A. Papers
Abstract
The problem of pricing European options based on multiple assets with transaction costs is considered. These options include, for example, quality options and options on the minimum of two or more risky assets. The value of these options is the solution of a nonlinear parabolic partial differential equation subject to a final condition given by the payoff function associated with the option. A computationally efficient method to solve this final-value problem is proposed. This method is based on an asymptotic expansion of the required solution with respect to the parameters related to the transaction costs followed by the numerical solution of the linear partial differential equations obtained at each order in perturbation theory. The numerical solution of these linear problems involves an implicit finite-difference scheme for the parabolic equation and the use of the fast Fourier sine transform to solve the resulting elliptic problems. Numerical results obtained on test problems with the method proposed here are shown and discussed.
Abstract
We consider a n-dimensional square-root process and we obtain a formula involving series expansions for the associated transition probability density. The process mentioned previously can be used to model interest rates. The formula that we propose for the transition probability density has been obtained using appropriately a perturbative expansion in the correlation coefficients of the square root process, the Fourier transform and the method of characteristics to solve first order hyperbolic partial differential equations. The computational effort needed to evaluate this formula is polynomial with respect to the dimension n of the space spanned by the square-root process when the order where all the series involved in the transition probability density formula are truncated is fixed. This strategy gives an accuracy that some numerical tests show approximately constant for a wide range of values of $n$. Some examples of prices of financial derivatives whose evaluation involves integrals in two, twenty and one hundred dimensions (i.e. n=2,20,100), that is derivatives on two, twenty and one hundred assets, where accurate results can be obtained are shown. An experiment shows that the formula derived here for the transition probability density is well suited for parallel computing. The website http://www.econ.univpm.it/recchioni/finance/w1 contains an interactive tool that helps the understanding of this paper and a portable software library that makes possible to the user to exploit the formula derived in this paper to evaluate the transition probability densities of its own models and the prices of the associated financial derivatives.
Abstract
In the paper the real options theory and its practical use as a tool to estimate the value of small companies, research projects and patent portfolios are described. In particular small or very small companies are considered. Here the term “small company" means a company that produces only one product or a few products and that can be satisfactorily modeled with a few variables. The economic value of these companies is of great interest in the Italian economic context since this context is characterized by a multitude of small companies. Furthermore due to the globalization process, the aggregation of these small companies is becoming necessary to compete in the international markets. Moreover the international context forces the Italian companies to re-qualify their product portfolios making relevant the problem of the quantitative evaluation of research projects, patent portfolios and so on. The real options methodology may be a practical tool to get these quantitative evaluations. The website http://www.econ.univpm.it/recchioni/finance/w2 contains some illustrative material and two interactive applets that are supposed to help the understanding of the paper.
Abstract
In this paper using a perturbative method a series expansion of the price of a (put up and out) barrier option with time dependent parameters in the Black and Scholes world is obtained. The first three terms of this series are written explicitly as formulae involving some elementary and non elementary transcendental functions. The formula obtained has been tested on some examples taken from the financial literature. The numerical experience shows that in the cases of practical interest considered the use of the first two or three terms of the series expansion mentioned above guarantees three or four correct significant digits in the prices computed. Similar formulae can be obtained for other types of barrier options. The website: http://www.econ.univpm.it/recchioni/finance/w3 contains some auxiliary material that helps the understanding of this paper and the computer programs needed to evaluate the formula obtained.
Abstract
We consider the problem of pricing European exotic path-dependent derivatives on an underlying described by the Heston stochastic volatility model. Lipton has found a closed form integral representation of the joint transition probability density function of underlying price and variance in the Heston model. We give a convenient numerical approximation of this formula and we use the obtained approximated transition probability density function to price discrete path-dependent options as discounted expectations. The expected value of the payoff is calculated evaluating an integral with the Monte Carlo method, using a variance reduction technique based on a suitable approximation of the transition probability density function of the Heston model. As a test case, we evaluate the price of a discrete arithmetic average Asian option, when the average over n = 12 prices is considered, that is when the integral to evaluate is a 2n = 24 dimensional integral. We show that the method proposed is computationally efficient and gives accurate results.
