Characteristic Function

Higher order approximation of option prices in Barndorff-Nielsen and Shephard models ArXiv ID: 2401.14390 “View on arXiv” Authors: Unknown Abstract We present an approximation method based on the mixing formula (Hull & White 1987, Romano & Touzi 1997) for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das & Langrené (2022). We obtain asymptotic results for the error of the $N^{"\text{th"}}$ order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein-Uhlenbeck process or a gamma Ornstein-Uhlenbeck process. ...

Notes on the SWIFT method based on Shannon Wavelets for Option Pricing – Revisited ArXiv ID: 2401.01758 “View on arXiv” Authors: Unknown Abstract This note revisits the SWIFT method based on Shannon wavelets to price European options under models with a known characteristic function in 2023. In particular, it discusses some possible improvements and exposes some concrete drawbacks of the method. Keywords: Shannon Wavelets, Option Pricing, Characteristic Function, Spectral Methods, Numerical Methods, Derivatives ...

The ATM implied skew in the ADO-Heston model ArXiv ID: 2309.15044 “View on arXiv” Authors: Unknown Abstract In this paper similar to [“P. Carr, A. Itkin, 2019”] we construct another Markovian approximation of the rough Heston-like volatility model - the ADO-Heston model. The characteristic function (CF) of the model is derived under both risk-neutral and real measures which is an unsteady three-dimensional PDE with some coefficients being functions of the time $t$ and the Hurst exponent $H$. To replicate known behavior of the market implied skew we proceed with a wise choice of the market price of risk, and then find a closed form expression for the CF of the log-price and the ATM implied skew. Based on the provided example, we claim that the ADO-Heston model (which is a pure diffusion model but with a stochastic mean-reversion speed of the variance process, or a Markovian approximation of the rough Heston model) is able (approximately) to reproduce the known behavior of the vanilla implied skew at small $T$. We conclude that the behavior of our implied volatility skew curve ${"\cal S"}(T) \propto a(H) T^{“b\cdot (H-1/2)”}, , b = const$, is not exactly same as in rough volatility models since $b \ne 1$, but seems to be close enough for all practical values of $T$. Thus, the proposed Markovian model is able to replicate some properties of the corresponding rough volatility model. Similar analysis is provided for the forward starting options where we found that the ATM implied skew for the forward starting options can blow-up for any $s > t$ when $T \to s$. This result, however, contradicts to the observation of [“E. Alos, D.G. Lorite, 2021”] that Markovian approximation is not able to catch this behavior, so remains the question on which one is closer to reality. ...

Fourier Neural Network Approximation of Transition Densities in Finance ArXiv ID: 2309.03966 “View on arXiv” Authors: Unknown Abstract This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet’s capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. We derive practical bounds for the $L_2$ estimation error and the potential pointwise loss of nonnegativity in FourNet for $d$-dimensions ($d\ge 1$), highlighting its robustness and applicability in practical settings. FourNet’s accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including Lévy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process. ...

Volatility of Volatility and Leverage Effect from Options ArXiv ID: 2305.04137 “View on arXiv” Authors: Unknown Abstract We propose model-free (nonparametric) estimators of the volatility of volatility and leverage effect using high-frequency observations of short-dated options. At each point in time, we integrate available options into estimates of the conditional characteristic function of the price increment until the options’ expiration and we use these estimates to recover spot volatility. Our volatility of volatility estimator is then formed from the sample variance and first-order autocovariance of the spot volatility increments, with the latter correcting for the bias in the former due to option observation errors. The leverage effect estimator is the sample covariance between price increments and the estimated volatility increments. The rate of convergence of the estimators depends on the diffusive innovations in the latent volatility process as well as on the observation error in the options with strikes in the vicinity of the current spot price. Feasible inference is developed in a way that does not require prior knowledge of the source of estimation error that is asymptotically dominating. ...