From an analysis of the time series of realized variance using recent high-frequency data, Gatheral, Jaisson and Rosenbaum noted in "Volatility is rough" that the logarithm of realized variance behaves substantially as a fractional Brownian motion with Hurst parameter H of order 0.1, at any reasonable time scale. The resulting rough fractional stochastic volatility (RFSV) model, obtained always in "Volatility is rough", is very compatible with financial time series data: the covariance function of the RFSV model is coherent with the empirical one. Furthermore, the implications of the RFSV model for option pricing are analysed following the work of Bayer, Friz and Gatheral in "Pricing under rough volatility" (under the pricing measure Q). A simple case of this model is investigated in detail. The Rough Bergomi model, introduced always in "Pricing under rough volatility", is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In the absence of analytical European option pricing methods for the model, Monte Carlo simulations are analyzed following the hybrid scheme introduced by Bennedsen, Lunde and Pakkanen in "Hybrid scheme for Brownian semistationary processes". McCrickerd and Pakkanen proposed in "Turbocharging Monte Carlo pricing for the rough Bergomi model" a control variable method to reduce the runtime-adjusted variance of Monte Carlo implied volatilities. Repeating the work of Jacquier, Martini and Muguruza in "On VIX futures in the rough Bergomi model", the dynamics of the VIX is also examined and efficient pricing algorithms for VIX futures and options are reproduced. Furthermore, the validity of the rough Bergomi model to jointly describe the VIX and the SPX is analyzed. Inspired always by the work of Jacquier, Martini e Muguruza in "On VIX futures in the rough Bergomi model", a possible variant is proposed for the calibration of the model. Using futures on the SP the paramenters concerning the volatility of the process are calibrated while, using plain vanilla options, the correlation between the process and the volatility is estimated. Although the futures sample is not particularly large, this variant allows to speed up remarkably the computational time making the calibration workable. Finally, the limitations of the model are shown and possible solutions to solve them are suggested following the work of Horvath, Jacquier and Tankov in "Volatility options in rough volatility models" and the one of Alòs, Garcia-Lorite and Muguruza in "On smile properties of volatility derivatives and exotic products: understanding the VIX skew". The thesis is organised as follows. In Section 1 an overview with advantages and drawbacks of the main models used for derivatives pricing is shown. Starting from the Black and Scholes framework, all the subsequent models are introduced until you come to the rough volatility models. In Section 2 the reasons why we can conclude volatility is rough are explained. As a consequence, in Section 3 and 4, the RFSV model is presented with its particular case, the Rough Bergomi model. Section 5 handles, therefore, the simulation of the model and the derivatives pricing. VIX derivatives are introduced in Section 6 and possible methods to compute their price are shown. Section 7, 8, 9 are devoted to the main result of this thesis: the calibration. Finally, in Section 10, the limitations of the model are discussed with possible ways to solve them.
Da un'analisi delle serie storiche sulla varianza realizzata usando dati ad alta frequenza, Gatheral, Jaisson e Rosenbaum hanno mostrato in "Volatility is rough" che il logaritmo della varianza realizzata si comporta essenzialmente come un moto Browniano frazionario con parametro di Hurst H dell'ordine di 0.1, per ogni scala temporale ragionevole. Il risultante rough fractional stochastic volatility (RFSV) model, ottenuto sempre in "Volatility is rough", è sorprendentemente consistente con i dati delle serie storiche finanziarie: la funzione di covarianza del modello (RFSV) è coerente con quella empirica. Successivamente, le implicazioni del modello RFSV per il pricing di opzioni sono esaminate seguendo il lavoro di Bayer, Friz e Gatheral in "Pricing under rough volatility" (sotto la misura neutrale al rischio Q). Viene analizzato in dettaglio un caso semplice del modello, il modello Rough Bergomi, introdotto sempre in "Pricing under rough volatility". E' uno dei recenti modelli a rough volatility ad essere consistenti con il fatto stilizzato che la superficie di volatilità implicita è tempo-invariante, e ad essere capaci di catturare la struttura a termine della skew osservata nell'equity market. Dal lato matematico, la Markovianità e, in parte, la semi-martingalità sono perse. In assenza di metodi analitici per prezzare opzioni Europee, ci si focalizza sulle simulazioni Montecarlo seguendo lo schema introdotto da Bennedsen, Lunde e Pakkanen in "Hybrid scheme for Brownian semistationary processes". McCricked e Pakkanen hanno anche proposto in "Turbocharging Monte Carlo pricing for the rough Bergomi model" un metodo con variabile di controllo per ridurre la run-time-adjusted variance delle volatilità implicite ottenute con Montecarlo. Seguendo il lavoro di Jacquier, Martini e Muguruza in "On VIX futures in the rough Bergomi model", anche la dinamica del VIX viene investigata ed efficienti algoritmi per prezzare opzioni e futures sul VIX sono riprodotti. In seguito, la validità del modello rough