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 cosupervisor 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 6. 5.1. Parameterization ............................................................................................................. 26 5.2. Interpolation – Based Calibration .................................................................................. 28 5.3. Non – Interpolation – Based Calibration........................................................................ 38 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 bidoffer 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 and 0 afterwards extended by time, i.e. it is equal to 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 Correlation between and and , i.e. when 2 Correlation between C and the j th factor of , i.e. 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 by 1. moves up (above) and down (below) 24 Table 5.2.1 EUR HICP ZC Cap prices, with maturities from 1yr to 10 yr and strikes from1% to 5%. Highlighted are market prices and others prices are interpolated. 31 Table 5.2.2 EUR HICP ZC Cap implied volatilities, with maturities from 1yr to 10 yr and strikes from1% to 5%. 31 Table 5.2.3 EUR HICP YoY Cap spot vol, with maturities from 1yr to 10 yr and strikes from1% to 5%. 32 Table 5.2.4 EUR HICP YoY Cap implied correlations, with maturities from 1yr to 9 yr and strikes from1% to 5%. 32 Table 5.2.5 EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and strikes from1% to 5%. New Interpolation. 33 Table 5.2.6 EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and strikes from1% to 5%. Old Interpolation. 33 Table 5.2.7 Model parameters of interpolation-based calibration. Left: volatility coefficients. Center: volatility factor loadings. Right: forward CPI / volatility correlations. 36 Table 5.2.8 Relative percentage error of ZC option prices with maturities from 1yr to 10 yr and strikes from 1% to 5%. 37 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 1yr to 10 yr and strikes from 1% to 5%. 42 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 linearly interpolated. 61 Table 7.4.3 EUR HICP YoY Cap prices. Bolded are quoted and others are from interpolated flat vols. 62 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 to 10 yr and strikes from1% to 5%. Left: new interpolation in this thesis; right: old interpolation (refer to figure 7.4.2 of Appendices). 34 Figure 5.2.2 Market and calibrated YoY implied volatility with x – axis representing the strikes (%). 35 Figure 5.2.3 Model parameters of interpolation-based calibration. Left: . Center left: . Center right: . Right: . 36 Figure 5.3.1 YoY implied correlation. Left: perturbed. Right: original. 38 Figure 5.3.2 EUR HICP YoY Caplet spot volatility. Left: perturbed; right: original. 39 Figure 5.3.3 Model parameters of non-interpolation-based calibration. Left: . Right: . 41 Figure 5.3.4 Model parameters of non-interpolation-based calibration. Left: . Center left: . Center right: . Right: . 41 Figure 5.3.5 Calibrated YoY spot volatility from non-interpolation-based calibration. 41 Figure 7.4.1 EUR HICP YoY Cap flat vol surface 62 Figure 7.4.2 EUR HICP YoY Cap resulted spot vol 63 6 1. Introduction Inflation-linked derivatives market was born around 2002 out of hedging needs of market makers. Currently, inflation – linked swap is the most liquid product whose volume has increased from almost zero in 2001 to $110 billion in 2007. Trading of inflation-linked options is also picking up gradually. Most inflation models so far have been derived from interest rate models. Currently, the pricing of inflation-linked options is addressed by resorting to a foreign currency analogy. In Jarrow and Yildrim (2003), the dynamics of nominal and real rates are modeled by one-factor Gaussian process in the framework proposed in Heath, Jarrow and Morton (1992), or the HJM framework. Inflation is then interpreted as the exchange rate between the nominal and the real economies. However, this model suffers two major drawbacks. Firstly, it is based on the market nonobservable of real interest rate. Secondly, it generates volatility skew at the expense of over – parameterization as remarked in Ungari (2008). As such, alternative approaches are gaining popularity. For example, Kazziha (1999), Belgrade et al (2004) and Mercurio (2005) considered a market model in which the underlying variables are forward CPIs evolving as driftless geometric Brownian motions. Mercurio and Moreni (2005) took one step further to incorporate a mean – reverting stochastic volatility process to the forward CPIs while Mercurio and Moreni (2010) built a more complete model with SABR stochastic volatility process. Details of SABR model can be found in Hagan (2002). In Liang (2010), a variation of Mercurio and Moreni (2005) model is built. All forward CPIs are assumed to share one common volatility process - differentiated only by respective factor loadings. The model is then implemented and calibrated with a boot-strapping algorithm. As 7 explained in Liang (2010) and presented again in Appendix III of this