Abstract
We consider the problem of pricing European interest rate derivatives based on the LIBOR Market Model (LMM) with one driving factor. We derive a closed-form approximation of the transition probability density functions associated to the stochastic dynamical systems that describe the behaviour of the forward LIBOR interest rates in the LMM. These approximate formulae are based on a truncated power series expansion of the solutions of the Fokker-Planck equations associated to the LMM. The approximate probability density functions obtained are used to price European interest rate derivatives using the method of discounted expectations. The resulting integrals are low-dimensional when the most commonly traded European interest rate derivatives are considered, and they can be computed efficiently using elementary numerical quadrature schemes (i.e. Simpson's rule). The algorithm obtained is very well suited for parallel computing and is tested on the problem of pricing several derivatives including an European swaption and an interest rate spread option. In both cases, the numerical method proposed appears to be accurate (i.e.: relative error of order 10^{-2}, 10^{-3}, or even 10^{-4}) and approximatively between 278 and 63000 times faster than previous methods based on the Monte Carlo simulation of the LMM stochastic dynamical systems. This website distributes to the potential users the software LIBOR_SWAPTION, a Fortran code that implements the numerical method proposed to compute the price of European swaptions.
Abstract
Let us suppose that the dynamics of the stock prices and of their stochastic variance is described by the Heston model, that is by a system of two stochastic differential equations with a suitable initial condition. The aim of this paper is to estimate the parameters of the Heston model and one component of the initial condition, that is the initial stochastic variance, from the knowledge of the stock and option prices observed at discrete times. The option prices considered refer to an European call on the stock whose prices are described by the Heston model. The method proposed to solve this problem is based on a filtering technique to construct a likelihood function and on the maximization of the likelihood function obtained. The estimated parameters and initial value component are characterized as being a maximizer of the likelihood function subject to some constraints. The solution of the filtering problem, used to construct the likelihood function, is based on an integral representation of the fundamental solution of the Fokker-Planck equation associated to the Heston model, on the use of the wavelet expansions presented in [1], [2], [3] to approximate the integral kernel appearing in the representation formula of the fundamental solution, on a simple truncation procedure to exploit the sparsifying properties of the wavelet expansions and on the use of the Fast Fourier Transform (FFT). The use of these techniques generates a very efficient and fully parallelizable numerical procedure to solve the filtering problem, this last fact makes possible to evaluate very efficiently the likelihood function and its gradient. As a byproduct of the solution of the filtering problem we have developed a stochastic variance tracking technique that gives very good results in numerical experiments. The maximum likelihood problem used in the estimation procedure is a low dimensional constrained optimization problem, its solution with ad hoc techniques is justified by the computational cost of evaluating the likelihood function and its gradient. We use parallel computing and a variable metric steepest ascent method to solve the maximum likelihood problem. Some numerical examples of the estimation problem using synthetic data obtained with a parallel implementation of the previous numerical method are presented. Very impressive speed up factors are obtained in the numerical examples using the parallel implementation of the numerical method proposed. This website contains animations and some auxiliary material that helps the understanding of this paper and makes available to the interested users the computer programs used to produce the numerical experience presented.
Abstract
This website presents a numerical method to price European options on realized variance. An European realized variance option is an option whose payoff depends on the time of maturity, on the variance of the log-returns of the stock price observed in a preassigned sequence of time values t_{i}, i = 0,1..,N. That is the realized variance is the variance observed in the sample of the log-returns, so that the value at maturity of the realized variance option depends on the discrete sample of the log-returns of the stock prices observed at the times ti, i = 0,1,¼,N. The method proposed to approximate the price of these options is based on the idea of approximating the discrete sum that gives the realized variance with an integral using as model of the log-return of the stock price dynamics the Heston stochastic volatility model. In this way the price of a realized variance option is approximated with the price of an integrated stochastic variance option whose payoff depends on the time of maturity and on the integrated stochastic variance. The integrated stochastic variance option is priced with the method of discounted expectations. We derive an integral representation formula for the price of this last kind of options. This integral formula reduces to an one dimensional Fourier integral in the case of the most commonly traded options that have a simple payoff function. The method has been validated on some test problems. The numerical experiments show that the approach suggested in this paper gives satisfactory approximations of the prices of realized variance options (relative error 10^{-2}, 10^{-3}) and allows substantial savings of computational time when compared with the Monte Carlo method used to evaluate realized variance options in the Heston stochastic volatility model with approximately the same accuracy. This website contains auxiliary material that helps the understanding of the paper [1] and makes available to the interested users the (FORTRAN) codes that implement the numerical method proposed here to price realized variance options. The use of one of these codes on a computing grid has been made user friendly developing a dedicated application using the software Symphony (that is a Service Oriented Architecture (SOAM) software of Platform Computing Toronto, Canada). This website makes available to the users this Symphony application.
