Approximate option pricing for jump-diffusion stochastic volatility models

Abstract

[eng] The following is a summary of the above-mentioned thesis. The thesis covers alternative stochastic models, risk management, and option price decomposition.

1.1 Alternative Stochastic Models Several alternative asset pricing models are explored, each designed to capture market dynamics better. The models we consider include: 1. The Hybrid Stochastic Local Volatility (SLV) model entitled the Heston-CEV model with jumps (HCEVJ). This model combines the strengths of the Constant Elasticity of Variance (CEV) and Heston models, making volatility dependent on time, asset price, and an underlying stochastic process. 2. The 2-factor stochastic volatility model with jumps (2FSVJ) with multiple factors to drive the volatility process, offering richer dynamics. 3. Lastly, we consider the infinite activity Heston-Lévy model to account for large changes in asset prices due to jumps, thus capturing large market movements. The HCEVJ model is a general model with the CEV, Heston, Heston-CEV, Bates, and Heston-CEV with jumps as special cases. We calibrate each of these models to the EURO STOXX 50 European option quotes of 30 September 2014 using a brute-force grid search algorithm to facilitate comparisons. Monte Carlo methods are employed to model the asset price movements and verify model properties. The analysis shows that the HCEVJ model deepens the volatility smile by introducing an additional parameter that controls the intensity of the volatility smile. Moreover, the HCEVJ model exhibits the leverage effect, volatility clustering and price jumps, offering a more comprehensive representation of asset price dynamics. The properties of the 2FSVJ and Heston-Lévy models are not tested, however.

1.2 Malliavin Calculus, Hedging, and Option Greeks A key element of options pricing is hedging and option sensitivities. In this thesis, we use Malliavin Calculus to obtain faster estimates of the first-order Greeks of the option prices for the HCEVJ model. Additionally, we use the Clark-Ocone formula to find an explicit formula to hedge a portfolio.

1.3 Pricing Methods The third and main objective is to explore option pricing methods. The thesis reviews existing approaches, such as Monte Carlo simulations, Fourier integral methods, and decomposition techniques, and compares them. Decomposition methods outperform Monte Carlo simulations and Fourier integral methods, particularly under simple jump structures. In the 2FSVJ case, a first-order and a second-order decomposition formula are derived under a general jump structure. We choose log-normal and double-exponential jumps and carry out numerical experiments. The double-exponential jump case shows poorer accuracy and speed performance. Furthermore, the decomposition pricing model was more accurate under short-maturity and out-of-the-money conditions. In the Heston-Lévy case, we build on existing research on decomposition formulas by considering infinite activity jumps in asset returns. The study addresses the pricing of options in models where the return process includes both stochastic volatility and jumps of infinite activity but finite variation. The key contribution of this work is two exact decomposition formulas for option pricing under this model. The first decomposition formula expands the pricing model using a series of expectations involving the underlying asset and volatility terms and correction terms for the jumps in the asset price. However, this approach remains computationally challenging, as obtaining a tractable version of the formula is an open problem. The second Decomposition Formula uses methods from Lévy process estimation to simplify the infinite activity problem by approximating it as a finite activity jump process. Error bounds are provided for this approximation, and numerical estimates are compared to benchmark prices. The method, while accurate, relies on Monte Carlo simulations making it computationally slow.

[cat] La tesis tracta diferents problemes de gestió del risc i valoració de derivats financers suposant models generals, alternatius als habituals per als processos de preus dels actius subjacents, cercant capturar millor la dinàmica del mercat. Concretament considerem els models Heston-CEV amb salts (HCEVJ), de volatilitat estocàstica de 2 factors amb salts (2FSVJ) i Heston-Lévy d'activitat infinita. El model HCEVJ és un model general que inclou la majoria de models estudiats a la literatura. Es calibren alguns d'aquests models a partir de dades reals mitjançant un algorisme de cerca de quadrícula i es comparen. L'anàlisi mostra que el model HCEVJ millora la modelització del somriure de volatilitat ja que introdueix un paràmetre addicional. A la tesi també s’utilitzen tècniques de Càlcul de Malliavin per a la valoració d’opcions i el càlcul de la seva sensibilitat a variacions dels paràmetres. Concretament, s’obtenen estimacions més ràpides de les sensibilitats (Gregues) de primer ordre dels preus de les opcions per al model HCEVJ. Pel que fa a la valoració d’opcions, es revisen enfocaments existents, com els basats en simulacions de Monte Carlo, els basats en la transformada de Fourier i les tècniques de descomposició, i es comparen. Els mètodes de descomposició superen els altres quan es tenen estructures de salt simples. En el cas 2FSVJ, una fórmula de descomposició de primer ordre i una de segon ordre es deriven sota una estructura de salts general. S’escullen salts lognormals i doble exponencial i es fan experiments numèrics. En el cas Heston-Lévy, s’aborda l'estudi del preu d’opcions sota models de preus amb volatilitat estocàstica i salts d'activitat infinita però variació finita. La contribució fonamental és una fórmula de descomposició basada en l’aproximació d’un procés de Lévy amb salts d'activitat infinita per un procés de Lévy amb salts d'activitat finita. Es proporcionen límits d'error per a aquesta aproximació i les estimacions numèriques es comparen amb èxit amb els preus de referència.