Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

Fusai, Gianluca; Germano, Guido; Marazzina, Daniele

doi:10.1016/j.ejor.2015.11.027

Cited by 70 publications

(37 citation statements)

References 59 publications

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“…So we propose a new representation of the characteristic function which is algebraically equivalent to all the previous expressions and does not show the discontinuities of Eqs. (10) and (14) for large maturities:…”

Section: 245 25mentioning

confidence: 99%

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Full and fast calibration of the Heston stochastic volatility model

Cui

Rollin

Germano

2017

European Journal of Operational Research

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This paper presents an algorithm for a complete and efficient calibration of the Heston stochastic volatility model. We express the calibration as a nonlinear least squares problem. We exploit a suitable representation of the Heston characteristic function and modify it to avoid discontinuities caused by branch switchings of complex functions. Using this representation, we obtain the analytical gradient of the price of a vanilla option with respect to the model parameters, which is the key element of all variants of the objective function. The interdependency between the components of the gradient enables an efficient implementation which is around ten times faster than a numerical gradient. We choose the Levenberg-Marquardt method to calibrate the model and do not observe multiple local minima reported in previous research. Two-dimensional sections show that the objective function is shaped as a narrow valley with a flat bottom. Our method is the fastest calibration of the Heston model developed so far and meets the speed requirement of practical trading.

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Section: 245 25mentioning

confidence: 99%

“…(23e) can be obtained by applying the chain rule to Eq. (14), and the intermediate terms for ∂φ(θ; u, T )/∂σ can be written in terms of those for ∂φ(θ; u, T )/∂ρ, that is…”

Section: Analytical Gradientmentioning

confidence: 99%

Full and fast calibration of the Heston stochastic volatility model

Cui

Rollin

Germano

2017

European Journal of Operational Research

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“…Oosterlee (2008, 2009) devised the COS method based on the Fouriercosine expansion. The Hilbert transform (King 2009) has also been successfully employed: by Feng and Linetsky (2008) to price barrier options using backward induction in the Fourier space and by Marazzina et al (2012) and Fusai et al (2016) to compute the factorisations required by the Spitzer identities (Spitzer 1956, Kemperman 1963) via the Plemelj-Sokhotsky relations. Feng and Linetsky showed that computing the Hilbert transform with the sinc expansion, as studied by Stenger (1993Stenger ( , 2011, gives errors that reduce exponentially as the number of fast Fourier transform (FFT) grid points increases.…”

Section: Introductionmentioning

confidence: 99%

“…Pricing derivatives, especially exotic options, is a challenging problem in the operations research literature. Fusai et al (2016) provide extensive references for this, as well as for many non-financial applications of the Hilbert transform and the related topics of Wiener-Hopf factorisation and Spitzer identities in insurance, queuing theory, physics, engineering, applied mathematics, etc.…”

Section: Introductionmentioning

confidence: 99%

Hilbert transform, spectral filters and option pricing

et al. 2018

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We show how spectral filtering techniques can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (2008) and Fusai, Germano and Marazzina (2016) to price discretely monitored barrier options where the underlying asset price is modelled by a Lévy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering.

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“…Fusai et al [37] combined this with Spitzer's identity for a general method of pricing discretely monitored exotic options, with an explicit formula for the fixed strike option. However, the method assumes equidistant monitoring windows.…”

Section: Past Pricing Model For Lookbacksmentioning

confidence: 99%

The Amnesiac Lookback Option: Selectively Monitored Lookback Options and Cryptocurrencies

Chang

2018

Front. Appl. Math. Stat.

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This study proposes a strategy to make the lookback option cheaper and more practical, and suggests the use of its properties to reduce risk exposure in cryptocurrency markets through blockchain enforced smart contracts and correct for informational inefficiencies surrounding prices and volatility. This paper generalizes partial, discretely-monitored lookback options that dilute premiums by selecting a subset of specified periods to determine payoff, which we call amnesiac lookback options. Prior literature on discretely-monitored lookback options considers the number of periods and assumes equidistant lookback periods in pricing partial lookback options. This study by contrast considers random sampling of lookback periods and compares resulting payoff of the call, put and spread options under floating and fixed strikes. Amnesiac lookbacks are priced with Monte Carlo simulations of Gaussian random walks under equidistant and random periods. Results are compared to analytic and binomial pricing models for the same derivatives. Simulations show diminishing marginal increases to the fair price as the number of selected periods is increased. The returns correspond to a Hill curve whose parameters are set by interest rate and volatility. We demonstrate over-pricing under equidistant monitoring assumptions with error increasing as the lookback periods decrease. An example of a direct implication for event trading is when shock is forecasted but its timing uncertain, equidistant sampling produces a lower error on the true maximum than random choice. We conclude that the instrument provides an ideal space for investors to balance their risk, and as a prime candidate to hedge extreme volatility. We discuss the application of the amnesiac lookback option and path-dependent options to cryptocurrencies and blockchain commodities in the context of smart contracts.

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Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

Cited by 70 publications

References 59 publications

Full and fast calibration of the Heston stochastic volatility model

Full and fast calibration of the Heston stochastic volatility model

Hilbert transform, spectral filters and option pricing

The Amnesiac Lookback Option: Selectively Monitored Lookback Options and Cryptocurrencies

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