Faculty of Science; School of Computational and Applied Maths; MSC Thesis / The assumption of constant volatility as an input parameter into the Black-Scholes option pricing formula is deemed primitive and highly erroneous when one considers the terminal distribution of the log-returns of the underlying process. To account for the `fat tails' of the distribution, we consider both local and stochastic volatility option pricing models. Each class of models, the former being a special case of the latter, gives rise to a parametrization of the skew, which may or may not re°ect the correct dynamics of the skew. We investigate a select few from each class and derive the results presented in the corresponding papers. We select one from each class, namely the implied trinomial tree (Derman, Kani & Chriss 1996) and the SABR model (Hagan, Kumar, Lesniewski & Woodward 2002), and calibrate to the implied skew for SAFEX futures. We also obtain prices for both vanilla and exotic equity index options and compare the two approaches.

本篇論文驗證SABR模型與SABR-LMM模型的動態設定與市場選擇權價格下標的未來價格之隱含分配是否一致，判斷準則為SABR模型與SABR-LMM模型校準出的參數是否符合市場直覺。根據實證結果答案是肯定的，所以在SABR模型與SABR-LMM模型下評價選擇權不需要再做任何的主觀判斷或調整。此外本篇論文對於SABR模型與SABR-LMM模型的參數校準方法做了詳細的分析，並且清楚的閳述SABR模型與SABR-LMM模型的模型直覺。

The project investigates the prices of barrier options from the constant underlying volatility in the Black-Scholes model to stochastic volatility model in SABR framework. The constant volatility assumption in derivative pricing is not able to capture the dynamics of volatility. In order to resolve the shortcomings of the Black-Scholes model, it becomes necessary to find a model that reproduces the smile effect of the volatility. To model the volatility more accurately, we look into the recently developed SABR model which is widely used by practitioners in the financial industry. Pricing a barrier option whose payoff to be path dependent intrigued us to find a proper numerical method to approximate its price. We discuss the basic sampling methods of Monte Carlo and several popular variance reduction techniques. Then, we apply Monte Carlo methods to simulate the price of the down-and-out put barrier options under the Black-Scholes model and the SABR model as well as compare the features of these two models.

This article presents a new approach for calculating sensitivities of European swaptions. The sensitivities are found by applying an adjoint method to a stochastic volatility model, namely the SABR/LIBOR market model. This market model predicts the volatility smile and follows the market fluctuations more accurately than earlier used deterministic volatility market models for complex derivatives. The new adjoint method involves not only sensitivity calculations, it also presents a way of estimating the time discretization error using an a posteriori approach. The error calculation is described in this document but not investigated further. The first step in order to calculate the sensitivities is to calibrate the SABR/LIBOR market model to some market data. In our calculations we used data from June 15 2011 with 6 month intervals between the maturity times. When this calibration is complete all of the parameters in the SABR/LIBOR market model are specified and we can continue with the sensitivity calculations using the new adjoint method. The results from these calculations show that the method is a good choice for estimating sensitivities if we consider a complex financial derivative like the European swaption. The method is quite computational so we recommend that it is only used on a small number of securities with respect to a large number of parameters. The method provides more market-driven price and sensitivity estimations than earlier used methods and can benefit hedging of portfolios.

Problem: The standard Black-Scholes framework cannot incorporate the volatility smiles usually observed in the markets. Instead, one must consider alternative stochastic volatility models such as the SABR. Little research about the suitability of the SABR model for Swedish market (swaption) data has been found. Purpose: The purpose of this paper is to account for and to calibrate the SABR model for swaptions trading on the Swedish market. We intend to alter the calibration techniques and parameter values to examine which method is the most consistent with the market. Method: In MATLAB, we investigate the model using two different minimization techniques to estimate the model’s parameters. For both techniques, we also implement refinements of the original SABR model. Results and Conclusion: The quality of the fit relies heavily on the underlying data. For the data used, we find superior fit for many different swaption smiles. In addition, little discrepancy in the quality of the fit between methods employed is found. We conclude that estimating the α parameter from at-the-money volatility produces slightly smaller errors than using minimization techniques to estimate all parameters. Using refinement techniques marginally increase the quality of the fit.