Abstract
In [1] we study an inverse problem for a parabolic
partial differential equation. The parabolic partial differential equation
considered is the Fokker Planck equation
associated to a system of stochastic
differential equations and the inverse problem studied consists in finding from
suitable data the values of the parameters that appear in the coefficients of
this Fokker Planck equation, that is of the parameters that define the system
of stochastic differential equations. The data used in the reconstruction of
the parameters are observations made at discrete times of the stochastic process solution of the
system of stochastic differential equations. That is, the data of the inverse
problem are a “sample” taken from some of the components of the vector solution
of the stochastic differential equations and not, as usual, observations made
on the solution of the parabolic equation. The choice of the system of
stochastic differential equations and of the data used in the inverse
problem are motivated by applications in
mathematical finance. The stochastic differential equations presented can be
used to model the dynamics of the log-returns
of the index of some classes of hedge funds, such as, for example, the so
called “long short equity” hedge funds
and of some auxiliary variables. The
solution of the inverse problem proposed is obtained through the solution
of a filtering and of an estimation
problem. The solution of the last two problems is based on the knowledge of the
joint probability density function of the state variables of the model
conditioned to the observations made and
to the initial condition. This joint
probability density function is solution of an initial value problem for the
Kushner equation that in the circumstances considered here can be written as a
sequence of initial value problems with appropriate
initial conditions for the Fokker Planck equation associated to the system of
stochastic differential equations. An integral representation formula for this
probability density function is derived and used to develop a numerical
procedure to solve the estimation problem using the maximum likelihood method.
The computational method proposed has been tested on synthetic data and the results obtained are presented. This
website contains some auxiliary material useful to understand [1]
and contains some animations and some numerical experiments. A more general
reference to the work of the authors and of their coauthors in mathematical
finance is the website: http://www.econ.univpm.it/recchioni/finance.
Abstract
In [1] we study the problem of obtaining accurate estimates of the parameters, of the initial stochastic variance and of the risk premium parameter of the risk neutral measure of the Heston stochastic volatility model from the observation at discrete times of the stock log-returns and of the prices of a European call option on the stock. This problem is an inverse problem known in the literature as calibration problem. As a byproduct of the solution of the calibration problem we develop a tracking procedure that can be used to forecast the stochastic variance and the stock price. From a mathematical point of view the problem considered is formulated as a constrained optimization problem where the objective function is the logarithm of the likelihood associated to the parameter, the initial stochastic variance and the risk premium parameter values given the observed stock log-returns and option prices, this function is called (log-)likelihood function. The evaluation of the (log-)likelihood function associated to a given choice of the parameter, of the initial stochastic variance and of the risk premium parameter values requires the solution of a filtering problem for the Heston model. An accurate and computationally efficient solution of this filtering problem is necessary for a satisfactory solution of the calibration problem for the Heston model. A similar problem has been considered in [2] and [3]. The aim of this paper is to extend and improve the results obtained in [2] with respect to the formulation of the problem, the accuracy of the solution obtained and the computational efficiency of the solution method. In particular in comparison with [2] we reformulate the problem adding to the quantities that must be estimated the risk premium parameter and to the observations the option price at the initial time. The addition of the risk premium parameter to the quantities that must be estimated makes the problem considered more realistic and makes interesting the analysis of time series of real financial data using the method proposed. The addition of the option price at the initial time to the data used in the solution of the problem is natural and improves significantly the estimate of the initial stochastic variance. Moreover we simplify the expression of some formulae used in [2] in the computation of the (log-)likelihood function and we improve the optimization method employed to solve the maximum likelihood problem introducing some ad hoc preliminary optimization steps. Finally a new, easy to compute, formula that gives the "Heston" option price is derived. The use of this new formula reduces substantially the computational cost of evaluating the (log-)likelihood function when compared to the cost of the (log-)likelihood function evaluation in [2]. Some numerical examples of the calibration problem using synthetic and real data are presented. As real data we use the 2005 historical data of the US S&P500 index and of the corresponding options prices. Very good results are obtained forecasting future values of the S&P500 index and of the corresponding option price using the solution of the calibration problem and the forecasting and tracking procedure mentioned above. In the website http://www.econ.univpm.it/recchioni/finance/w6 some auxiliary material useful in the understanding of [1] including some animations and some numerical experiments is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
Abstract
