented in the paper by Appleby, Riedle and Swords [2]. They studied the asymptotic behaviour of an stochastic delay dierential equation modeling the evolution of the cumulative return of a risky security.…”

Section: Some Motivational Applications For Delayed Modelsmentioning

confidence: 99%

“…The following results can be founded in Appleby, Riedle and Swords [2]. Let (S(t)) t≥0 denote the prices of a risky asset which satisfy (1.1.3) dS(t) = S(t) dR(t) t ≥ 0,…”

Section: Some Motivational Applications For Delayed Modelsmentioning

confidence: 99%

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Asymptotic inference for linear stochastic differential equations with time delay

Benke¹

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Res z=λ (f (z)) residue of the function f at λ LAQ local asymptotic quadratic, see Denition 2.1.4 LAMN local asymptotic mixed normal, see Denition 2.1.5 LAN local asymptotic normal, see Denition 2.1.6 PLAMN periodic local asymptotic mixed normal, see Denition 2.1.7 MLE maximum likelihood estimator N (µ, σ 2) the normal (or Gaussian) distribution with expected value µ ∈ R and variance σ 2 ∈ R ++ N d (µ, Σ) the d-dimensional normal (or Gaussian) distribution with expected value µ ∈ R d and covariance matrix Σ ∈ R d×d v

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Bubbles and crashes in a Black–Scholes model with delay

Cited by 8 publications

References 22 publications

On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

Large-Scale Empirical Tests of the Markov Tree Model

Asymptotic inference for linear stochastic differential equations with time delay

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