thesis, the effect of the factor loading of stochastic volatility on volatility surface is relatively limited, the model has encountered structural difficulty in generating both skew and smile configurations in one volatility surface. Moreover, we note that inflation - linked zero – coupon option resembles an equity vanilla option while YoY option is nothing but a series of forward starting options. In Bergomi (2004) and Fonseca, Grasselli and Tebaldi (2008), it is well explained that there is a structural limitation which prevents one – factor stochastic volatility models to price consistently forward starting options with vanilla options. This thesis corrects the original model deficiency encountered in Liang (2010) and documented in other literatures detailed above and addresses related issues such as hedging, convexity adjustment and calibration. As a whole, we strive to build a comprehensive framework under which an inflation-linked pricing model can be calibrated and applied. Works on multi – factor stochastic volatility model, such as Bates (2000) and Christoffersen (2007) have inspired us to extend the model in Liang (2010) to two-factor setting. Our calibration shows that one factor can have relatively fast mean-reversion to determine short-run variance while the other can have relatively slow mean-reversion to determine long-run variance. Despite the seemingly straight-forward extension from one to two-factor setting, it has significant implication to the calibration scheme. The boot-strapping style schemes proposed in Liang (2010) will no longer work. Calibration taking into consideration of the global configuration of the volatility surface is employed in this thesis. The goal of consistently pricing YoY and ZC options further complicates the calibration. Belgrade et al. (2004) attempted to address this consistency, though their model setting was too simplistic and there was no 8 calibration against actual data to measure the accuracy. We have, thus, designed a number of schemes, which can be broadly categorized as interpolation- and non-interpolation-based schemes. Variable reduction is important in global optimization as we seek to avoid over-parameterization as well as to increase the efficiency of optimization. To this end, we have adopted various parameterization functions for different parameter sets. For example, the parameterization of correlations is based on Mercurio and Moreni (2010) while that of the factor loadings of volatility is as proposed in Zhu (2007). The interpolation-based scheme conforms more to the current practice in the interest rate market. The advantage of this scheme is that there is more control and more information extracted on the individual caplets and floorlets. However, the parsimony of the model and the regularity of the parameters are sacrificed, rendering the model parameters sometimes arbitrary. The non-interpolation-based scheme does not carry out any form of interpolation and depends solely on market-quoted prices. It improves on the deficiencies of interpolated scheme. However, the model is sensitive to the choice of the parameterization functional. The contributions of the thesis are the followings: 1. We extend and explore the inflation-linked stochastic volatility market model under a multi - factor setting. 2. The hedging analysis is conducted from a new perspective that ZC options are used to hedge YoY options – a strategy of great practical interest. 9 3. While most articles treat the subject of calibration as if it is no different from that of interest rate models, it is actually more delicate in an inflation-linked context. This thesis fills the gap by proposing and testing two original calibration schemes. The thesis is structured as follows. In chapter 2, the extended two-factor stochastic volatility model and the derived pricing formula for inflation-linked options are presented. Following that, two related topics – hedging of inflation-linked options and convexity adjustment of YoY swaps and options – are addressed in chapters 3 and 4 respectively. Chapter 5 deals with calibration schemes under the new model. It starts with a review of the original calibration scheme in Liang (2010) and a discussion on parameterization. The next subsection focuses on how to maintain the consistency between ZC and YoY option market and proposes an alternative schemeinterpolation-based calibration scheme. Non-interpolation-based calibration scheme is presented in the last subsection. Conclusions are presented in chapter 6. For self-contained purpose, mathematical derivations and illustrative examples are presented in Appendices. 