In [3] we study an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two different time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions the transition probability density function of the stochastic process solution of the model is represented as a one dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. In mathematical finance multiscale stochastic volatility models are used for several purposes such as, for example: i) the study of commodity prices, in fact usually commodity prices are characterized by spikes that can be modeled using a fast time scale volatility together with a standard time scale volatility, ii) the study of financial products that live for long time periods (such as life insurance contracts) that can be modeled using a long time scale volatility together with a standard time scale volatility. In alternative to multiscale stochastic volatility models the stochastic jump models, such as, for example, Lévy's processes with stochastic volatility, can be used. However this last type of models is explicitly solvable only in very special circumstances and in general the transition probability density functions associated to them are defined as solutions of integro-differential equations that must be solved numerically. So that the use of jump models in practical situations is, in general, cumbersome when compared to the use of the model proposed here. In order to use the multiscale stochastic volatility model proposed here to compute option prices we define its risk neutral measure and the associated risk premium parameters. In [3] we derive a formula for the price of the European vanilla (call and put) options in the multiscale stochastic volatility model considered. The formulae obtained are given as one dimensional integrals of explicitly known integrands and in a special case reduce to the corresponding formulae in the Heston model [7]. This last formula in the case of a European vanilla call option improves the formula given in Heston [7] formula (10) in a sense made clear later. We use the option price formulae obtained to study the values of the model parameters, of the correlation coefficients of the Wiener processes defining the model and of the initial stochastic variances implied by the ``observed" option prices using both synthetic and real data. The real data analyzed are those relative to the S&P 500 index in the year 2005. That is we generalize to the model proposed here the so called term structure analysis of the implied volatility used in the Black Scholes model. This analysis is translated in the solution of a constrained nonlinear optimization problem with an objective function containing an integral that must be evaluated numerically. The real data analysis presented shows that the multiscale stochastic volatility model gives a satisfactory explanation of the observed option prices including those of the out of the money options and that the implied values of the model parameters, of the correlation coefficients and of the initial stochastic variances can be used to obtain high quality forecasts of the option prices. This website contains some auxiliary material including some animations that helps the understanding of [3]. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance/ .
Abstract
In this paper we test the ability of a stochastic differential model proposed in [12] of forecasting the returns of a long-short equity hedge fund index and of a market index, that is of the HFRI-Equity index and of the S&P 500 index respectively. The model is based on the assumptions that the value of the variation of the log-return of the hedge fund index (HFRI Equity) is proportional up to an additive stochastic error to the value of the variation of the log-return of a market index (S&P 500) and that the log-return of the market index can be satisfactorily modeled using the Heston stochastic volatility model. The model consists in a system of three stochastic differential equations, two of them are the Heston stochastic volatility model and the third one is the equation that models the behaviour of the hedge fund index and its relation with the market index. The model is calibrated on observed data using a method based on filtering and maximum likelihood proposed in [22] and further developed in [12], [13]. The data analyzed go from January 1990 to June 2007, and are monthly data. For each observation time they consist in the value at the observation time of the log-returns of the HFRI-Equity and of the S&P 500 indices. The calibration procedure uses appropriate subsets of data, that is the data observed in a six months time period. The values of the HFRI-Equity and S&P 500 indices forecasted by the calibrated models are compared to the values of the indices observed. The result of the comparison is very satisfactory. The website http://www.econ.univpm.it/recchioni/finance/w8 contains some auxiliary material including some animations that helps the understanding of this paper. A more general reference to the work of some of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.
Abstract
A software package named Option Engine that implements 27 modules of the QuantLib software library in a grid environment consisting in the Platform Symphony Enterprise Grid package is presented. The Option Engine package can be downloaded from the website http://www.ceri.uniroma1.it/ceri/zirilli/w5. The QuantLib modules evaluate the prices of several kinds of traded financial options. The titles of the modules are: 01. Black-Scholes, 02. Barone-Adesi/Whaley, 03. Bierksund/Stensland, 04. Integral, 05. Finite Differences, 06. Binomial Jarrow-Rudd, 07. Binomial Cox-Ross-Rubistein, 08. Binomial equiprobabilities, 09. Binomial Trigeorgis, 10. Binomial Tian, 11. Binomial Leisen-Reimer, 12. Binomial Joshi, 13. MC (crude), 14. QMC (Sobol), 15. MC (Longstaff Schwartz), 16. Digital payoff, 17. Discrete dividends, 18. Bates, 19. Ju, 20. FD for dividend options, 21. Heston, 22. Black-Scholes with Hull-White, 23. Heston with Hull-White, 24. MC (crude) Heston, 25. QMC (Sobol) Heston, 26. Heston Variance Option, 27. Perturbative Barrier Option . The modules 26 Heston Variance Option and 27 Perturbative Barrier Option are derived from the research work of the authors and of their coauthors, see [1], [2] respectively. The main novelty of the software package Option Engine is that it works in the Symphony environment. Symphony tries the optimization of the use of the computing resources of a parallel machine or of a computing grid.
