Delft University of Technology Pricing and calibration with stochastic local volatility models in a monte carlo setting van der Stoep, Anton DOI 10.4233/uuid:a2314b15-6fee-4be1-bf77-f78b4356d11a Publication date 2019 Document Version Final published version Citation (APA) van der Stoep, A. (2019). Pricing and calibration with stochastic local volatility models in a monte carlo setting. https://doi.org/10.4233/uuid:a2314b15-6fee-4be1-bf77-f78b4356d11a Important note To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10. PRICINGAND CALIBRATION WITH STOCHASTIC LOCAL VOLATILITY MODELSINA MONTE CARLO SETTING PRICINGAND CALIBRATION WITH STOCHASTIC LOCAL VOLATILITY MODELSINA MONTE CARLO SETTING Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr. ir. T.H.J.J. van der Hagen, voorzitter van het College voor Promoties, in het openbaar te verdedigen op dinsdag 26 maart 2019 om 12:30 uur door Anthonie Willem VAN DER STOEP Master of Science in Toegepaste Wiskunde, Universiteit Twente, Nederland geboren te Barendrecht, Nederland. Dit proefschrift is goedgekeurd door de promotoren. Samenstelling promotiecommissie: Rector Magnificus voorzitter Prof. dr. ir. C.W. Oosterlee Technische Universiteit Delft, promotor Dr. ir. L.A. Grzelak Technische Universiteit Delft, copromotor Onafhankelijke leden: Prof. dr. C. Vázquez Cendón Universidade da Coruña, Spanje Prof. dr. K.J. In ’t Hout Universiteit Antwerpen, België Prof. dr. P.J.C. Spreij Universiteit van Amsterdam Prof. dr. ir. A.W. Heemink Technische Universiteit Delft Drs. W.F.van Raaij Rabobank Reservelid: Prof. dr. ir. G. Jongbloed Technische Universiteit Delft Pricing and Calibration with Stochastic Local Volatility Models in a Monte Carlo Setting Dissertation at Delft University of Technology Copyright © 2019 by A.W. van der Stoep Printed by: Gildeprint ISBN 978-94-6323-540-2 An electronic version of this dissertation is available at http://repository.tudelft.nl/. SUMMARY A general purpose of mathematical models is to accurately mimic some observed phe- nomena in the real world. In financial engineering, for example, one aim is to reproduce market prices of financial contracts with the help of applied mathematics. In the Foreign Exchange (FX) market, the so-called implied volatility smile plays a key role in the pricing and hedging of financial derivative contracts. This volatility smile is a phenomenon that reflects the prices of European-type options for different strike prices; the implied volatility tends to be higher for options that are deeper In The Money and Out of The Money than options that are approximately At The Money. In order for a pricing model to be accepted in the financial industry, it should at least be able to ac- curately price back the most simple financial derivative contracts, namely European call and put options. In other words, the model should calibrate well to the implied volatility smile observed in the financial market. The calibration should not only be accurate, but also reasonably fast. Another feature we wish the financial asset model to possess, is an accurate pric- ing of so-called exotic financial products. Exotic products are not traded on regular exchanges, but over-the-counter, i.e. directly between two parties without the super- vision of an exchange. An example is a barrier option, which is a financial contract of which its payoff depends on the possible event that the underlying asset price hits a certain pre-determined level. The model prices of these path-dependent contracts are determined by the transition densities of the relevant underlying asset(s) between future time-points. These transition densities are reflected by the forward volatility smile the model implies; in order for the model to accurately price exotic products, it should yield realistic forward volatilities. The models discussed in this thesis can be considered as enhancements of Dupire’s classical and famous Local Volatility (LV) model [34,35], which by its non-parametric lo- cal volatility component yields a perfect calibration to any set of arbitrage-free European- type options prices. We will consider the addition to the LV model of stochastic volatil- ity, resulting in the stochastic local volatility (SLV) model [75, 80, 82], and we also add stochastic interest rates – for both extensions, a perfect calibration is preserved. As an alternative to the LV model, Hagan et al. [63] introduced the SABR model. We further enrich this model by time-dependent parameters and propose an efficient calibration procedure. Also, we introduce a novel asset model dynamics class, the Collocating Lo- cal Volatility (CLV) model. Similarly to an SLV model, by construction the CLV model is perfectly calibrated to liquid (i.e. heavily traded) market