10 2. Two Factor Stochastic Volatility LMM Model 2.1. Forward CPI and Forward Risk Neutral Measure We denote time t, the Consumer Price Index at time . In Kazziha (1999), is defined as the fixed amount X to be exchanged at time - forward CPI at for the CPI , so that the swap has zero value at time t. With the description of zero coupon inflation linked swap in Liang (2010), X at time zero is then written as where denotes the swap breakeven of a zero-coupon inflation-linked swap. More specifically, the fixed leg of the swap is priced at which is equal to the floating leg, priced at above denotes the price at time t of nominal zero coupon bond with maturity Since . we note that the floating leg of the swap can be priced at So we derive an important property of the forward CPI forward measure, i.e. that it is a martingale under - 11 where 2.2. defines the canonical filtration from time t. Two Factor Model and Derivation of Pricing Formula We present the model directly under - forward measure. where we denote the factor loading of ; forward rate between times and as seen at time t, ignoring day count conventions; ; volatility of and where will be defined shortly. Before we proceed to define stochastic volatilities and the corresponding correlation structures, we remark that terminates as at . To avoid the confusion in time, we define Then, the two forward CPIs are matched in time dimension. 12 And the stochastic process of volatility V is The correlation structures are defined as With j = 1 and 2, k = i and i – 1 and < , > denotes the quadratic co-variation of stochastic processes. The technical aspect of quadratic variation can be found in Brigo and Mercurio (2006). And at the same time, we also have, in the nominal market, with By applying the same drift-freezing technique and fast Fourier transformation as in Liang (2010) and Mercurio and Moreni (2005)1, we derive the pricing formula of YoY caplet as 1 The two articles are on one-factor setting, but the extension to two-factor setting is straight-forward as we have shown in Appendix I. 13 where And functions A and B above are solutions of the following general Riccati equation: 14 which take the specific form in equations below where And the YoY caplet can thus be evaluated by computing numerically the integral. Detailed derivation is presented in Appendix I. Similarly but more easily, ZC caplet can be priced as where We will end this section with a few remarks, which serve to deepen our understanding of the model. A more thorough discussion can be found in Christofferson (2007). Since 15 thus, and So finally, we see that The simple calculation above shows that the effective correlation is now stochastic. By adding one more volatility factor, the model is not only extended, but is also fundamentally changed. The richer volatility surface configurations result not only from a mere increase of the number of parameters, but from a more complex model structure. 2.3. Implementation Issues As pointed out in many articles, Heston integrals (1) and (3) involve a complex logarithm which is inherently discontinuous. This discontinuity causes numerical integration to be difficult and sometimes generates mispricing for long maturities, as documented in Albrecher et al. (2006). In Liang (2010), the method proposed in Kahl and Jäckel (2006) is adopted to alleviate, if not resolved the problem. However, it has been pointed out but left unresolved that the method is difficult to extend to inflation – linked context. It turns out that Albrecher et al. (2006) provides a more feasible correction for the purpose of this thesis. 16 They noted that complex root of has two possible values. By conventions, the principal value is used in most formulas of Heston characteristic function and is also returned by most software packages. But using the principal values causes a branch cut – a curve in the complex across which a function is discontinuous. Function A in equation (2) above jumps discontinuously each time the imaginary part of the argument of the logarithm crosses the negative real axis. They followed by proving that choosing the second value of would circumvent the problem. Contrary to Liang (2010), we adopt this proposition for both YoY and ZC pricing formulas throughout the thesis. Another benefit of this remedy is that the numerical integration becomes much more stable, in the sense that it yields valid value for much wider range of parameters. Consequently, the calibration process improves in efficiency. This is because when optimization algorithm scans through the cost function across a range of parameter sets, many more points contain a valid numerical value. 17 3. Hedging of Inflation - Linked Options Although fast Fourier transformation enables us to derive a closed formula to price inflationlinked options, this approach yields little information on hedging strategies. In this section, we apply Itô‟s lemma to obtain a hedging strategy. According to conventional wisdom, we normally dynamically hedge a derivative with its underlying. In the present context, a YoY option should, thus, be hedged via a combination of ZC swaps. However, in order to leverage on the synergy of a bank‟s trading book, which contains both YoY and ZC options, we aim to hedge YoY options with ZC options. In this way, we save transaction cost compared to usual strategies. be the price of a YoY caplet, then by Itô‟s lemma, Let where denotes as before the quadratic co – variation of stochastic processes and we abbreviate further as We propose a hedging portfolio, consisting