Analytically tractable stochastic volatility models in asset and option pricing

Questa tesi si compone di quattro saggi sui modelli stocastici di volatilità in asset e option pricing. Più precisamente, questa tesi si concentra sul tasso di interesse stocastico e sui modelli di volatilità stocastica multiscala, con applicazioni in vari prodotti finanziari. Nel primo saggio viene presentato un modello ibrido Heston-CIR (HCIR) con un tasso di interesse stocastico. In questo saggio, sono state dedotte formule elementari esplicite per i momenti relativi alle distribuzioni dei prezzi delle azioni, nonché formule efficaci per approssimare i prezzi delle opzioni. Utilizzando i prezzi di call e put options europei sul l'indice S&P 500 americano, questo studio empirico dimostra che il modello HCIR è migliore del modello di Heston nell'interpretazione e nella previsione dei prezzi di call e put options. Il secondo saggio è un ulteriore estensione del modello HCIR con due diverse applicazioni. La prima applicazione utilizza il modello HCIR per interpretare la struttura a termine dei rendimenti e per prevedere la loro tendenza al rialzo /ribasso. La seconda analisi è basata sui valori della “long-term health endowment policy”. L'analisi empirica mostra che il tasso di interesse stocastico gioca un ruolo cruciale come fattore di volatilità e fornisce un modello multi-fattore che supera il modello di Heston nel predire prezzo della polizza assicurativa sanitaria. Nel terzo saggio è stato proposto un modello ibrido di tipo Heston Hull-White (HHW) per descrivere le dinamiche di un prezzo di asset con volatilità e tasso di interesse stocastici che tiene conto di valori negativi. Nel lavoro sono state dedotte sia formule elementari esplicite per la funzione di densità di probabilità di transizione della variabile prezzi degli asset che formule in forma chiusa per stimare il prezzo delle opzioni. Nella prima analisi empirica è stato calibrato il modello HHW utilizzando la cosiddetta implied volatility. La seconda analisi empirica si concentra sui prezzi di futures Eurodollar e i corrispondenti prezzi delle opzioni europee con una generalizzazione del modello di Heston con tassi di interesse stocastici. I risultati relativi all’approssimazione e alla previsione sono molto interessanti. Ciò conferma l'efficienza del modello di HHW e la necessità di considerare possibili valori negativi di tasso di interesse. Il quarto saggio descrive un modello di Heston ibrido e multiscala per il tasso di cambio a pronti FX per poter considerare tassi di interesse stocastici. Il trattamento analitico del modello è descritto in dettaglio sia considerando delle misure legate alla fisica che misura di neutralità al rischio. In particolare, è stata derivata una formula per la funzione di densità di transizione di probabilità usando un integrale a una dimensione di una funzione integrale elementare che viene utilizzato per il prezzaggio delle opzioni call e put European Vanilla.This dissertation consists of four related essays on stochastic volatility models in asset and option pricing. More precisely, this dissertation focuses on stochastic interest rate and multiscale stochastic volatility models, with applications in various financial products. In first essay, a hybrid Heston-CIR (HCIR) model with a stochastic interest rate process is presented. In this essay, explicit elementary formulas for the moments of the asset price variables as well as efficient formulas to approximate the option prices are deduced. Using European call and put option prices on U.S. S&P 500 index, empirical study shows that the HCIR model outperforms Heston model in interpreting and predicting both call and put option prices. The second essay is a further extension of the HCIR model with two different applications. The first application is using HCIR model to interpret bond yield term structure and to forecast their upward/downward trend. The second analysis is based on the values of the long-term health endowment policy. The empirical analysis shows that the stochastic interest rate plays a crucial role as a volatility factor and provides a multi-factor model that outperforms the Heston model in predicting health endowment policy price. In the third essay, a hybrid Heston Hull-White (HHW) model is designed to describe the dynamics of an asset price under stochastic volatility and interest rate that allows negative values. Explicit elementary formulas for the transition probability density function of the asset price variable and closed-form formulas to approximate the option prices are deduced. In first empirical analysis, the HHW model is calibrated by using implied volatility. The second empirical analysis focuses on the Eurodollar futures prices and the corresponding European options prices with a generalization of the Heston model in the stochastic interest rate framework. Both the results are impressive for approximation and prediction. This confirms the efficiency of HHW model and the necessary to allow for negative values of interest rate. The fourth essay describes a multiscale hybrid Heston model of the spot FX rate which is an extension of the model De Col, Gnoatto and Grasselli 2013 in order to allow stochastic interest rate. The analytical treatment of the model is described in detail both under physical measure and risk neutral measure. In particular, a formula for the transition probability density function is derived as a one dimensional integral of an elementary integral function which is used to price European Vanilla call and put options