References
[1] L. Fatone, M.C. Recchioni, F. Zirilli: "A perturbative formula to price barrier options with time dependent parameters in the Black and Scholes world", Journal of Risk 10 (2008) 131-146. (http://www.econ.univpm.it/recchioni/finance/w3)
[2] L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli: "Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model", Discrete and Continuous Dynamical Systems Supplement 2007 (2007) 354-363. (http://www.econ.univpm.it/recchioni/finance/w4)
Abstract
In this paper we use filtering and maximum likelihood methods to solve a calibration problem for a multiscale stochastic volatility model. The multiscale stochastic volatility model considered has been introduced in [1], [2] and describes the dynamics of the asset price using as auxiliary variables two stochastic variances varying on two different time scales. The aim of this paper is to estimate the parameters of this multiscale model (including the risk premium parameters when necessary) and its two initial stochastic variances from the knowledge, at discrete times, of the asset price and, eventually, of the prices of call and/or put European options on the asset. This problem is translated in a maximum likelihood problem with the likelihood function defined through the solution of a filtering problem. The estimated values of the parameters and of the two initial stochastic variances are characterized as being a constrained maximizer of a likelihood function. Furthermore we develop a tracking procedure that is able to track the asset price and the values of its two stochastic variances for time values where there are no data available. The solution of the calibration problem and the tracking procedure are used to do the analysis of data time series. Numerical examples of the solution of the calibration problem and of the performance of the tracking procedure using synthetic and real data are presented. The synthetic data time series is analyzed using data samples made of observations done with a given time frequency. We consider observation frequencies ranging from "high" to "low" frequencies and we study the performance of the calibration procedure as a function of the observation frequency. The real data studied are two time series of electric power price data taken from the U.S. electricity market and the 2005 data relative to the US S&P 500 index and to the prices of a call and a put European options on the S&P 500 index. In the study of real data we consider daily data. The results obtained from the analysis of the synthetic and of the real data are very satisfactory. Moreover in the real data case we use the estimated values of the parameters and of the initial stochastic variances obtained solving the calibration problem to produce through the tracking procedure mentioned above forecasts of the asset prices (electric power price and S&P 500 index), of the associated stochastic variances and of the option prices. The forecasts of the asset prices and of the option prices are compared with the prices actually observed. This comparison shows that the forecasts are of very high quality even when we consider "spiky" electric power price data. The website: http://www.econ.univpm.it/recchioni/finance/w9 contains some auxiliary material including some animations that helps the understanding of this paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
References
[1] L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli: " Calibration of a multiscale stochastic volatility model using as data European option prices", Mathematical Methods in Economics and Finance 3 (2008) 49-61.
[2] L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli: " An explicitly solvable multi-scale stochastic volatility model: option pricing and calibration ",Journal of Futures Markets 29(9) (2009) 862-893. (http://www.econ.univpm.it/recchioni/finance/w4)
Abstract
In this paper we use filtering and maximum likelihood methods to solve a calibration problem for a stochastic dynamical system used to model spiky asset prices. The data used in the calibration problem are the observations at discrete times of the asset price. The model considered has been introduced by V.A. Kholodnyi in [1], [2] and describes spiky asset prices through a stochastic process that can be represented as the product of two independent Markov processes: the spike process and the process that represents the asset prices in absence of spikes. A Markov chain is used to regulate the transitions between presence and absence of spikes. The calibration problem for this model is translated in a maximum likelihood problem with the likelihood function defined through the solution of a filtering problem as suggested in [3]. The estimated values of the model parameters are a constrained maximizer of the likelihood function. Furthermore, given the calibrated model, we develop a sort of tracking procedure able to forecast the forward asset prices. Numerical examples using synthetic and real data of the solution of the calibration problem and of the performance of the tracking procedure are presented. The real data studied are electric power prices data taken from the U.K. electricity market. The forward prices forecasted with the tracking procedure and the observed forward prices are compared to evaluate the quality of the model and of the forecasting procedure. The numerical results obtained are satisfactory. This website contains some auxiliary material including animations that helps the understanding of this paper. A more general reference to the work of the authors and of their co-authors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
References
[1] V.A. Kholodnyi: " Valuation and hedging of European contingent claims on power with spikes: A non-Markovian approach", Journal of Engineering Mathematics 49(3) (2004) 233-252.
[1] V.A. Kholodnyi: " The non-Markovian approach to the valuation and hedging of European contingent claims on power with scaling spikes", Nonlinear Analysis: Hybrid Systems 2 (2008) 285-304.
[2] F.Mariani, G.Pacelli, F.Zirilli: " Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: an application of nonlinear filtering theory", Optimization Letters 2 (2008) 177-222.