of ZC swaps as well as ZC options with maturities of and , i.e. 18 where is the date of the reference CPI fixing, which is assumed to be identical for both swaps and caps for simplification purpose; and denote ZC caps with maturities of and respectively; represents ZC swap with reference CPI fixing at . Again, by Itô‟s lemma, each component satisfies the following: Putting everything together, we have Thus, in order to hedge all the risks, we must have and maturity at 19 Solving this simultaneous equation give us the hedging ratios: We remark finally that important prerequisites of this section is that there is a liquid market for ZC options and that the model must be able to price both YoY and ZC options accurately. However, the latter is a question not entirely trivial and few articles explicitly address it. We will discuss this in chapter 5 on calibration. 20 4. Convexity Adjustment A peculiar feature of inflation market is that there are two structures co-existing in both swap and option markets, which brings us to the important notion of convexity adjustment. When calculating YoY swap and YoY option, the main problem lies in calculating the YoY forward value. A simplistic view would be to calculate the YoY forward ratio as the ratio of forward CPIs, which is principally derived from ZC swaps. By doing so, we implicitly assume that the forward value of a ratio is the ratio of the forward value. This is generally not true. Thus, we need to compute convexity adjustment to correct this error. More precisely, we want to compute u such that Recall expression (8) from Appendix I that Adopting the similar drift-freezing technique as before, but freezing further the volatility terms too such that the drift is completely deterministic, we get 21 where we set to simplify the notation. Define , then is a martingale as below As a result, Having determined the convexity adjustment, we digress a little to the pricing of YoY swap and the capturing of its convexity. As YoY swap is nothing but a strip of forward starting ZC 22 inflation linked swaps, here we only focus on forward starting ZC swaps for simplification‟s sake. Refer to Figure 4.1 below for a forward starting ZC swap. We enter the trade at time T0. It involves exchanging a fixed payment with realized inflation from time T1 to T2. Contrary to standard ZC swap which can be priced in a model – independent fashion, the pricing of forward starting ZC swap is model-dependent. By non-arbitrage argument and the definition of Ti - forward measure, we know that the inflation leg is evaluated to Fixed leg T0 T1 T2 Figure 4.1. Structure of forward starting ZC swap Letting and by Itô‟s lemma, we obtain the following hedging strategy: 23 So to hedge a long position of a forward starting ZC swap, we short swap and long units of units of ZC ZC swap. In Table 4.1 and 4.2 below, we detail the hedging strategies under different scenarios and how we capture the convexity along the way. Position Rebalancing Rebalancing P&L 0 swap swap P&L 0 N/A N/A Table 4.1. Dynamic hedging when moves by 1. We note that the hedging shown in Table 4.1 is exact because the swap is linear in . 24 Position Rebalancing P&L swap 0 swap Position Rebalancing P&L swap 0 swap Table 4.2. Dynamic hedging when moves up (above) and down (below) by 1. We remark the “buy low sell high” pattern in the rebalancing as bolded in Table 4.2; this is where we capture convexity just like how gamma is captured through dynamic hedging of vanilla options. As a result, the greater the volatility of through our dynamic hedging. , the more convexity we can capture 25 5. Calibration Before we move on to calibration of two-factor model, we review first the boot – strapping calibration scheme proposed in Mercurio and Moreni (2005) and Liang (2010)2. Suppose we have already interpolated market quoted prices and a matrix of prices with no missing maturities is available3. The first year caplet prices are used to obtain . Next, with the first six parameters fixed, the second year caplet prices are inputted to obtain The next four parameters . are obtained similarly with third year caplet prices and so on. The advantage of this algorithm is that each time, we only need to run an optimization with four (six for the first) parameters. Assuming we have ten maturities, ten optimizations each with four parameters (six for the first) will take much less time than one with 42 parameters. As remarked in Liang (2010), this scheme works fine if the volatility surface displays similar pattern across maturities. It encounters structural difficulty while handling more irregular volatility surface, i.e. surface with both skew and smile configurations. A detailed analysis quoted from Liang (2010) on this deficiency can be found in Appendix III. As we have noted earlier, the model is extended so that it can generate richer volatility surface configurations. The extension seems to be straight forward. However, it entails certain changes to the calibration scheme. Firstly, the original boot-strapping calibration scheme is no longer available. Should the bootstrapping scheme still be applied and volatility parameters calibrated from the first year price, the greater flexibility of the two factor model is lost. All the volatility parameters will still only 2 Parameters in this paragraph follow those in Liang (2010). They are the same as in this thesis, except not indexed by k = 1 & 2 due to one-factor setting. 