quotes, while maintaining the flexibility in accurately capturing the forward volatility smile. In some more detail, in Chapter2 we explain for a general SLV model the local volatil- ity component, which consists of Dupire’s local volatility component and a non-trivial conditional expectation. We present a Monte Carlo approach for the efficient evalua- tion of a general SLV model. The approach is based on an intuitive and easy to imple- v vi SUMMARY ment non-parametric approximation of the conditional expectation, which consists of assigning asset path realizations to appropriate ‘bins’. This approximation is embedded in a simulation scheme that is strongly based on a so-called Quadratic Exponential (QE) scheme [5], which introduces less bias than more common Euler schemes. We show by means of numerical experiments and an error analysis that our approach yields accurate prices for European-type options. Also, we price forward-start options and determine the corresponding forward volatility smile; we observe that the Heston-SLV model pre- serves the current shape of the implied volatility smile, which is typically more in line with financial market observations. The time-dependent FX-SABR model is discussed in Chapter3. In contrast to the constant-parameter SABR model introduced by Hagan et al. [63], the time-dependent model can be calibrated to an implied volatility surface; in this way, it captures as much market information as possible. However, the calibration of time-dependent parame- ters is non-trivial. We propose an efficient calibration approach that is based on effective parameters, which can be considered as ‘sophisticated averages’ of the corresponding time-dependent parameters. By considering the qualitative effects of the SABR parame- ters on the shape of the implied volatility smile, we derive ‘mappings’ between the time- dependent parameters and their ‘constant counterparts’. The mappings allow for an ef- ficient calibration of the time-dependent parameters. Numerical experiments confirm that both the separate and combinated performance of the effective parameters are ac- curate. Also, we numerically show that the effective parameters derived yield highly sat- isfactory calibration results. In a barrier option pricing experiment, the time-dependent FX-SABR model yields more accurate prices than the traditional LV and SABR models. We consider two types of hybrid local volatility models in Chapter4, namely the SABR and Heston models enhanced by Dupire’s local volatility component, and the Lo- cal Volatility model enriched with stochastic interest rates. For both model classes one needs to determine an, above mentioned, non-trivial (conditional) expectation, which is expensive to compute and cannot be extracted from the market quotes. In this chap- ter, we propose a second generic efficient Monte Carlo approach to these hybrid models, which consists of two projection steps. The first step is based on stochastic collocation, where in general a certain ‘expensive’ variable of interest Y is approximated by a function of a ‘cheaper to evaluate’ random variable X – more concrete, a Lagrange interpolation is established through so-called optimal collocation points that are determined based on the distribution of X . The second projection step relies on standard regression tech- niques. We numerically show that our approach yields a fast Monte Carlo evaluation and highly accurate pricing results for European-type options. Also, we provide an er- ror analysis for a model consisting of two correlated Geometric Brownian Motions, the ‘2D-GBM’ model. In Chapter5 we introduce a novel asset price model class, namely the Collocating Local Volatility (CLV) model. The CLV model consists of two elements, a kernel pro- cess and a local volatility function. The kernel process can be chosen freely and deter- mines the forward volatility smile the model implies. The local volatility function, which is constructed based on stochastic collocation, ensures a perfect calibration to finan- cial market quotes. We compare three different kernel processes within the CLV model framework, namely the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ros (CIR) process SUMMARY vii and the Heston model. For each of these kernels, we consider the effect of the kernel parameters on the shape of the forward volatility smile. Subsequently, we calibrate the OU-CLV, CIR-CLV and Heston-CLV models to FX barrier option prices observed in the market by means of a Monte Carlo simulation, where we make use of Brownian bridge techniques. With the work in this thesis, one is able to price complicated FX derivatives by ef- ficient Monte Carlo methods, while an accurate calibration to liquid market quotes is preserved.