scholarly journals Generalized Mean-Reverting 4/2 Factor Model

2019 ◽  
Vol 12 (4) ◽  
pp. 159 ◽  
Author(s):  
Yuyang Cheng ◽  
Marcos Escobar-Anel ◽  
Zhenxian Gong
Keyword(s):  
Value At Risk ◽  
Implied Volatility ◽  
Factor Model ◽  
Risk Measures ◽  
Risk Measure ◽  
Common Factor ◽  
Principal Component ◽  
Characteristic Functions ◽  
Implied Volatility Surface ◽  
Volatility Surface

This paper proposes and investigates a multivariate 4/2 Factor Model. The name 4/2 comes from the superposition of a CIR term and a 3/2-model component. Our model goes multidimensional along the lines of a principal component and factor covariance decomposition. We find conditions for well-defined changes of measure and we also find two key characteristic functions in closed-form, which help with pricing and risk measure calculations. In a numerical example, we demonstrate the significant impact of the newly added 3/2 component (parameter b) and the common factor (a), both with respect to changes on the implied volatility surface (up to 100%) and on two risk measures: value at risk and expected shortfall where an increase of up to 29% was detected.

scholarly journals From characteristic functions to implied volatility expansions

2015 ◽  
Vol 47 (03) ◽  
pp. 837-857 ◽  
Author(s):  
Antoine Jacquier ◽  
Matthew Lorig
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Characteristic Functions ◽  
Volatility Smile ◽  
Model Free ◽  
Implied Volatility Surface ◽  
Volatility Model ◽  
Volatility Surface

For any strictly positive martingaleS= eXfor whichXhas a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

A closed-form approximation formula for pricing European options under a three-factor model

2021 ◽  
pp. 1-27
Author(s):  
Hye-mee Kil ◽  
Jeong-Hoon Kim
Keyword(s):  
Closed Form ◽  
Implied Volatility ◽  
Factor Model ◽  
Closed Form Solution ◽  
Form Solution ◽  
Approximation Formula ◽  
European Options ◽  
Implied Volatility Surface ◽  
Volatility Surface ◽  
Three Factor

Abstract The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

scholarly journals From characteristic functions to implied volatility expansions

2015 ◽  
Vol 47 (3) ◽  
pp. 837-857 ◽  
Author(s):  
Antoine Jacquier ◽  
Matthew Lorig
Keyword(s):  
Characteristic Function ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Characteristic Functions ◽  
Volatility Smile ◽  
Model Free ◽  
Implied Volatility Surface ◽  
Volatility Model ◽  
Volatility Surface

For any strictly positive martingale S = eX for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

scholarly journals An Intuitive Introduction to Fractional and Rough Volatilities

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 994
Author(s):  
Elisa Alòs ◽  
Jorge A. León
Keyword(s):  
Malliavin Calculus ◽  
Numerical Experiments ◽  
Implied Volatility ◽  
Natural Approach ◽  
Volatility Models ◽  
Implied Volatility Surface ◽  
Short Memory ◽  
White Type ◽  
Volatility Surface ◽  
Theoretical Results

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

scholarly journals Portfolio Optimization Constrained by Performance Attribution

2021 ◽  
Vol 14 (5) ◽  
pp. 201
Author(s):  
Yuan Hu ◽  
W. Brent Lindquist ◽  
Svetlozar T. Rachev
Keyword(s):  
Portfolio Optimization ◽  
Asset Allocation ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Selection Effect ◽  
Conditional Value At Risk ◽  
Positive Role ◽  
Performance Attribution ◽  
Dow Jones Industrial Average

This paper investigates performance attribution measures as a basis for constraining portfolio optimization. We employ optimizations that minimize conditional value-at-risk and investigate two performance attributes, asset allocation (AA) and the selection effect (SE), as constraints on asset weights. The test portfolio consists of stocks from the Dow Jones Industrial Average index. Values for the performance attributes are established relative to two benchmarks, equi-weighted and price-weighted portfolios of the same stocks. Performance of the optimized portfolios is judged using comparisons of cumulative price and the risk-measures: maximum drawdown, Sharpe ratio, Sortino–Satchell ratio and Rachev ratio. The results suggest that achieving SE performance thresholds requires larger turnover values than that required for achieving comparable AA thresholds. The results also suggest a positive role in price and risk-measure performance for the imposition of constraints on AA and SE.

scholarly journals Sharing Risk – An Economic Perspective

2009 ◽  
Vol 39 (2) ◽  
pp. 591-613 ◽  
Author(s):  
Andreas Kull
Keyword(s):  
Value At Risk ◽  
Risk Sharing ◽  
Risk Measures ◽  
Insurance Industry ◽  
Risk Measure ◽  
Capital Allocation ◽  
Risk Capital ◽  
Risk Class ◽  
Capital Costs ◽  
Risk Transfer

AbstractWe revisit the relative retention problem originally introduced by de Finetti using concepts recently developed in risk theory and quantitative risk management. Instead of using the Variance as a risk measure we consider the Expected Shortfall (Tail-Value-at-Risk) and include capital costs and take constraints on risk capital into account. Starting from a risk-based capital allocation, the paper presents an optimization scheme for sharing risk in a multi-risk class environment. Risk sharing takes place between two portfolios and the pricing of risktransfer reflects both portfolio structures. This allows us to shed more light on the question of how optimal risk sharing is characterized in a situation where risk transfer takes place between parties employing similar risk and performance measures. Recent developments in the regulatory domain (‘risk-based supervision’) pushing for common, insurance industry-wide risk measures underline the importance of this question. The paper includes a simple non-life insurance example illustrating optimal risk transfer in terms of retentions of common reinsurance structures.

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