Search

We calibrate a novel multifactor stochastic volatility model that includes as special cases the Heston-based model of De Col et al. (J Bank Finance 37(10):3799–3818, 2013) and the 3/2-based model of Baldeaux et al. (J Bank Finance 53:34–48, 2015). Using a dataset on vanilla option quotes in a triangle of currencies, we find that the risk neutral approach typically fails for the calibrated model, in line with the results of Baldeaux et al. (2015).

We propose a quantization-based numerical scheme for a family of decoupled FBSDEs. We simplify the scheme for the control in Pag\`es and Sagna (2018) so that our approach is fully based on recursive marginal quantization and does not involve any Monte Carlo simulation for the computation of conditional expectations. We analyse in detail the numerical error of our scheme and we show through some examples the performance of the whole procedure, which proves to be very effective in view of financial applications., Comment: 22 pages

In this article, we focus upon a family of matrix valued stochastic processes and study the problem of determining the smallest time such that their Laplace transforms become infinite. In particular, we concentrate upon the class of Wishart processes, which have proved to be very useful in different applications by their ability in describing non-trivial dependence. Thanks to this remarkable property we are able to explain the behavior of the explosion times for the Laplace transforms of the Wishart process and its time integral in terms of the relative importance of the involved factors and their correlations., SCOPUS: ar.j, info:eu-repo/semantics/published

We solve a family of fractional Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility behavior. We first consider the case of a zero initial value corresponding to the characteristic function of the log-price. Then we investigate the case of a general starting value associated to a transform also involving the volatility process. The solution to the fractional Riccati equation takes the form of power series, whose convergence domain is typically finite. This naturally suggests a hybrid numerical algorithm to explicitly obtain the solution also beyond the convergence domain of the power series representation. Our numerical tests show that the hybrid algorithm turns out to be extremely fast and stable. When applied to option pricing, our method largely outperforms the only available alternative in the literature, based on the Adams method., Comment: 48 pages, 4 figures

We analyze the VIX futures market with a focus on the exchange-traded notes written on such contracts, in particular we investigate the VXX notes tracking the short-end part of the futures term structure. Inspired by recent developments in commodity smile modelling, we present a multi-factor stochastic-local volatility model that is able to jointly calibrate plain vanilla options both on VIX futures and VXX notes, thus going beyond the failure of purely stochastic or simply local volatility models. We discuss numerical results on real market data by highlighting the impact of model parameters on implied volatilities.

We propose a framework for modeling in a consistent manner the VIX index and the VXX, an exchange-traded note written on the VIX. Our study enables to link the properties of VXX to those of the VIX in a tractable way. In particular, we quantify the systematic loss observed empirically for VXX when the VIX futures term-structure is in contango and we derive option prices, implied volatilities and skews of VXX from those of VIX in infinitesimal developments. We also perform a calibration on real data which highlights the flexibility of our model in fitting the futures and the vanilla options market of VIX and VXX. Our framework can be used to model other exchange-traded notes on the VIX as well as any market where exchange-traded notes have been introduced on a reference index, hence providing tools to better anticipate and quantify systematic behavior of an exchange-traded note with respect to the underlying index.

We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for partial differential equations; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model, which is useful for numerical applications.

In this paper we define a new dynamic approach for measuring the Cash- Flow-at-Risk of a firm. Starting from the assumption that the balance sheet evolves according to a system of difference equations involving the most important accounting records, we define a new risk measure, tailored on our dynamic approach, which takes full advantage of its focus on the liquidity process. A numerical example based on a real case study shows the flexibility of our approach in describing distress and default events.

We propose a framework for modeling in a consistent manner the VIX index and the VXX, an exchange-traded note written on the VIX. Our study enables to link the properties of VXX to those of the VIX in a tractable way. In particular, we quantify the systematic loss observed empirically for VXX when the VIX futures term-structure is in contango and we derive option prices, implied volatilities and skews of VXX from those of VIX in infinitesimal developments. We also perform a calibration on real data which highlights the flexibility of our model in fitting the futures and the vanilla options market of VIX and VXX. Our framework can be used to model other exchange-traded notes on the VIX as well as any market where exchange-traded notes have been introduced on a reference index, hence providing tools to better anticipate and quantify systematic behavior of an exchange-traded note with respect to the underlying index.

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.