3 Interpolation will be dealt with specifically in the subsection of interpolation-based calibration 26 reflect the information of volatility surface captured in the first year price. As a result, calibration is to be carried out with the global configuration of the volatility surface taken into consideration. Secondly, the number of variables is an important factor in optimization. By extending the model from one to two factors, we increase the total number of parameters from 42 to 84, if the final maturity is taken to be 10 yr. Over – fitting should be avoided in modeling and running a global optimization with 84 parameters is a computationally expensive. Thus, parameterization is adopted to reduce the number of variables. 5.1. Parameterization We parameterize correlation structures as proposed in Mercurio and Moreni (2010). In this subsection, M denotes the total number of maturities against which we are calibrating. For k = 1 and 2, and with It is proven in Mercurio and Moreni (2010) that this parameterization is well defined in the sense that the correlation matrix is positive semi-definite. 27 The main intuitions of such parameterization are: firstly in (4), decreases as i and j become further apart and, thus, intuitively get less correlated while on the other hand, there will always be an asymptotic correlation secondly in (5), maximum coupling is between This is because monetary policy over period response to inflation behavior over period [ , summarized by and . , is considered as a ]. is parameterized by where A more delicate task is to parameterize ‟s. Instead of viewing the model as one common stochastic volatility process differentiated by different volatility coefficient, we propose to view it as different stochastic volatility processes: Similar to that we think that volatility surface is smooth; we also propose to parameterize the volatility of volatility by a smooth function, e.g. 28 We remark that the proposed parameterization is by no means unique. For example, was originally tested. It yielded satisfactory result though equation (6) produces an even more accurate calibration and is therefore adopted. This set of parameterization has a few repercussions: first and foremost, we reduce the total number of parameters from 84 to 24; secondly, the smoothness of the parameter sets is guaranteed. A calibration with smooth and regular parameter sets is more reliable than that with an arbitrary and highly irregular parameter sets. However, as we will observe in the next subsection, some of the parameterizations have to be loosened to achieve an accurate calibration under certain circumstances. 5.2. Interpolation – Based Calibration What we term „interpolated – based calibration‟ in this thesis is very similar to the existing methodology in interest rate market. Flat volatility is backed out from market quoted prices and then interpolated – linearly or in a more complicated fashion – to obtain volatilities and hence prices for other missing maturities. Calibration is finally conducted against this matrix of market quoted as well as interpolated prices. This scheme is reasonable when we only calibrate a single instrument, e.g. YoY option. As a comparison benchmark for following contents, Appendix IV illustrates the procedure on YoY caps with data from EUR HICP on 13 Jan 2010. Nevertheless, the structure of inflation-linked market adds one complication. Market quoted prices start from maturity of 2yr, 5yr and so on. If the focus is solely on YoY option market, simple extrapolation from 2yr to 1yr suffices. As our scope encompasses both YoY and ZC 29 option markets, it then must be noted that first year YoY option is the same as first year ZC option. So the first year prices of YoY options have to be imported from first year ZC options. Furthermore, as we would like to price both YoY and ZC options, the interpolated prices of YoY options must be consistent with the interpolated prices of ZC options. Belgrade et al. (2004) represented so far the only attempt to address this issue of consistency. However, there is no calibration against actual data to measure the accuracy. In this subsection, we improve on Belgrade et al. (2004) to design a more accurate and robust interpolation scheme and, consequently, a calibration algorithm based on the interpolated prices. Based on market model approach, we present in the following parts an interpolation scheme that will achieve theoretical consistency across two markets and at the same time taking into consideration of market quoted prices. The core notion of „implied correlation‟ is introduced here just like the notion of „implied volatility‟ was introduced to reconcile Black-Scholes model and market quoted option prices. Introduce a simplified market model with constant volatility as follows: with indicating the correlation structure. a stochastic drift frozen to time 0. 