Abstract
A new method to solve the calibration problem for the Black-Scholes asset price dynamics model is proposed. The data used in the calibration problem are the observations of the asset price on a finite set of equispaced (known) discrete time values. Statistical tests are used to obtain estimates with statistical significance of the two parameters of the Black-Scholes model, that is of the volatility and of the drift. The consequences of these estimates on the option pricing problem are investigated. In particular the pricing problem for options with uncertain volatility in the Black-Scholes framework is revisited and a statistical significance is associated to the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments with synthetic and real data are presented. The real data considered are the daily closing values of the S&P500 index and of the associated European call and put option prices in the year 2005. The method proposed to calibrate the Black-Scholes dynamics model can be extended to other models containing stochastic dynamical systems used in science and engineering. In this website some auxiliary material, including animations, that helps the understanding of this paper is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance/.
Abstract
In this paper we consider the problem of pricing life insurance contracts using ideas and methods taken from mathematical finance, more specifically taken from the solution given in mathematical finance to the option pricing problem. The methodology developed is rather general, however, to fix the ideas, we focus our attention on the problem of pricing a pure endowment policy also known as Guaranteed Minimum Income Benefit Contract. In particular we consider a pure endowment policy that has its life contingent payout linked to the performance of a risky asset (such as, for example, a stock, a market index, or an interest rate). The behaviour of the value of the risky asset is modeled using a multiscale stochastic volatility model given by a system of three stochastic differential equations. This model has been introduced in [1] in the context of mathematical finance. The mortality risk is modeled via a stochastic differential equation suggested in [2], [3] that describes the time evolution of the mortality rate. That is the time dynamics of the state variables of the model used to price a pure endowment policy is defined by the system of four stochastic differential equations obtained considering simultaneously the two models mentioned above. Under the assumption that the financial and the mortality risks are independent a formula to price the simplest pure endowment policy is derived as the product of the discounted expected value of the money that must be paid at maturity of the contract to the policy holder if the policy holder is alive at maturity time and of the relevant survival probability. The relevant survival probability is the probability that an individual belonging to the ``cohort" of the policy holder is alive at maturity time given the fact that he is alive at the time when the contract is subscribed. A ``cohort" is defined as the subset of the population made by the individuals born in a given year. A computational method to evaluate the pricing formula derived is developed. This computational method largely overcomes in accuracy and computational efficiency the performance of the Monte Carlo method used in a straightforward way to evaluate the same formula. The computational method proposed is based on two formulae: a semi-explicit formula to price in the multiscale stochastic volatility model adopted a European call option [1] and an explicit formula for a family of conditioned transition probability densities associated to the hazard rate in the mortality risk model considered. This last formula leads to an ad hoc Monte Carlo method to evaluate the survival probability. Furthermore, in the limit when the volatility parameter of the mortality risk model goes to zero an asymptotic formula for the survival probability is derived. In its region of validity the numerical evaluation of this asymptotic formula leads to a method to price the pure endowment policy that greatly outperforms the computational methods mentioned above based on Monte Carlo. Some numerical experiments on test problems are presented. Elementary test problems that can be solved explicitly are used to show the performance of the numerical methods proposed and the accuracy of the results obtained. More complex test problems are considered. On these last test problems a comparison between the results obtained with the numerical methods presented is carried out. Finally the range of validity of the asymptotic formula for the survival probability is studied. Auxiliary material that assists an understanding of this work is on this website. A more general reference to the work of some of the authors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance/.
References
[1] L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli: " An explicitly solvable multi-scale stochastic volatility model: option pricing and calibration ",Journal of Futures Markets 29(9) (2009) 862-893. (http://www.econ.univpm.it/recchioni/finance/w4)
[2] R. Giacometti, S. Ortobelli, M.I. Bertocchi: "A stochastic model for mortality rate on Italian data ", submitted to Journal of Optimization Theory and Applications (2010).
[3] M.A. Milevsky, S.D. Promislow: "Mortality derivatives and the option to annuitise", Insurance: Mathematics and Economics 29 (2001) 299-318.