The Heston model under stochastic interest rates

Tese de mestrado, Matemática Financeira, Faculdade de Ciências, Universidade de Lisboa,2008In this dissertation the Heston (1993) model is considered, but using, instead of a constant interest rate, stochastic interest rates according to Vasicek (1977) and to Cox, Ingersoll and Ross (1985) models. Under this framework, a closed-form solution is determined for the price of European standard calls, which, by using a manipulation implemented by Attari (2004), only require the evaluation of one characteristic function. For forward-start European calls, starting from the result for standard calls and using analytic characteristic functions, it is determined a closed-form solution that only requires one numerical integration. In the end, the results of these closedform solutions are compared with the results presented by Monte Carlo simulations for the considered models.Nesta dissertação é considerado o modelo de Heston (1993), mas em vez de utilizar uma taxa de juro constante, considera-se taxas de juro estocásticas segundo os modelos de Vasicek (1977) e de Cox, Ingersoll e Ross (1985). Neste contexto, é determinada uma solução fechada para a avaliação de standard calls Europeias, que, por ter sido usada uma manipulação implementada por Attari (2004), apenas necessitará da avaliação de uma função característica. Para calls forward-start Europeias, partindo do resultado apresentado para standard calls e utilizando funções característica analíticas, é determinada uma solução fechada que também recorrerá a apenas uma integração numérica. No final, os resultados destas fórmulas fechadas são comparados com os resultantes de simulações de Monte Carlo para os modelos considerados

Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

Pricing options and computing implied volatilities using neural networks

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data set generated by a sophisticated financial model, and runs the trained ANN as an agent of the original solver in a fast and efficient way. We test this approach on three different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and Brent's iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly

Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

Random Time Forward Starting Options

We introduce a natural generalization of the forward-starting options, first discussed by M. Rubinstein. The main feature of the contract presented here is that the strike-determination time is not fixed ex-ante, but allowed to be random, usually related to the occurrence of some event, either of financial nature or not. We will call these options {\bf Random Time Forward Starting (RTFS)}. We show that, under an appropriate "martingale preserving" hypothesis, we can exhibit arbitrage free prices, which can be explicitly computed in many classical market models, at least under independence between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally a credit value adjustment formula for these OTC options is computed for the unilateral counterparty credit risk.Comment: 19 pages, 1 figur

An Analysis of the Heston Stochastic Volatility Model: Implementation and Calibration using Matlab

This paper analyses the implementation and calibration of the Heston Stochastic Volatility Model. We first explain how characteristic functions can be used to estimate option prices. Then we consider the implementation of the Heston model, showing that relatively simple solutions can lead to fast and accurate vanilla option prices. We also perform several calibration tests, using both local and global optimization. Our analyses show that straightforward setups deliver good calibration results. All calculations are carried out in Matlab and numerical examples are included in the paper to facilitate the understanding of mathematical concepts.Comment: 34 page

An analytic multi-currency model with stochastic volatility and stochastic interest rates

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 effciently through the FFT methodology thanks to the affinity of the model 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 fit simultaneously FX implied volatilities while being coherent with interest rate products