View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarBank@NUS Inflation – linked Option Pricing: a Market Model Approach with Stochastic Volatility LIANG LIFEI (B.Sc. (Hons), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2010 i Acknowledgements I have been interested in the area of financial modeling for the whole of my undergraduate and graduate years, and I am very glad that this thesis has given me the chance to gain more modeling knowledge of inflation-linked derivatives. I would like to extend my deepest appreciation to the following people whose support has made my research project an enjoyable experience. First and foremost, I am very lucky to be a student of my supervisor Prof. Xia Jianming and co- supervisor Dr. Oliver Chen. They have provided me with valuable suggestions and encouragement for the research, and their insights have inspired me and broadened my knowledge in this field. I am very grateful to have them as my supervisors. I would like to thank Mr. Pierre Lalanne as well as inflation trading desk of UBS who have answered my queries with practical knowledge and helped me generously throughout the thesis. I would also like to thank Mr. Zhang Haibo and Ms. Zhang Chi from Department of Chemical and Biomolecular Engineering, NUS, whose expertise in optimization has helped me greatly. Finally, I would like to thank Department of Mathematics and National University of Singapore for providing necessary resources and financial support; my friends for their cheerful company and my parents who have always been supportive throughout the years. ii Table of Contents Acknowledgements ........................................................................................................................ i Summary ....................................................................................................................................... iv List of Symbols .............................................................................................................................. 1 List of Tables ................................................................................................................................. 3 List of Figures ................................................................................................................................ 5 1. Introduction ........................................................................................................................... 6 2. Two Factor Stochastic Volatility LMM Model ................................................................. 10 2.1. Forward CPI and Forward Risk Neutral Measure .......................................................... 10 2.2. Two Factor Model and Derivation of Pricing Formula.................................................. 11 2.3. Implementation Issues .................................................................................................... 15 3. Hedging of Inflation - Linked Options .............................................................................. 17 4. Convexity Adjustment ......................................................................................................... 20 5. Calibration ........................................................................................................................... 25 5.1. Parameterization ............................................................................................................. 26 5.2. Interpolation – Based Calibration .................................................................................. 28 5.3. Non – Interpolation – Based Calibration........................................................................ 38 6. Conclusions........................................................................................................................... 44 Bibliography ................................................................................................................................ 46 Appendices ................................................................................................................................... 49 iii Appendix I Derivation of YoY caplet price under one factor stochastic volatility................... 49 Appendix II Riccati equation .................................................................................................... 59 Appendix III Structural deficiency of one factor model ........................................................... 60 Appendix IV Interpolation based on flat volatilities ................................................................. 61 iv Summary Inflation-linked derivatives‟ modeling is a relatively new branch in financial modeling. Originally it was adapted from interest rate models; but attention is currently turning to market model. In this thesis, we extend stochastic volatility market model to two-factor setting. The analysis in this thesis shows that two-factor model offers more profound structure and greater flexibility of fitting volatility surface while retaining the tractability of one-factor model. We then apply the two-factor model to two related issues. Hedging analysis is conducted from a new perspective where zero-coupon (ZC) options are used to hedge year-on-year (YoY) options. This can be of great practical interest as it leverages on a complicated trading book and saves on transaction cost. Convexity adjustment is also approximated under the model. Furthermore, we have illustrated in detail how it can be captured via concrete trading activities. The new two-factor model regime and broader scope which aims to calibrate both ZC and YoY options with one model, call for new calibration procedures. In this thesis, two approaches have been proposed. Firstly, we devise an interpolation scheme that yields a market consistent interpolation. A calibration against these interpolated prices can reveal mispricing and, thus, arbitrage opportunities between the two options markets. However, a more thorough analysis is necessary to determine if a misprice can really constitute an arbitrage opportunity. Secondly, to mitigate the arbitrary nature of interpolation, we propose a non-interpolation-based calibration scheme. In this approach, only market-quoted prices are inputs of calibration. ZC and YoY option prices are weighted differently to reflect their respective market liquidity and bid- offer spreads. v With this thesis, we fulfilled the aim to build a comprehensive framework under which an inflation-linked option pricing model can be calibrated and applied. 1 List of Symbols The Consumer Price Index at time - forward CPI at time t Swap break-even of a zero – coupon inflation-linked swap Price at time t of nominal zero coupon bond with maturity Short rate at time s Period between and Forward rate between times and as seen at time t Volatility of The j th factor loading of , j = 1 & 2 The j th factor loading of extended by time, i.e. it is equal to when and 0 afterwards The j th common variance process of forward CPIs, j = 1 & 2 Mean reversion of Long-term variance of Volatility of variance Brownian motion associated to of Brownian motion that drives Brownian motion that drives Correlation between the j th factor of and , i.e. Correlation between and 2 Correlation between and the j th factor of , i.e. C YoY caplet price Price of ZC caplet with maturity at Weight assigned to errors of ZC caps in non-interpolation-based calibration Weight assigned to errors of YoY caps in non-interpolation-based calibration 3 List of Tables Table 4.1 Dynamic hedging when moves by 1. 23 Table 4.2 Dynamic hedging when moves up (above) and down (below) 24 by 1. Table 5.2.1 EUR HICP ZC Cap prices, with maturities from 1yr to 10 yr and 31 strikes from1% to 5%. Highlighted are market prices and others prices are interpolated. Table 5.2.2 EUR HICP ZC Cap implied volatilities, with maturities from 1yr to 31 10 yr and strikes from1% to 5%. Table 5.2.3 EUR HICP YoY Cap spot vol, with maturities from 1yr to 10 yr and 32 strikes from1% to 5%. Table 5.2.4 EUR HICP YoY Cap implied correlations, with maturities from 1yr 32 to 9 yr and strikes from1% to 5%. Table 5.2.5 EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and 33 strikes from1% to 5%. New Interpolation. Table 5.2.6 EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and 33 strikes from1% to 5%. Old Interpolation. Table 5.2.7 Model parameters of interpolation-based calibration. Left: volatility 36 coefficients. Center: volatility factor loadings. Right: forward CPI / volatility correlations. Table 5.2.8 Relative percentage error of ZC option prices with maturities from 37 1yr to 10 yr and strikes from 1% to 5%. Table 5.3.1 YoY implied correlation. Above: perturbed. Below: original. 38/39 4 Table 5.3.2 Relative percentage error of YoY cap prices. Bolded are relative error 40 of extended caps. Table 5.3.3 Relative percentage error of ZC option prices with maturities from 42 1yr to 10 yr and strikes from 1% to 5%. Table 7.4.1 EUR HICP YoY Cap prices. 61 Table 7.4.2 EUR HICP YoY Cap flat vol. Bolded are quoted and others are 61 linearly interpolated. Table 7.4.3 EUR HICP YoY Cap prices. Bolded are quoted and others are from 62 interpolated flat vols. 5 List of Figures Figure 4.1 Structure of forward starting ZC swap 22 Figure 5.2.1 EUR HICP YoY Caplet spot volatility, with maturities from 1yr 34 to 10 yr and strikes from1% to 5%. Left: new interpolation