Abstract
This paper shows on two examples how the analysis of option pricing problems can lead to computational methods efficiently implemented in parallel that outperform the “general purpose” methods (i.e., for example, Monte Carlo, finite differences). In particular this paper studies the GPU implementation of two numerical algorithms to price two derivatives: continuous barrier options and realized variance options. These algorithms are implemented in CUDA subroutines ready to run on Graphics Processing Units (GPUs) and their performance is studied. The implementation of these subroutines is motivated by the extensive use of the derivatives considered in the financial markets to hedge or to take risk and by the interest of the financial institutions in the use of state of the art hardware and software to speed up the decision process. The performance of these algorithms is measured using the (CPU/GPU) speed up factor, that is using the ratio between the (wall clock) times required to execute the code on a CPU and on a GPU. The choice of the reference CPU and GPU used to evaluate the speed up factors presented is stated. The outstanding performance of the algorithms developed is due to the mathematical properties of the pricing formulae used and to the ad hoc software implementation. The performance of these algorithms is measured using the (CPU/GPU) speed up factor, that is using the ratio between the (wall clock) times required to execute the code on a CP Uand on a GP U. The choice of the reference CPU and GPU used to evaluate the speed up factors presented is stated. In the case of realized variance options when the computation is done in single precision the comparisons between CPU and GPU (wall clock) execution times gives speed up factors of the order of a few hundreds. For barrier options the corresponding speed up factors are of about fifteen, twenty. The CUDA subroutines to price barrier options and realized variance options can be downloaded from the website http://www.econ.univpm.it/recchioni/finance/w13.
Abstract
A multiscale SABR model that describes the dynamics of forward prices/rates is presented. New closed form formulae for the transition probability density functions of the normal and lognormal SABR and multiscale SABR models and for the prices of the corresponding European call and put options are deduced. The technique used to obtain these formulae is rather general and can be used to study other stochastic volatility models. A calibration problem for these models is formulated and solved. Numerical experiments with real data are presented.
Abstract
We investigate the idea of solving calibration problems for stochastic dynamical systems using statistical tests. We consider a specific stochastic dynamical system: the normal SABR model. The SABR model has been introduced in mathematical finance in 2002 by Hagan, Kumar, Lesniewski, Woodward [1]. The model is a system of two stochastic differential equations whose independent variable is the time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The normal SABR model is a special case of the SABR model. The calibration problem for the normal SABR model is an inverse problem that consists in determining the values of the parameters of the model from a set of data. We consider as set of data two different sets of forward prices/rates and we study the resulting calibration problems. Ad hoc statistical tests are developed to solve these calibration problems. Estimates with statistical significance of the parameters of the model are obtained. Let T>0 be a constant, we consider multiple independent trajectories of the normal SABR model associated to given initial conditions assigned at time t = 0. In the first calibration problem studied the set of the forward prices/rates observed at time t=T in this set of trajectories is used as data sample of a statistical test. The statistical test used to solve this calibration problem is based on some new formulae for the moments of the forward prices/rates variable of the normal SABR model. The second calibration problem studied uses as data sample the observations of the forward prices/rates made on a discrete set of given time values along a single trajectory of the normal SABR model. The statistical test used to solve this second calibration problem is based on the numerical evaluation of some high dimensional integrals. The results obtained in the study of the normal SABR model are easily extended from mathematical finance to other contexts in science and engineering where stochastic models involving stochastic volatility or stochastic state space models are used. This website contains some auxiliary material that helps the understanding of this paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance/.
References
[1] P. S. Hagan, D. Kumar, A. S. Lesniewski, D. E. Woodward, "Managing smile risk ",Wilmott Magazine September 2002 (2002) 84-108. (http://www.wilmott.com/pdfs/021118_smile.pdf)
Abstract
We study two calibration problems for the lognormal SABR model using the moment method and some new formulae for the moments of the logarithm of the forward prices/rates variable. The lognormal SABR model is a special case of the SABR model [1] and is a system of two stochastic differential equations widely used in mathematical finance whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The lognormal SABR model depends on three parameters. A calibration problem is an inverse problem that consists in determining the values of these three parameters starting from a set of data. We consider two different sets of data, that is: i) the set of the forward prices/rates observed at a given time on multiple independent trajectories of the lognormal SABR model, ii) the set of the forward prices/rates observed on a discrete set of known time values along a single trajectory of the lognormal SABR model. The calibration problems corresponding to these two sets of data are formulated as constrained nonlinear least squares problems and solved numerically. Note that in the financial markets the first set of data considered is hardly available while the second set of data is of common use and corresponds simply to the time series of the observed prices. As a consequence the first calibration problem although realistic in several contexts of science and engineering is of limited interest in finance while the second calibration problem is of practical use in finance (and elsewhere). The formulation of the calibration problems and the methods used to solve them are tested on synthetic and on real data. The real data studied are the data belonging to a time series of exchange rates between currencies (euro/U.S. dollar exchange rates). In this website some auxiliary material that helps the understanding of this paper is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance/.