Target volatility options (TVO) are a new class of derivatives whose payoff depends on some measure of volatility. These options allow investors to take a joint exposure to the evolution of the underlying asset, as well as to its realized volatility. In equity options markets the slope of the skew is largely independent of the volatility level. A single‐factor Heston based volatility model can generate steep skew or flat skew at a given volatility level but cannot generate both for a given parameterization. Since the payoff corresponding to TVO is a function of the joint evolution of the underlying asset and its realized variance, the consideration of stochastic skew is a relevant question for the valuation of TVO. In this sense, this article studies the effect of considering a multifactor stochastic volatility specification in the valuation of the TVO as well as forward‐start TVO. © 2015 Wiley Periodicals, Inc. Jrl Fut Mark 36:174–193, 2016

Explicitly taking into account the risk incurred when borrowing at a shorter tenor versus lending at a longer tenor ("roll-over risk"), we construct a stochastic model framework for the term structure of interest rates in which a frequency basis (i.e. a spread applied to one leg of a swap to exchange one floating interest rate for another of a different tenor in the same currency) arises endogenously. This roll-over risk consists of two components, a credit risk component due to the possibility of being downgraded and thus facing a higher credit spread when attempting to roll over short-term borrowing, and a component reflecting the (systemic) possibility of being unable to roll over short-term borrowing at the reference rate (e.g., LIBOR) due to an absence of liquidity in the market. The modelling framework is of "reduced form" in the sense that (similar to the credit risk literature) the source of credit risk is not modelled (nor is the source of liquidity risk). However, the framework has more structure than the literature seeking to simply model a different term structure of interest rates for each tenor frequency, since relationships between rates for all tenor frequencies are established based on the modelled roll-over risk. We proceed to consider a specific case within this framework, where the dynamics of interest rate and roll-over risk are driven by a multifactor Cox/Ingersoll/Ross-type process, show how such model can be calibrated to market data, and used for relative pricing of interest rate derivatives, including bespoke tenor frequencies not liquidly traded in the market.

In this paper we introduce a novel pricing methodology for a broad class of models for which the characteristic function of the log-asset price can be efficiently computed. The new method avoids the numerical integration required by the Fourier-based approaches and reveals to be fast and accurate, to the point that we can calibrate the models on real data. Our approach allows to price also American- style options, as it is possible to compute the transition probabilities for the underlying. This is accomplished through an efficient multinomial lattice discretization of the asset price based on a new quantization procedure which exploits the knowledge of the Fourier transform of the process at a given time. As a motivating example, we price an American Put option in a Tempered Stable model, with constitutes the first application of quantization to a pure jump process.

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.

We present a flexible approach for the valuation of interest rate derivatives based on affine processes. We extend the methodology proposed in Keller-Ressel et al. (in press) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in this multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002) . A numerical exercise illustrates the flexibility of Wishart Libor model in describing the movements of the implied volatility surface.

In this paper we show how to hedge a zero coupon bond with a smaller amount of initial capital than required by the classical risk neutral paradigm, whose (trivial) hedging strategy does not suggest to invest in the risky assets. Long dated zero coupon bonds we derive, invest first primarily in risky securities and when approaching more and more the maturity date they increase also more and more the fraction invested in fixed income. The conventional wisdom of financial planners suggesting investor to invest in risky securities when they are young and mostly in fixed income when they approach retirement, is here made rigorous. The paper provides a strong warning for life insurers, pension fund managers and long term investors to take the possibility of less expensive products seriously to avoid the adverse consequences of the low interest rate regimes that many developed economies face., 42 pages

We propose a model for the instantaneous risk-free spot rate and for the spot LIBOR, driven by a time-homogeneous Markovian process. We introduce deterministic time-shifts in order to match any initial term-structure. By doing so, the model automatically becomes an exogenous term-structure model, in the spirit of Brigo and Mercurio (2001) who proposed this approach in the single curve case. A calibration exercise based on real data illustrates the flexibility of our approach for some typical speci fications used in the literature and in the bank industry.

We investigate PDEs of the form u t = 1 2 σ 2 ( t , x ) u x x − g ( x ) u which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can be used to obtain Fourier transforms of fundamental solutions of the PDE. We detail explicit computations in the separable volatility case when σ ( t , x ) = h ( t ) ( α + β x + γ x 2 ) , g = 0 , corresponding to the so called Quadratic Normal Volatility Model. We give financial applications and also show how symmetries can be used to compute first hitting distributions.

