At present, the most liquidly traded listed options on the crypto underlying are so-called invers... more At present, the most liquidly traded listed options on the crypto underlying are so-called inverse vanilla options. In this note we show how such contacts are just regular vanilla options considered under the well-known martingale measure making underlying the numeraire.
We provide a generalization of the price surface introduced in Carr and Pelts (2015), and use it ... more We provide a generalization of the price surface introduced in Carr and Pelts (2015), and use it to develop a new way to interpolate implied volatility in time.
A commonly used interpolation of implied volatility is investigating from the calendar arbitrage ... more A commonly used interpolation of implied volatility is investigating from the calendar arbitrage viewpoint.
The “exponential forgetting” (or ”weighting”) method of discarding the old information in regress... more The “exponential forgetting” (or ”weighting”) method of discarding the old information in regressions typically yields increased predictive performance in practice, often noticeably so. In this note we introduce the exponential forgetting into (the temporal dimension of) the HOPLS multilinear regression model.
Motivated by the recent Log Moment formula of \cite{raval_jacquier23}, we look at the power of ... more Motivated by the recent Log Moment formula of \cite{raval_jacquier23}, we look at the power of absolute Log Contract
$|\log(S_T/F_T)|^q$, $q\geq 1$, and derive its corresponding replicating strip of Vanilla options. Unlike the standard Log Contract where the Delta of the replicating portfolio synthesizes a pure short position in realized asset variance, here
the position changes between short and long depending on the asset level. We also derive a generalization of the standard Log Contract replication, where the payoff $K\mapsto (\ln(K))^q I_{K\geq 1}-(-\ln(K))^q I_{K< 1}$ is parameterized by $q>0$.
We describe a heuristic method for active learning of a regression model, in particular of a deep... more We describe a heuristic method for active learning of a regression model, in particular of a deep neural network model, e.g. [6, 2, 7, 3, 4]. In the following, an automated “oracle” Z capable of providing real-valued supervision (a regression target) for samples, for purposes of training, is assumed available. A parameterized model Y for regression, which could be a deep neural network (DNN), is typically trained to approximate the oracle based on the mean squared error (MSE) training objective, i.e.,
In this work we address some of the fundamental properties related to no-arbitrage conditions fo... more In this work we address some of the fundamental properties related to no-arbitrage conditions for the implied volatility surface.
Firstly, we extend the result on monotonicity of the normalizing volatility transforms due to \cite{Fukasawa2012} to the general case of continuous
implied volatility.
We then present a parameterization of all volatility slices for which the
normalizing volatility transforms are increasing, and give
some new implied volatility slope bounds.
Finally, we look at interpolation of implied volatility in time for fixed Delta (so-called ``Delta volatility surface'') and establish a link between absence of calendar
arbitrage and absence of strike arbitrage in that case.
This is utilized to evaluate commonly used schemes for interpolating Delta volatility surface in time.
Inspired by initially proposed IBOR fallback mechanisms, we show how deep learning can be used to... more Inspired by initially proposed IBOR fallback mechanisms, we show how deep learning can be used to quickly and accurately compute the {expected median} of a time series at future inference dates with varying amounts of observed data. While the IBOR fallback spreads were ultimately fixed, the technique outlined here showcases the ability of neural networks to tackle financial problems over seemingly impossibly large domains.
We revisit the problem addressed in Gatarek et al. and derive an algorithm for determining the Ch... more We revisit the problem addressed in Gatarek et al. and derive an algorithm for determining the Chayette local volatility based on the options marked for fixed-tenor rolling maturity swaptions. Apart from the discretization error inherent in calibration-simulation procedures, the formula presented is exact.
We revisit the classical problem of convergence of the maximum of cumulative sums of IID random v... more We revisit the classical problem of convergence of the maximum of cumulative sums of IID random variables by introducing ideas from the Karamata's celebrated proof of the Hardy-Littlewood Tauberian Theorem [J. Karamata, Uber die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), 319-320]. As a main result we present an extension of the Darling-Heyde convergence device, and use it to provide a direct analytic proof of convergence in distribution of normalized cumulative sums of IID random variables. The introduction of Karamata's ideas makes the proof self-contained and relying only on basic regular variation results.