30 Under this simplified model, ZC option can be easily priced with a Black – Scholes formula. To . Then by Itô‟s lemma, price YoY option, we let By non – arbitrage argument, a YoY caplet is written as This simple model provides us with an interpolation method. First of all, market quoted prices of ZC options are used to obtain ‟s. We then interpolate to get the implied volatility surface of ZC options. With formula (7), a consistent implied volatility of YoY option can be derived as a function of correlation prices of YoY options. ‟s. we finally derive the „implied correlation‟ from market quoted 31 We will illustrate the proposed approach with again the market data of EUR HICP on 13 Jan 2010. The ZC Cap prices are shown in Table 5.2.1 with highlighted being market prices and others interpolated. Table 5.2.2 shows the corresponding implied volatilities. 1 2 3 4 5 6 7 8 9 10 1% 0.0101573 0.02119 0.03372882 0.04723139 0.06074 0.07387372 0.08681 0.0990976 0.11069907 0.12209 2% 0.00536902 0.0101 0.01563763 0.02190822 0.02841 0.03485188 0.04141 0.04767158 0.05366521 0.05974 3% 0.00285556 0.00434 0.00606033 0.00807568 0.01021 0.01230405 0.01451 0.01651061 0.01843999 0.02048 4% 0.00168913 0.00197 0.002350818 0.002844145 0.00338 0.003861447 0.00439 0.004794616 0.005182388 0.00562 5% 0.001103933 0.001 0.001017436 0.001109173 0.00123 0.001322455 0.00144 0.001485382 0.001531798 0.0016 Table 5.2.1. EUR HICP ZC Cap prices, with maturities from 1yr to 10 yr and strikes from 1% to 5%. Highlighted are market prices and others prices are interpolated (same below). 1 2 3 4 5 6 7 8 9 10 1% 2.06% 2.28% 2.51% 2.73% 2.96% 3.08% 3.20% 3.25% 3.30% 3.35% 2% 2.02% 2.22% 2.42% 2.62% 2.83% 2.97% 3.12% 3.23% 3.34% 3.45% 3% 2.14% 2.33% 2.51% 2.70% 2.89% 3.04% 3.18% 3.30% 3.41% 3.53% 4% 2.36% 2.55% 2.74% 2.93% 3.12% 3.28% 3.43% 3.55% 3.67% 3.79% 5% 2.62% 2.83% 3.04% 3.25% 3.46% 3.63% 3.80% 3.93% 4.06% 4.19% Table 5.2.2. EUR HICP ZC Cap implied volatilities, with maturities from 1yr to 10 yr and strikes from 1% to 5%. With Table 5.2.2 and formula (7), corresponding YoY option implied volatilities can be expressed as a function of correlations. As for each strike, there are only four market-quoted prices and nine correlations available. To avoid over – fitting and ensure regularity of result, 32 correlations of each strike are parameterized by a cubic – spline. Finally, calibration against market data with this cubic spline is presented below in Table 5.2.3 and 5.2.4. 1 2 3 4 5 6 7 8 9 10 1% 2.06% 1.33% 1.12% 1.06% 1.01% 0.90% 0.77% 0.64% 0.73% 0.81% 2% 2.02% 1.30% 1.07% 0.98% 0.92% 0.84% 0.73% 0.62% 0.67% 0.79% 3% 2.14% 1.35% 1.08% 0.98% 0.91% 0.83% 0.72% 0.59% 0.61% 0.83% 4% 2.36% 1.47% 1.16% 1.05% 0.96% 0.88% 0.76% 0.60% 0.64% 0.87% 5% 2.62% 1.62% 1.29% 1.16% 1.07% 0.97% 0.84% 0.64% 0.68% 0.96% Table 5.2.3. EUR HICP YoY Cap spot vol, with maturities from 1yr to 10 yr and strikes from 1% to 5%. ρ(I, i, i - 1) 1 2 3 4 5 6 7 8 9 1% 118.38% 111.51% 107.61% 105.93% 105.15% 105.22% 104.99% 103.56% 102.34% 2% 117.74% 111.76% 108.23% 106.52% 105.67% 105.53% 105.30% 104.20% 102.80% 3% 118.12% 112.23% 108.71% 106.95% 105.94% 105.72% 105.56% 104.66% 102.64% 4% 118.61% 112.59% 108.97% 107.16% 106.08% 105.83% 105.75% 104.75% 102.75% 5% 118.72% 112.64% 109.01% 107.19% 106.12% 105.88% 105.88% 104.88% 102.78% Table 5.2.4. EUR HICP YoY Cap implied correlations, with maturities from 1yr to 9 yr and strikes from 1% to 5%. Table 5.2.4 shows correlation greater than one, which is certainly against mathematical intuition. However, we would like to note that 1. The implied correlations only play an intermediary role, that lead to a smooth and accurate calibration of YoY option prices. We do not seek to draw any conclusion on correlation structure with it. 33 2. The model set – up of constant volatility and frozen drift is simplistic and approximate; the implied correlation thus does not purely reflect information on correlation structure but, at the same time, contains information of other parameters not represented in the model. In consequence, despite its absurd appearance, we believe that when exercised with caution, implied correlations do provide a reasonable and promising means to a consistent interpolation. To further demonstrate the effects of the new interpolation, we compare the interpolated caplet prices and spot volatilities from the new (Table 5.2.5) and the old (Table 5.2.6) interpolation. Maturity 1 2 3 4 5 6 7 8 9 10 1% 0.010157 0.013712 0.015037 0.016078 0.016164 0.016016 0.015196 0.014394 0.014561 0.015104 2% 0.005369 0.007631 0.008588 0.009507 0.009724 0.009739 0.009161 0.008434 0.008760 0.009996 3% 0.002856 0.003915 0.004407 0.005067 0.005275 0.005369 0.004882 0.004070 0.004338 0.006613 4% 0.001689 0.002071 0.002270 0.002702 0.002868 0.002936 0.002514 0.001707 0.002121 0.004252 5% 0.001104 0.001206 0.001288 0.001587 0.001725 0.001757 0.001403 0.000703 0.001009 0.002988 Table 5.2.5. EUR HICP YoY Caplet prices, with maturities