In this paper, we quantify the impact on the representative agent's welfare of the presence of derivative products spanning covariance risk. In an asset allocation framework with stochastic (co)variances, we allow the agent to invest not only in the stocks but also in the associated variance swaps. We solve this optimal portfolio allocation program using the Wishart Affine Stochastic Correlation framework, as introduced in Da Fonseca, Grasselli and Tebaldi (2007): it shares the analytical tractability of the single-asset counterpart represented by the [36] model and it seems to be the natural framework for studying multivariate problems when volatilities as well as correlations are stochastic. What is more, this framework shows how variance swaps can implicitly span the covariance risk. We provide the explicit solution to the portfolio optimization problem and we discuss the structure of the portfolio loadings with respect to model parameters. Using real data on major indexes, we find that the impact of covariance risk on the optimal strategy is huge. It first leads to a portfolio that is mostly driven by the market price of volatility-covolatility risks. It is then strongly leveraged through variance swaps, thus leading to a much higher utility, when compared to the case when investing in such derivatives is not possible.

Pricing of contingent claims in the Affine Term Structure Mod- els (ATSM) can be reduced to the solution of a set of Riccati-type Ordinary Differential Equations (ODE), as shown in Duffie, Pan and Singleton (2000) and in Duffie, Filipovic and Schachermayer (2001). We discuss the solvability of these equations. While admissibility is a necessary and sufficient condition in order to express their general so- lution as an analytic series expansion, we prove that, when the factors are restricted to have continuous paths, these ODE admit a funda- mental system of solutions if and only if all the positive factors are in- dependent. Finally, we classify and solve all the consistent polynomial term structure models admitting a fundamental system of solutions.

In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM.

In this paper we develop a novel market model where asset variances–covariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variance–covariance matrix. Analytical tractability is preserved since the model is linear-affine and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices for vanilla options consistent with observed smile and skew effects, while making it possible to detect and quantify the correlation risk in multiple-asset derivatives like basket options. In particular, it can reproduce and quantify the asymmetric conditional correlations observed on historical data for equity markets. As an illustrative example, we provide explicit pricing formulas for rainbow “Best-of” options.

© 2014 Elsevier B.V. This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic volatility model that recovers not only the typically observed implied volatility smiles and skews for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies, lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main reason for this important feature is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the proposed model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still works.

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.

In this paper we apply a new methodology based on quantization to price options in stochastic volatility models. This method can be applied to any model for which an Euler scheme is available for the underlying process and it allows for pricing vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein (1991) model and the SABR model introduced in Hagan and Woodward (2002). A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.

This paper provides the pricing for a new class of derivatives with different affine stochastic volatility models. These products are characterized by payoffs depending on both stock and its volatility. We provide closed-form solutions for different products and two multivariate Wishart-based stochastic volatility models. The methodology is independent of the dimension of the problem. Overall, our results highlight the great flexibility and tractability of Wishart-based stochastic volatility models to develop multivariate extensions of the Heston model.

In the financial literature, the problem of maximizing the expected utility of the terminal wealth has been investigated extensively (for a survey, see, e.g., Karatzas and Shreve (1998), p. 153, and references therein) by using different approaches. In this paper, we extend the existing literature in two directions. First, we let the utility function U(.) of the financial agent (who is a price taker) be implicitly defined through I(.)=(U ′ (.))–1, which is assumed to be additively separable, i.e., I(.)=∑ k=1 N I k (.). Second, we solve the investment problem in the general affine term structure model proposed by Duffie and Kan (1996) in which the functions I k (.), k=1,...,N are associated to HARA utility functions (with possibly different risk aversion parameters), and we show that the utility maximization problem leads to a Riccati ODE. Moreover, we extend to the multi-factor framework the stability result proved in Grasselli (2003), namely, the almost-sure convergence of the solution with respect to the parameters of the utility function. Mathematics Subject Classification (2000): 91B28 Journal of Economic Literature Classification: G11

In this paper, we analyse the Affine Term Structure Model (ATSM) proposed by Balduzzi, Das, Foresi and Sundaram (BDFS, 1996) and provide the closed-form expression of the bond price. In addition, we extend the notion of Impulse Response Function to the class of ATSM. We show that it is closely related to the duration measure, and we compute it explicitly in the BDFS model. (J.E.L.: C61, C63, G11).

