1,720,979 research outputs found
Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes
No full textIn this paper we propose a general derivative pricing framework that
employs decoupled time-changed (DTC) Lévy processes to model the underlying
assets of contingent claims. A DTC Lévy process is a generalized time-changed Lévy
process whose continuous and pure jump parts are allowed to follow separate random
time scalings; we devise the martingale structure for a DTC Lévy-driven asset and
revisit many popular models which fall under this framework. Postulating different
time changes for the underlying Lévy decomposition allows the introduction of asset
price models consistent with the assumption of a correlated pair of continuous and
jump market activity rates; we study one illustrative DTC model of this kind based
on the so-called Wishart process. The theory we develop is applied to the problem of
pricing not only claims that depend on the price or the volatility of an underlying asset,
but also more sophisticated derivatives whose payoffs rely on the joint performance
of these two financial variables, such as the target volatility option.We solve the pricing
problem through a Fourier-inversion method. Numerical analyses validating our
techniques are provided. In particular, we present some evidence that correlating th
Volatility Targeting Using Delayed Diffusions
No full textA target volatility strategy (TVS) is a risky asset-riskless bond dynamic portfolio allocation which makes use of the risky asset historical volatility as an allocation rule with the aim of maintaining the instantaneous volatility of the investment constant at a target level. In a market with stochastic volatility, we consider a diffusion model for the value of a target volatility fund (TVF) which employs a system of stochastic delayed differential equations (SDDEs) involving the asset realized variance. First we prove that under some technical assumptions, contingent claim valuation on a TVF is approximately of Black-Scholes type, which is consistent with and supports the standing market practice. In second place, we develop a computational framework using recent results on Markovian approximations of SDDEs systems, which we then implement in the Heston variance model using an ad hoc Euler scheme. Our framework allows for efficient numerical valuation of derivatives on TVFs, whose typical purpose is the assessment of the guarantee costs of such funds for insurers
On the convolution equivalence of tempered stable distributions on the real line
No full textWe show the convolution equivalence property of univariate tempered stable distributions in the sense of Rosinski (2007). This makes rigorous various classic heuristic arguments on the asymptotic similarity between the probability and Levy densities of such distributions. Some specific examples from the literature are discussed
Thorin processes and their subordination
No full textA Thorin process is a stochastic process with independent and stationary increments whose laws are weak limits of finite convolutions of gamma distributions. Many popular L & eacute;vy processes fall under this class. The Thorin class can be characterized by a representing triplet that conveys more information on the process compared to the L & eacute;vy triplet. In this paper, we investigate some relationships between the Thorin structure and the process properties, and find that the support of the Thorin measure characterizes the existence of the critical exponential moment, as well as the asymptotic equivalence between the L & eacute;vy tail function and the complementary distribution function. Furthermore, it is illustrated how univariate Brownian subordination with respect to Thorin subordinators produces Thorin processes whose representing measure is given by a pushforward with respect to a hyperbolic function, leading to arguably easier formulae compared to the Bochner integral determining the L & eacute;vy measure. We provide a full account of the theory of multivariate Thorin processes, starting from the Thorin-Bondesson representation for the characteristic exponent, and highlight the roles of the Thorin measure in the analysis of density functions, moments, path variation and subordination. Various old and new examples are discussed. We finally detail a treatment of subordination of gamma processes with respect to negative binomial subordinators which is made possible by the Thorin-Bondesson representation
The effect of an instantaneous dependency rate on the social equitability of hybrid PAYG public pension schemes
Full text linkThe defined convex combination (DCC) pay-as-you-go public pension systems recently introduced in the literature are a form of hybridization between defined benefit (DB) and defined contribution (DC) designed to maintain intergenerational social equitability by reacting to demographic shocks in an optimal way. In this paper, we augment DCC schemes with the assumption that the dependency ratio between pensioners and workers is driven by an exogenously modelled instantaneous stochastic rate of change. This assumption enjoys support from the empirical data and allows explicit solutions for the contribution and replacement rate processes which make transparent the nature of the dynamic evolution of a DCC system, as well as the role of the variables involved. The analysis of intergenerational social equitability measures under the assumption of an instantaneous dependency rate confirms the view expressed in previous literature that neither DB nor DC achieves social fairness, and that DCC plans have the potential to improve on both. We perform a calibration test, and our findings seem to indicate that in ageing economies the DC system might indeed be superior to the DB one in terms of intergenerational fairness
The effect of an instantaneous dependency rate on the social equitability of hybrid PAYG public pension schemes
No full textThe defined convex combination (DCC) pay-as-you-go public pension systems recently introduced in the literature are a form of hybridization between defined benefit (DB) and defined contribution (DC) designed to maintain intergenerational social equitability by reacting to demographic shocks in an optimal way. In this paper, we augment DCC schemes with the assumption that the dependency ratio between pensioners and workers is driven by an exogenously modelled instantaneous stochastic rate of change. This assumption enjoys support from the empirical data and allows explicit solutions for the contribution and replacement rate processes which make transparent the nature of the dynamic evolution of a DCC system, as well as the role of the variables involved. The analysis of intergenerational social equitability measures under the assumption of an instantaneous dependency rate confirms the view expressed in previous literature that neither DB nor DC achieves social fairness, and that DCC plans have the potential to improve on both. We perform a calibration test, and our findings seem to indicate that in ageing economies the DC system might indeed be superior to the DB one in terms of intergenerational fairness
Pricing joint claims on an asset and its realized variance in stochastic volatility models
No full textIn the setting of a stochastic volatility model, we find a general pricing equation for the
class of payoffs depending on the terminal value of a market asset and its final quadratic
variation. This provides a pricing tool for European-style claims paying off at maturity
a joint function of the underlying and its realized volatility or variance. We study the
solution under various specific stochastic volatility models, give a formula for the computation
of the delta and gamma of these claims, and introduce some new interesting
payoffs that can be valued by means of the general pricing equation. Numerical results
are given and compared to those from plain vanilla derivatives
Additive logistic processes in option pricing
Full text linkIn option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an lp vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input
An Analytical Valuation Framework for Financial Assets with Trading Suspensions
Full text linkIn this paper we propose a derivative valuation framework based on Lévy processes which takes into account the possibility that the underlying asset is subject to information-related trading halts/suspensions. Since such assets are not traded at all times, we argue that the natural underlying for derivative risk-neutral valuation is not the asset itself but rather a contract that, when the asset is in trade suspensions upon maturity, cash settles the last quoted price plus the interest accrued since the last quote update. Combining some elements from semimartingale time changes and potential theory, we devise martingale dynamics and no-arbitrage relations for such a price process, provide Fourier transform–based pricing formulae for derivatives, and study the asymptotic behavior of the obtained formulae as a function of the halt parameters. The volatility surface analysis reveals that the short-term skew of our model is typically steeper than that of the underlying Lévy models, indicating that the presence of a trade suspension risk is consistent with the well-documented stylized fact of volatility skew persistence/explosion. A simple calibration example to market option prices is provided
On the implied volatility skew outside the at-the-money point
Full text linkThe small-maturity implied volatility of an asset pricing model is fully determined by the asymptotics of traded option prices, and thus model-free expressions are available. We show how by sharpening one such expression it is possible to derive a novel general formula for the leading order of the in-the-money and out-of-the money (ITM/OTM) implied volatility skew. We apply this formula to find expressions of the small maturity limiting skew of the Heston stochastic volatility model, of exponential L & eacute;vy models and their time changes, as well as that of some recently proposed pricing models with independent log returns
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