We demonstrate a backdoor attack on deep regression, with an example from financial derivatives p... more We demonstrate a backdoor attack on deep regression, with an example from financial derivatives pricing. The backdoor attack is localized based on training-set data poisoning wherein the mislabeled samples are surrounded by correctly supervised ones. We demonstrate how such localization is necessary for attack success. We also study the performance of a backdoor defense using gradient-based discovery of local error maximizers. Local error maximizers which are associated with significant (interpolation) error, and are proximal to many training samples, are suspicious. This method is also used to accurately train for deep regression in the first place by active (deep) learning leveraging an “oracle” capable of providing real-valued supervision (a regression target) for samples. Such oracles, including traditional numerical solvers of the pricing PDEs are far more computationally costly at (test-time) inference, compared to deep regression.
This thesis is a contribution to the theory of noniinear filtering, and is concerned with hvo bas... more This thesis is a contribution to the theory of noniinear filtering, and is concerned with hvo basic issues. The first of these deals with pathwise uniqueness and uniqueness in law for solutions of the measure-valued stochâstic differentid equations of nonlinear filtering, in the case mhere there is genuine dependence of the signal on the observation process. The second issue uses this uaiqueness to establish convergence of nonlinear filters corresponding to a class of djmamical systems governed by sigularly perturbed stochastic differential equations, in which the perturbation is wide-band random process with certain ergodic properties. Acknowledgements I would Like to thank my thesis advisor, Prof. h d r e w Heunis, for inspiring rny interest in Probability and being a constant source of knowledge, encouragement, and support dong the way. His enthusiasm for research and cornmitment to excellence have had a profound effect on my academic career. 1 would also like to thank the MIT-4C...
Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995
ABSTRACT The l1-optimal linear time-invariant (LTI) compensator may have an order significantly h... more ABSTRACT The l1-optimal linear time-invariant (LTI) compensator may have an order significantly higher than that of the plant. Here the authors consider the square discrete time multivariable setting in which T3-1 is stable, and show, under a technical assumption, that there always exists a low order nonlinear stabilizing controller which provides the same level of performance as a given LTI stabilizing controller. Furthermore, for the scalar case the authors present an algorithm which can be used to compute the optimal performance which can be achieved by a nonlinear time-varying stabilizing controller
At present, the most liquidly traded listed options on the crypto underlying are so-called invers... more At present, the most liquidly traded listed options on the crypto underlying are so-called inverse vanilla options. In this note we show how such contacts are just regular vanilla options considered under the well-known martingale measure making underlying the numeraire.
We provide a generalization of the price surface introduced in Carr and Pelts (2015), and use it ... more We provide a generalization of the price surface introduced in Carr and Pelts (2015), and use it to develop a new way to interpolate implied volatility in time.
A commonly used interpolation of implied volatility is investigating from the calendar arbitrage ... more A commonly used interpolation of implied volatility is investigating from the calendar arbitrage viewpoint.
The “exponential forgetting” (or ”weighting”) method of discarding the old information in regress... more The “exponential forgetting” (or ”weighting”) method of discarding the old information in regressions typically yields increased predictive performance in practice, often noticeably so. In this note we introduce the exponential forgetting into (the temporal dimension of) the HOPLS multilinear regression model.
Motivated by the recent Log Moment formula of \cite{raval_jacquier23}, we look at the power of ... more Motivated by the recent Log Moment formula of \cite{raval_jacquier23}, we look at the power of absolute Log Contract
$|\log(S_T/F_T)|^q$, $q\geq 1$, and derive its corresponding replicating strip of Vanilla options. Unlike the standard Log Contract where the Delta of the replicating portfolio synthesizes a pure short position in realized asset variance, here
the position changes between short and long depending on the asset level. We also derive a generalization of the standard Log Contract replication, where the payoff $K\mapsto (\ln(K))^q I_{K\geq 1}-(-\ln(K))^q I_{K< 1}$ is parameterized by $q>0$.