The thesis is dealing with option pricing. The basic Black-Scholes model is described, along with the reasons that led to the development of stochastic volatility models. SABR model and Heston model are described in detail. These models are then applied to equity options in the times of high volatility. The models and their application are then evaluated.

För att prissätta optioner är val av modell en viktig fråga. I denna kandidatuppsats beskrivs både Black & Scholes modell och SABR-modellen. Förstnämnda modell är enklare än SABR-modellen men bygger på antaganden som inte stämmer överens med verkligheten. Den ger heller inte någon explicit formel för den implicita volatiliteten och predikterar inte heller på ett korrekt sätt fenomenet volatility smile vilket observeras på marknaden. Syftet med uppsatsen är att utvärdera prestandan hos SABR-modellen och användarvänligheten, samt att undersöka lite av teorin bakom modellen och vissa av dess egenskaper. Till grund för beräkningarna ligger datamaterial hämtat från Nasdaq OMX Nordic. Enligt mina beräkningar är resultatet att SABR-modellen endast presterar marginellt bättre än Black & Scholes-modellen. Dock kan även små förbättringar spela stor roll i dessa sammanhang.

Thesis (MSc)--Stellenbosch University / ENGLISH ABSTRACT : In this thesis, we first review the traditional pre-credit crunch approach that considers a single curve to consistently price all instruments. We review the theoretical pricing framework and introduce pricing formulas for plain vanilla interest rate derivatives. We then review the curve construction methodologies (bootstrapping and global methods) to build an interest rate curve using the instruments described previously as inputs. Second, we extend this work in the modern post-credit framework. Third, we review the calibration of the SABR model. Finally we present applications that use interest rate curves and SABR model: stripping implied volatilities, transforming the market observed smile (given quotes for standard tenors) to non-standard tenors (or inversely) and calibrating the market volatility smile coherently with the new market evidences. / AFRIKAANSE OPSOMMING : Geen Afrikaanse opsomming geskikbaar nie

<p>För att prissätta optioner är val av modell en viktig fråga. I denna kandidatuppsats</p><p>beskrivs både Black & Scholes modell och SABR-modellen. Förstnämnda modell är</p><p>enklare än SABR-modellen men bygger på antaganden som inte stämmer överens med</p><p>verkligheten. Den ger heller inte någon explicit formel för den implicita volatiliteten</p><p>och predikterar inte heller på ett korrekt sätt fenomenet volatility smile vilket</p><p>observeras på marknaden.</p><p>Syftet med uppsatsen är att utvärdera prestandan hos SABR-modellen och</p><p>användarvänligheten, samt att undersöka lite av teorin bakom modellen och vissa av</p><p>dess egenskaper. Till grund för beräkningarna ligger datamaterial hämtat från Nasdaq</p><p>OMX Nordic.</p><p>Enligt mina beräkningar är resultatet att SABR-modellen endast presterar marginellt</p><p>bättre än Black & Scholes-modellen. Dock kan även små förbättringar spela stor roll i</p><p>dessa sammanhang.</p>

The United States' fleet of Light Water Reactors (LWRs) produces a large amount of spent fuel each year; all of which is presently intended to be stored in a fuel repository for disposal. As these LWRs continue to operate and more are built to match the increasing demand for electricity, the required capacity for these repositories grows. Georgia Tech's Subcritical Advanced Burner Reactor (SABR) has been designed to reduce the capacity requirements for these repositories and thereby help close the back end of the nuclear fuel cycle by burning the long-lived transuranics in spent nuclear fuel. SABR's design is based heavily off of the Integral Fast Reactor (IFR). It is important to understand whether the SABR design retains the passive safety characteristics of the IFR. A full safety analysis of SABR's transient response to various possible accident scenarios needs to be performed to determine this. However, before this safety analysis can be performed, it is imperative to model all components of the reactivity feedback mechanism in SABR. The purpose of this work is to develop a calculational model for the fuel bowing reactivity coefficients that can be used in SABR's future safety analysis. This thesis discusses background on fuel bowing and other reactivity coefficients, the history of the IFR, the design of SABR, describes the method that was developed for calculating fuel bowing reactivity coefficients and its validation, and presents an example of a fuel bowing reactivity calculation for SABR.