References
[1] P. S. Hagan, D. Kumar, A. S. Lesniewski, D. E. Woodward, "Managing smile risk ",Wilmott Magazine September 2002 (2002) 84-108. (http://www.wilmott.com/pdfs/021118_smile.pdf)
Abstract
An explicit formula for the transition probability density function of the Hull and White stochastic volatility model in presence of nonzero correlation between the stochastic differentials of the Wiener processes on the right hand side of the model equations is presented. This formula gives the transition probability density function as a two dimensional integral of an explicitly known integrand. Previously an explicit formula for this probability density function was known only in the case of zero correlation. In the case of nonzero correlation from the formula for the transition probability density function we deduce formulae (expressed by integrals) for the price of European call and put options and closed form formulae (that do not involve integrals) for the moments of the asset price logarithm. These formulae are based on recent results on the Whittaker functions [1] and generalize similar formulae for the SABR and multiscale SABR models [2]. Using the option pricing formulae derived and the least squares method a calibration problem for the Hull and White model is formulated and solved numerically. The calibration problem uses as data a set of option prices. Experiments with real data are presented. The real data studied are those belonging to a time series of the USA S&P 500 index and of the prices of its European call and put options. The quality of the model and of the calibration procedure is established comparing the forecast option prices obtained using the calibrated model with the option prices actually observed in the financial market. The website: http://www.econ.univpm.it/recchioni/finance/w17 contains some auxiliary material including animations and interactive applications that helps the understanding of this paper.
References
[1] R. Szmytkowki, S. Bielski, "An orthogonality relation for the Whittaker functions of the second kind of imaginary order ", Integral Transforms and Special Functions 21(10) (2010) 739-744.
[2] L. Fatone, F. Mariani, M.C. Recchioni, F. Zirilli, " Some explicitly solvable SABR and multiscale SABR models: option pricing and calibration ", Journal of Mathematical Finance 3 (2013) 10-32.
Abstract
The SABR stochastic volatility model with b-volatility b Î (0, 1) is studied. The model is further specified imposing an absorbing barrier in zero to the forward prices/rates stochastic process. This model is considered in presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations. A series expansion in powers of the correlation coefficient of these stochastic differentials of the transition probability density function of the SABR model is presented. Explicit formulae for the first three terms of this expansion are given. These formulae are integrals of known integrands. A new expression of the transition probability density function of the SABR model when the correlation coefficient is zero is obtained as zero-th order term of the expansion. The expansion is deduced starting from the final value problem for the backward Kolmogorov equation of the SABR model satisfied by the transition probability density function. Each term of the expansion is the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and the Kontorovich-Lebedev transforms. From the series expansion for the probability density function we deduce the corresponding expansions for the prices of European call and put options. The terms of the series expansions of the option prices are integrals of known integrands. For the moments of the forward prices/rates variable of the SABR model closed form formulae that do not involve integrals and series expansions are obtained. These formulae are expressed as the product of an elementary function times a polynomial in the correlation coefficient. The option pricing formulae obtained are used to study synthetic and real data. In particular they are used to study a time series of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations and interactive applications that helps the understanding of the paper.
Abstract
In this paper we present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative'” investors that: i) behave in a similar way, ii) try to avoid abrupt changes in their trading strategies, iii) want to limit the risk due to the fact of having open positions on the asset shares, iv) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and a stochastic optimal control problem. These models are coupled by the asset share price dynamic equation. In particular this equation describes how the asset shares price depends from the trading activity of the retail traders and of the big trader. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. The knowledge of this strategy is necessary to formulate the optimal control problem that determines the strategy adopted by the big trader. The mathematical models considered are solved using the dynamic programming principle. Under some hypotheses the solution of the mean field game model is reduced to the solution of a (constrained) two point boundary value problem for a system of six Riccati ordinary differential equations. The optimal control problem associated to the big trader is a linear quadratic control problem. Under some hypotheses an explicit formula for the optimal trading execution strategy of the big trader is deduced. A numerical study of the trading execution model proposed is presented. The trading execution strategies obtained with the model developed in this paper and those obtained using alternative models are compared. This comparison confirms the quality of the model presented. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
Abstract
This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order trucated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the series expansions of the option prices and of the corresponding free boundaries outperform in accuracy and/or in computational cost the results obtained with alternative methods ([1], [2], [3]). For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the free boundary obtained using the Barone-Adesi, Whaley method [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: website: http://www.econ.univpm.it/recchioni/finance.
References
[1] G. Barone-Adesi, R.E. Whaley, "Efficient analytical approximation of American option values", The Journal of Finance, 42(2) (1987) 301-320.
[2] N. Ju, R. Zhong, "An approximate formula for pricing American options", The Journal of Derivatives, 7(2) (1999) 31-40.
[3] T. Little, V. Pant, C. Hou, "A new integral representation of the early exercise boundary for American put options", Journal of Computational Finance, 3 (2000) 73-96.
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