We describe a heuristic method for active learning of a regression model, in particular of a deep... more We describe a heuristic method for active learning of a regression model, in particular of a deep neural network model, e.g. [6, 2, 7, 3, 4]. In the following, an automated “oracle” Z capable of providing real-valued supervision (a regression target) for samples, for purposes of training, is assumed available. A parameterized model Y for regression, which could be a deep neural network (DNN), is typically trained to approximate the oracle based on the mean squared error (MSE) training objective, i.e.,
In this work we address some of the fundamental properties related to no-arbitrage conditions fo... more In this work we address some of the fundamental properties related to no-arbitrage conditions for the implied volatility surface.
Firstly, we extend the result on monotonicity of the normalizing volatility transforms due to \cite{Fukasawa2012} to the general case of continuous
implied volatility.
We then present a parameterization of all volatility slices for which the
normalizing volatility transforms are increasing, and give
some new implied volatility slope bounds.
Finally, we look at interpolation of implied volatility in time for fixed Delta (so-called ``Delta volatility surface'') and establish a link between absence of calendar
arbitrage and absence of strike arbitrage in that case.
This is utilized to evaluate commonly used schemes for interpolating Delta volatility surface in time.
Inspired by initially proposed IBOR fallback mechanisms, we show how deep learning can be used to... more Inspired by initially proposed IBOR fallback mechanisms, we show how deep learning can be used to quickly and accurately compute the {expected median} of a time series at future inference dates with varying amounts of observed data. While the IBOR fallback spreads were ultimately fixed, the technique outlined here showcases the ability of neural networks to tackle financial problems over seemingly impossibly large domains.
We revisit the problem addressed in Gatarek et al. and derive an algorithm for determining the Ch... more We revisit the problem addressed in Gatarek et al. and derive an algorithm for determining the Chayette local volatility based on the options marked for fixed-tenor rolling maturity swaptions. Apart from the discretization error inherent in calibration-simulation procedures, the formula presented is exact.
We revisit the classical problem of convergence of the maximum of cumulative sums of IID random v... more We revisit the classical problem of convergence of the maximum of cumulative sums of IID random variables by introducing ideas from the Karamata's celebrated proof of the Hardy-Littlewood Tauberian Theorem [J. Karamata, Uber die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), 319-320]. As a main result we present an extension of the Darling-Heyde convergence device, and use it to provide a direct analytic proof of convergence in distribution of normalized cumulative sums of IID random variables. The introduction of Karamata's ideas makes the proof self-contained and relying only on basic regular variation results.
We demonstrate a backdoor attack on deep regression, with an example from financial derivatives p... more We demonstrate a backdoor attack on deep regression, with an example from financial derivatives pricing. The backdoor attack is localized based on training-set data poisoning wherein the mislabeled samples are surrounded by correctly supervised ones. We demonstrate how such localization is necessary for attack success. We also study the performance of a backdoor defense using gradient-based discovery of local error maximizers. Local error maximizers which are associated with significant (interpolation) error, and are proximal to many training samples, are suspicious. This method is also used to accurately train for deep regression in the first place by active (deep) learning leveraging an “oracle” capable of providing real-valued supervision (a regression target) for samples. Such oracles, including traditional numerical solvers of the pricing PDEs are far more computationally costly at (test-time) inference, compared to deep regression.
This thesis is a contribution to the theory of noniinear filtering, and is concerned with hvo bas... more This thesis is a contribution to the theory of noniinear filtering, and is concerned with hvo basic issues. The first of these deals with pathwise uniqueness and uniqueness in law for solutions of the measure-valued stochâstic differentid equations of nonlinear filtering, in the case mhere there is genuine dependence of the signal on the observation process. The second issue uses this uaiqueness to establish convergence of nonlinear filters corresponding to a class of djmamical systems governed by sigularly perturbed stochastic differential equations, in which the perturbation is wide-band random process with certain ergodic properties. Acknowledgements I would Like to thank my thesis advisor, Prof. h d r e w Heunis, for inspiring rny interest in Probability and being a constant source of knowledge, encouragement, and support dong the way. His enthusiasm for research and cornmitment to excellence have had a profound effect on my academic career. 1 would also like to thank the MIT-4C...
Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995
ABSTRACT The l1-optimal linear time-invariant (LTI) compensator may have an order significantly h... more ABSTRACT The l1-optimal linear time-invariant (LTI) compensator may have an order significantly higher than that of the plant. Here the authors consider the square discrete time multivariable setting in which T3-1 is stable, and show, under a technical assumption, that there always exists a low order nonlinear stabilizing controller which provides the same level of performance as a given LTI stabilizing controller. Furthermore, for the scalar case the authors present an algorithm which can be used to compute the optimal performance which can be achieved by a nonlinear time-varying stabilizing controller
In this note we revisit the polynomial representation for the derivatives of Mills' ratio, $R^{(n... more In this note we revisit the polynomial representation for the derivatives of Mills' ratio, $R^{(n)}(x)=P_n(x)R(x)-Q_n(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$, where $R(\cdot)$ stands for the Mills ratio and $P_n(\cdot)$, $Q_n(\cdot)$ are polynomials of order $n$. (\cite{kouba06},
\cite{Kreinin14}, \cite{gasull14}, \cite{Lu2018}). By combining properties of Laplace transform with representation of monomials by Hermite polynomials, the formula in question is follows directly.
In this work we provide generalization and unification of several moment formulae: the Lee Moment... more In this work we provide generalization and unification of several moment formulae: the Lee Moment Formula [11], Log Moment Formula [17], and the modified Piterbarg conjecture [7]. We approach the problem via investigating asymptotic behavior of the normalizing volatility transforms behavior of the normalizing volatility transforms
rather than the implied volatility itself.
Our derivations of are elementary and do not rely on Regular Variation Theory.
An example illustrating the theory was given.
At present, the most liquidly traded listed options on the crypto underlying are so-called invers... more At present, the most liquidly traded listed options on the crypto underlying are so-called inverse vanilla options. In this note we show how such contacts are just regular vanilla options considered under the well-known martingale measure making underlying the numeraire.
We provide a generalization of the price surface introduced in Carr and Pelts (2015), and use it ... more We provide a generalization of the price surface introduced in Carr and Pelts (2015), and use it to develop a new way to interpolate implied volatility in time.
A commonly used interpolation of implied volatility is investigating from the calendar arbitrage ... more A commonly used interpolation of implied volatility is investigating from the calendar arbitrage viewpoint.
We provide an alternative proof of monotonicity of normalizing volatility transforms (NVTs) due t... more We provide an alternative proof of monotonicity of normalizing volatility transforms (NVTs) due to Fukasawa (2012), and then obtain a general formula for volatility surface for which the NVTs are increasing. This is used to obtain several results related to strike arbitrage and asymptotic behavior of implied volatility for large strikes.
Motivated by the recent Log Moment formula of Raval and Jacquier (2021), we look at the power of ... more Motivated by the recent Log Moment formula of Raval and Jacquier (2021), we look at the power of absolute Log Contract | log(S T /F T)| q , q ≥ 1, and derive its corresponding replicating strip of Vanilla options. Unlike the standard Log Contract where the Delta of the replicating portfolio synthesizes a pure short position in realized asset variance, here the position changes between short and long depending on the asset level.
The “exponential forgetting” (or ”weighting”) method of discarding the old
information in regress... more The “exponential forgetting” (or ”weighting”) method of discarding the old information in regressions typically yields increased predictive performance in practice, often noticeably so. In this note we introduce the exponential forgetting into (the temporal dimension of) the HOPLS multilinear regression model.
