Fractional Ornstein-Uhlenbeck processes

Cheridito, Patrick; Kawaguchi, Haruko; Maejima, Makoto

doi:10.1214/ejp.v8-125

Cited by 322 publications

(359 citation statements)

References 6 publications

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“…The decay of the autocovariance function of {Y t ; t ∈ R + } is similar to that of the increments of the fBm, and thus it exhibits long-range dependence. See Cheridito et al [6] for more details.…”

Section: Fractional Ornstein-uhlenbeck Processmentioning

confidence: 99%

Estimation and pricing under long-memory stochastic volatility

Chronopoulou

Viens

2010

Ann Finance

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We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein-Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.

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Section: Fractional Ornstein-uhlenbeck Processmentioning

confidence: 99%

Estimation and pricing under long-memory stochastic volatility

Chronopoulou

Viens

2010

Ann Finance

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“…Cheridito et al [7]). Thus, by (a) and (b), it suffices to investigate the case γ = σ = 1, H = 1/2 and h ↓ 0.…”

Section: Appendix a Proof Of Lemma 21mentioning

confidence: 98%

Maxima of stochastic processes driven by fractional Brownian motion

Buchmann

Klüppelberg

2005

Adv. Appl. Probab.

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We study stationary processes given as solutions to SDEs driven by fractional Brownian motion (FBM). This class includes the fractional Ornstein-Uhlenbeck process (FOUP), but is a much richer class of processes, which can be obtained by state space transformations of the FOUP. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of FOUP near zero, which, by an application of Berman's condition, guarantees that the FOUP is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions of FBM-driven SDEs.

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“…In Section 4, we discuss two important examples: the stable Ornstein-Uhlenbeck processes (which was the starting point of this research) where f (x) = e x in the representation (1.1), and the fractional OrnsteinUhlenbeck processes as recently introduced in [5,17,21], where f (x) = x c e x in the representation (1.1). These new fractional processes seem important for applications, especially in network traffic [21].…”

Section: K(t X) Z(dx)mentioning

confidence: 99%

“…These processes were introduced in [5] as the solution to the generalised Langevin equation In [5], it is also remarked that except in the Brownian case, X defined in (4.3) cannot be retrieved from a fBM by the Lamperti transformation. As above, we will consider the case ξ = 0 viz.…”

Section: Fractional Ornstein-uhlenbeck Processesmentioning

confidence: 99%