We study an investment problem where the interest rates follow the Cox–Ingersoll–Ross dynamics. The optimal investment strategy is obtained in explicit form under the hypotheses that financial markets are complete and that the utility functions belong to the HARA, exponential and logarithmic classes. We show that the solution for the HARA utility is stable when the parameters vary in a suitable way: more precisely, we find that the optimal investment strategy corresponding to the HARA function converges almost surely to the one corresponding to the exponential and logarithmic utility functions.

In this paper, we present and discuss the estimation of the Wishart Affine Stochastic Correlation (WASC) model introduced in Da Fonseca et al. (2006) under the historical measure. We review the main estimation possibilities for this continuous time process and provide elements to show that the utilization of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We thus propose to use the estimation strategy presented in Carrasco et al. (2003), using a continuum of moment conditions based on the characteristic function. We investigate the behavior of the estimates through Monte Carlo simulations. Then, we present the estimation results obtained using a dataset of equity indexes: SP500, FTSE, DAX and CAC. On the basis of these results, we show that the WASC captures many of the known stylized facts associated with financial markets, including the negative correlation between stock returns and volatility. It also helps reveal interesting patterns in the studied indexes'covariances and their correlation dynamics.

We consider non mean-reverting Wishart processes and we study the problem of determining the smallest time such that the Laplace transforms of the process and its integral become infinite. Thanks to the remarkable property of (affine) Wishart processes to reproduce non-trivial dependence among the positive factors, we are able to explain the behavior of the explosion times in terms of the relative importance of the involved factors and their correlations. In this way, we go far beyond the known results that can be recovered in the classical affine framework.

Target volatility options (TVO) are a new class of derivatives whose payoff depends on some measure of volatility. These options allow investors to take a joint exposure to the evolution of the underlying asset, as well as to its realized volatility. For instance, a target volatility call can be viewed as a European call whose notional amount depends on the ratio of the target volatility (a fixed quantity representing the investor's expectation of the future realized volatility) and the realized volatility of the underlying asset over the life of the option. In equity options markets the slope of the skew is largely independent of the volatility level. A single-factor Heston based volatility model can generate steep skew or flat skew at a given volatility level but cannot generate both for a given parameterization. Since the payoff corresponding to TVO is a function of the joint evolution of the underlying asset and its realized variance, the consideration of stochastic skew is a relevant question for the valuation of TVO. In this sense, this article studies the effect of considering a multifactor stochastic volatility specification in the valuation of the TVO. To this end, we consider the two-factor Heston-based model of Christoffersen et al. (2009) in order to investigate TVO, as well as forward-start TVO, that is, TVO where the strike is determined at a later date.

In this paper we propose the first calibration exercise based on quantization methods. Pricing and calibration are typically difficult tasks to accomplish: pricing should be fast and accurate, otherwise calibration cannot be performed efficiently. We apply in a local volatility context the recursive marginal quantization methodology to the pricing of vanilla and barrier options. A successful calibration of the Quadratic Normal Volatility model is performed in order to show the potentiality of the method in a concrete example, while a numerical exercise on barrier options shows that quantization overcomes Monte-Carlo methods.

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known in closed-form. Moreover, the basket value is not required to be positive. We test our new bounds on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms, hence they do not suffer from the curse of dimensionality. In particular, our new lower bound turns out to be fast and accurate.

We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for PDEs; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model; this is useful in view of the numerical applications.

We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed efficiently through the FFT methodology thanks to the affinity of the model. A joint calibration exercise of the implied volatility surfaces of a triangle of FX rates shows the flexibility of our framework in dealing with the typical symmetries that characterize the FX market. Our framework is also able to describe many non trivial links between FX rates and interest rates: a second calibration exercise highlights the ability of the model to fi t simultaneously FX implied volatilities while being coherent with interest rate products.

We provide analytic pricing formulas for Fixed and Floating Range Accrual Notes within the multifactor Wishart affine framework which extends significantly the standard affine model. Using estimates for three short rate models, two of which are based on the Wishart process whilst the third one belongs to the standard affine framework, we price these structured products using the FFT methodology. Thanks to the Wishart tractability the hedge ratios are also easily computed. As the models are estimated on the same dataset, our results illustrate how the fit discrepancies (meaning differences in the likelihood functions) between models translate in terms of derivatives pricing errors, and we show that the models can produce different price evolutions for the Range Accrual Notes. The differences can be substantial and underline the importance of model risk both from a static and a dynamic perspective. These results are confirmed by an analysis performed at the hedge ratios level.