We look into problems related to volatility modelling, focusing on general properties of implied ... more We look into problems related to volatility modelling, focusing on general properties of implied volatility surface and valuation of volatility products. This work contributes to the large body of existing literature by presenting several new results and providing alternative proofs of some existing ones. The paper is organised as a series of facts, centred around a common theme.
We look at the real positive (semi)definite matrix completion problem from the convex optimizatio... more We look at the real positive (semi)definite matrix completion problem from the convex optimization viewpoint. The problem is introduced via relative entropy minimization, transformed into the standard max-det from, and conditions are sought for existence of positive definite and positive semidefinite completions. Using basic tools of convex optimization, a unifying view of the existence and uniqueness problem for positive (semi)definite matrix completions is presented. Some results previously established using functional-analytic techniques are recovered and some new are given. In particular, the maximum determinant completion is generalized to the positive semidefinte matrices with an arbitrary sparsity pattern.
We revisit the classical problem of convergence of the maximum of cumulative sums of IID random v... more We revisit the classical problem of convergence of the maximum of cumulative sums of IID random variables by introducing ideas from the Karamata's celebrated proof of the Hardy-Littlewood Tauberian Theorem [J. Karamata, ¨ Uber die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), 319-320]. As a main result we present an extension of the Darling-Heyde convergence device ([4], [3]), and use it to provide a direct analytic proof of convergence in distribution of normalized cumulative sums of IID random variables. The introduction of Karamata's ideas makes the proof self-contained and relying only on basic regular variation results.
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$|\log(S_T/F_T)|^q$, $q\geq 1$, and derive its corresponding replicating strip of Vanilla options. Unlike the standard Log Contract where the Delta of the replicating portfolio synthesizes a pure short position in realized asset variance, here
the position changes between short and long depending on the asset level. We also derive a generalization of the standard Log Contract replication, where the payoff $K\mapsto (\ln(K))^q I_{K\geq 1}-(-\ln(K))^q I_{K< 1}$ is parameterized by $q>0$.
Firstly, we extend the result on monotonicity of the normalizing volatility transforms due to \cite{Fukasawa2012} to the general case of continuous
implied volatility.
We then present a parameterization of all volatility slices for which the
normalizing volatility transforms are increasing, and give
some new implied volatility slope bounds.
Finally, we look at interpolation of implied volatility in time for fixed Delta (so-called ``Delta volatility surface'') and establish a link between absence of calendar
arbitrage and absence of strike arbitrage in that case.
This is utilized to evaluate commonly used schemes for interpolating Delta volatility surface in time.
$|\log(S_T/F_T)|^q$, $q\geq 1$, and derive its corresponding replicating strip of Vanilla options. Unlike the standard Log Contract where the Delta of the replicating portfolio synthesizes a pure short position in realized asset variance, here
the position changes between short and long depending on the asset level. We also derive a generalization of the standard Log Contract replication, where the payoff $K\mapsto (\ln(K))^q I_{K\geq 1}-(-\ln(K))^q I_{K< 1}$ is parameterized by $q>0$.
Firstly, we extend the result on monotonicity of the normalizing volatility transforms due to \cite{Fukasawa2012} to the general case of continuous
implied volatility.
We then present a parameterization of all volatility slices for which the
normalizing volatility transforms are increasing, and give
some new implied volatility slope bounds.
Finally, we look at interpolation of implied volatility in time for fixed Delta (so-called ``Delta volatility surface'') and establish a link between absence of calendar
arbitrage and absence of strike arbitrage in that case.
This is utilized to evaluate commonly used schemes for interpolating Delta volatility surface in time.
\cite{Kreinin14}, \cite{gasull14}, \cite{Lu2018}). By combining properties of Laplace transform with representation of monomials by Hermite polynomials, the formula in question is follows directly.
rather than the implied volatility itself.
Our derivations of are elementary and do not rely on Regular Variation Theory.
An example illustrating the theory was given.
information in regressions typically yields increased predictive performance
in practice, often noticeably so. In this note we introduce the exponential
forgetting into (the temporal dimension of) the HOPLS multilinear regression model.