from 1 to 10 yr and strikes from 1% to 5%. New Interpolation Maturity 1 2 3 4 5 6 7 8 9 10 1% 0.009105135 0.014764865 0.016155617 0.016146946 0.014977 0.016024223 0.015185777 0.015564375 0.014595249 0.013900375 2% 0.004286642 0.008713358 0.009852065 0.00963961 0.008328 0.009817785 0.009082215 0.009763408 0.008994644 0.008431948 3% 0.001863572 0.004906428 0.005713865 0.005239501 0.003797 0.00546183 0.00478817 0.005567226 0.004960705 0.004492069 4 4% 0.000865558 0.002894442 0.003472092 0.002877243 0.001491 0.003027956 0.002422044 0.003137214 0.002665026 0.00227776 5% 0.000450534 0.001859466 0.002313825 0.001742613 0.000544 0.001843234 0.001316766 0.001917624 0.001550129 0.001232247 Table 5.2.6. EUR HICP YoY Caplet prices, with maturities from 1yr to 10 yr and strikes from1% to 5%. Old Interpolation4. 4 This table can be directly derived from table 7.4.3 of Appendix. 34 The first observation is that old interpolation does not take into consideration the ZC option market. Hence, the first year prices do not correspond to the first year ZC option prices. New interpolation scheme, on the other hand, corrects this error. Moreover, even though flat volatilities are linearly interpolated and extrapolated in the case of Liang (2010), the resulted spot volatility surface is still irregular, as we can see in the right of figure 5.2.1. But new interpolation generates a much smoother volatility surface, e.g. figure 5.2.1 left. With a full matrix of prices extracted, we are now ready to conduct the calibration, during which the square sum of errors of market and model prices is minimized. Given the proposed 3.00% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 2.00% 1.00% 0.00% 1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 10 8 9 10 Figure 5.2.1. EUR HICP YoY Caplet spot volatility, with maturities from 1yr to 10 yr and strikes from1% to 5%. Left: new interpolation in this thesis; right: old interpolation (refer to figure 7.4.2 of Appendices). parameterization functional, the following algorithm is designed. I. We first start with the most parsimonious structure, i.e. both volatility coefficients and correlation parameters are parameterized by 24 variables as enumerated below: 35 II. If the above does not generate a good calibration, we will loosen some parameterization. ‟s, as this condition appears to be the most We start with volatility coefficients ‟s determined by arbitrary. We will take the values of as the initial guess, keep the rest of the parameters constant and run a calibration with the ten parameters. III. Similarly, if this still does not generate satisfactory result, we loosen the regularity condition imposed on As and ‟s are more important than and in terms of the impact on pricing, by loosening these two restrictions, we can normally obtain a fairly good calibration. IV. If the algorithm does not terminate at III, then in the final step, we optimize with : There are altogether 56 parameters involved. Intimidating as it appears, this does not pose great computational challenge as we are already close to the minimal and, thus, the optimization terminates reasonably fast with few iterations. We report the calibration result of YoY options in Figure 5.2.2 and parameters obtained in Table 5.2.7 and Figure 5.2.3. 1.80% 2yr Market Vol 1.60% 2yr Calibrated Vol 1.40% 5yr Market Vol 5yr Calibrated Vol 1.20% 7yr Market Vol 7yr Calibrated Vol 1.00% 10yr Market Vol 0.80% 10yr Calibrated Vol 0.60% 1 2 3 4 5 Figure 5.2.2. Market and calibrated YoY implied volatility with x – axis representing the strikes (%) 36 α θ ε V(0) 4.5771823 0.8190538 0.2781674 0.0760959 0.9634653 1 0.2316799 0.0773175 σ1 0.013163 0.017208 0.005775 0 0.00598 0.007727 0.002213 0.00932 0.009372 0.006246 σ2 0.082623 0.066601 0.069499 0.079097 0.077454 0.058275 0.038511 0.043456 0.058504 0.029766 ρ1(I, i, V) -1 -0.69911151 1 1 -1 -1 -1 1 1 -1 ρ2(I, i, V) -0.06325 -0.02557 -0.10504 -0.11698 -0.13001 -0.14053 -0.11102 -0.22586 -0.27116 -0.09706 Table 5.2.7. Model parameters. Left: volatility coefficients. Center: volatility factor loadings. Right: forward CPI / volatility correlations It is worth noting that the speed of reversion α is about 4.6 for one volatility process and about 0.8 for the other. Thus, we can interpret that the factor with smaller reversion is determining the long – run variance while that with greater reversion is determining the short – run variance. Also we note that the calibrated correlation structures are well – behaved. 1 1 0.5 0 1 3 5 7 9 -0.48 0 0.95 0.9 0.85 0.8 1 3 Figure 5.2.3. Model parameters. Left: 5 7 9 -0.2 1 3 5 7 9 -0.4 -0.485 1 3 5 7 9 -0.49 -0.6 -0.495 -0.8 -0.5 . Center left: . Center right: . Right: . The rational for using global optimization and the proposed algorithm are on the hypothesis that information of the initial point is not available. This is so when there is a big shift to the market. For example, when an inflation index turns out to be very different from expected. When market 37 model becomes more popular and regularly used and market is stable, simple local optimization suffices. The final step consists of calculating ZC options prices from the calibrated parameters. We present the relative percentage errors in Table 5.2.8 below. 