Bergomi a descrivere congiuntamente il VIX e lo SPX viene analizzata. Ispirandosi sempre al lavoro di Jacquier, Martini e Muguruza in "On VIX futures in the rough Bergomi model", viene proposta una possibile variante per la calibrazione del modello. Usando i futures sullo SP, vengono calibrati i parametri riguardanti la volatilità del processo mentre, mediante opzioni plain vanilla, viene stimata la correlazione tra il processo e la volatilità. Sebbene il campione dei futures non sia particolarmente ampio, questa variante permette di velocizzare notevolmente i tempi di calcolo rendendo la calibrazione praticabile. Infine, sono mostrate le limitazioni del modello e sono proposte possibili soluzioni per risolverle seguendo il lavoro di Horvath, Jacquier e Tankov in "Volatility options in rough volatility models" e quello di Alòs, Garcia-Lorite and Muguruza in "On smile properties of volatility derivatives and exotic products: understanding the VIX skew". La tesi è strutturata nel modo seguente. Nella Sezione 1 viene effettuata una panoramica dei principali modelli per calcolare il prezzo di derivati mostrando i loro vantaggi/svantaggi. Partendo dal modello di Black and Scholes, vengono introdotti tutti i modelli successivi fino ad arrivare a quelli con volatilità ruvida. Nella Sezione 2 sono spiegate le ragioni per cui la volatilità può essere considerata ruvida. Di conseguenza, nelle Sezioni 3 e 4, il modello RFSV viene presentato insieme al suo caso particolare, il modello Rough Bergomi. La Sezione 5 si occupa della simulazione del modello e del pricing di derivati. I derivati sul VIX sono introdotti nella Sezione 6 e sono mostrati dei possibili metodi per calcolare il loro prezzo. Le sezioni 7, 8, 9 sono dedicate al risultato principale della tesi: la calibrazione. Infine, nella sezione 10, sono presentate le limitazioni del modello con possibili modi per risolverle.
The rough Bergomi model
RUOTOLO, RICCARDO PAOLO
2017/2018
Abstract
From an analysis of the time series of realized variance using recent high-frequency data, Gatheral, Jaisson and Rosenbaum noted in "Volatility is rough" that the logarithm of realized variance behaves substantially as a fractional Brownian motion with Hurst parameter H of order 0.1, at any reasonable time scale. The resulting rough fractional stochastic volatility (RFSV) model, obtained always in "Volatility is rough", is very compatible with financial time series data: the covariance function of the RFSV model is coherent with the empirical one. Furthermore, the implications of the RFSV model for option pricing are analysed following the work of Bayer, Friz and Gatheral in "Pricing under rough volatility" (under the pricing measure Q). A simple case of this model is investigated in detail. The Rough Bergomi model, introduced always in "Pricing under rough volatility", is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In the absence of analytical European option pricing methods for the model, Monte Carlo simulations are analyzed following the hybrid scheme introduced by Bennedsen, Lunde and Pakkanen in "Hybrid scheme for Brownian semistationary processes". McCrickerd and Pakkanen proposed in "Turbocharging Monte Carlo pricing for the rough Bergomi model" a control variable method to reduce the runtime-adjusted variance of Monte Carlo implied volatilities. Repeating the work of Jacquier, Martini and Muguruza in "On VIX futures in the rough Bergomi model", the dynamics of the VIX is also examined and efficient pricing algorithms for VIX futures and options are reproduced. Furthermore, the validity of the rough Bergomi model to jointly describe the VIX and the SPX is analyzed. Inspired always by the work of Jacquier, Martini e Muguruza in "On VIX futures in the rough Bergomi model", a possible variant is proposed for the calibration of the model. Using futures on the SP the paramenters concerning the volatility of the process are calibrated while, using plain vanilla options, the correlation between the process and the volatility is estimated. Although the futures sample is not particularly large, this variant allows to speed up remarkably the computational time making the calibration workable. Finally, the limitations of the model are shown and possible solutions to solve them are suggested following the work of Horvath, Jacquier and Tankov in "Volatility options in rough volatility models" and the one of Alòs, Garcia-Lorite and Muguruza in "On smile properties of volatility derivatives and exotic products: understanding the VIX skew". The thesis is organised as follows. In Section 1 an overview with advantages and drawbacks of the main models used for derivatives pricing is shown. Starting from the Black and Scholes framework, all the subsequent models are introduced until you come to the rough volatility models. In Section 2 the reasons why we can conclude volatility is rough are explained. As a consequence, in Section 3 and 4, the RFSV model is presented with its particular case, the Rough Bergomi model. Section 5 handles, therefore, the simulation of the model and the derivatives pricing. VIX derivatives are introduced in Section 6 and possible methods to compute their price are shown. Section 7, 8, 9 are devoted to the main result of this thesis: the calibration. Finally, in Section 10, the limitations of the model are discussed with possible ways to solve them.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/146037