1 2 3 4 5 6 7 8 9 10 1% -1.72% -9.82% -9.57% -5.62% -5.27% -6.09% -6.20% -4.41% -3.01% -3.10% 2% -0.72% -25.17% -35.80% -28.16% -28.02% -35.37% -42.76% -34.99% -28.36% -35.17% 3% 2.16% -39.14% -55.83% -52.39% -58.69% -76.83% -92.05% -89.13% -83.34% -96.41% 4% 2.44% -44.87% -60.99% -58.46% -68.40% -87.98% -98.06% -97.94% -96.39% -99.77% 5% -2.91% -49.38% -64.37% -61.50% -73.36% -92.63% -99.39% -99.45% -98.92% -99.98% Table 5.2.8. Relative percentage error of ZC option prices with maturities from 1yr to 10 yr and strikes from 1% to 5%. Results are acceptable for short maturities and for at-the-money (ATM) caplets, but deteriorate rapidly for longer maturities and greater strikes. We remark here a few points on how to further assess the quality of the results: 1. Relative errors must be assessed with respect to bid-offer spreads. ZC option market is rather illiquid compared to YoY option market and bid-offer spreads are much larger. An accurate calibration against mid-prices might not be more significant compared to a calibration with moderate relative error, when bid-offer spreads dominate. 2. Calibration should be carried out across a period. Illiquid market is always subject to structural factors that result in consistent mispricing from model “fair” price. Such a phenomenon is well documented in nominal bond market, though few articles tackle this issue in inflation – linked market. 38 5.3. Non – Interpolation – Based Calibration Non – interpolation – based calibration is introduced because of the observation that interpolated YoY caplet prices are sensitive to implied correlation. By choosing an equally smooth but slightly different implied correlation structure, we end up with a very different interpolated YoY caplet prices. An example is presented below. The implied correlation structures are almost identical but the resulted interpolated caplet prices and spot volatility surfaces vary by a large extent as shown in Figure 5.3.2. 120.00% 120.00% 110.00% 110.00% 100.00% 100.00% 90.00% 90.00% 1 2 3 4 5 6 1 7 8 2 3 9 4 5 6 7 Figure 5.3.1. YoY implied correlation. Left: perturbed. Right: original. ρ(I, i, i - 1) 1 2 3 4 5 1 118.38% 117.74% 118.12% 118.61% 118.70% 2 111.50% 113.82% 114.26% 112.09% 112.16% 3 108.23% 108.23% 107.98% 109.16% 109.00% 4 105.50% 105.56% 106.57% 107.23% 107.37% 5 105.21% 105.50% 106.10% 106.30% 106.34% 6 105.18% 105.65% 105.60% 105.66% 105.71% 7 104.40% 104.70% 104.79% 104.71% 104.79% 8 103.48% 103.92% 104.16% 104.24% 104.38% 9 102.94% 103.60% 103.87% 104.12% 104.02% 8 9 39 ρ(I, i, i - 1) 1 1 118.38% 117.74% 118.12% 118.61% 118.72% 2 111.51% 111.76% 112.23% 112.59% 112.64% 3 107.61% 108.23% 108.71% 108.97% 109.01% 4 105.93% 106.52% 106.95% 107.16% 107.19% 5 105.15% 105.67% 105.94% 106.08% 106.12% 6 105.22% 105.53% 105.72% 105.83% 105.88% 7 104.99% 105.30% 105.56% 105.75% 105.88% 8 103.56% 104.20% 104.66% 104.75% 104.88% 9 102.34% 102.80% 102.64% 102.75% 102.78% 2 3 4 5 Table 5.3.1. YoY implied correlation. Above: perturbed. Below: original. 3.0000% 3.00% 2.0000% 2.00% 1.0000% 1.00% 0.00% 0.0000% 1 2 3 4 5 1 6 7 8 9 10 2 3 4 5 6 7 8 9 10 Figure 5.3.2. EUR HICP YoY Caplet spot volatility. Left: perturbed; right: original. It is thus logical to suspect that interpolation introduces some degree of arbitrary information and contaminates calibration. This brings us naturally to the idea of non – interpolation – based calibration. In this scheme, interpolation step is omitted and only market quoted cap prices are used for calibration. A crucial issue in non – interpolation – based calibration is how to ensure the regularity of resulted caplet prices. But as it turns out, calibration with the most parsimonious structure, i.e. 40 the first step of the calibration algorithm proposed above, yields a satisfactory result. Consequently, the regularity of parameters guarantees that final prices and implied volatility surface is also smooth. Finally, to avoid the problem of not taking into consideration of the information of ZC option market, we calibrate against market quoted prices of both ZC and YoY options. As ZC option market exhibits a much larger bid – offer spread, ZC option prices should not retain the same significance as YoY option prices in terms of providing information to calibration. As a result, we apply different weightings to different prices. Intuitively, the ratio of the two weights can be inversely proportional to the ratio of the bid – offer spreads of the market. The optimization, thus, becomes where With denotes market quoted ZC option prices. Other notations are interpreted similarly. 10% and 2 5 7 10 12 1% -0.39% -0.91% -1.21% -2.97% -4.82% 90%5, the result is reported in Table 5.3.2. 2% 0.83% -0.49% 0.04% -0.96% -2.66% 3% 1.05% -1.73% -0.18% 0.29% -0.86% 4% 1.15% -2.26% -1.38% -0.68% -1.51% 5% -2.02% 0.95% 0.53% -0.65% -2.21% 6% -8.08% 4.24% 4.69% 2.49% 0.04% Table 5.3.2. Relative percentage error of YoY cap prices. Bolded are relative error of extended caps. 5 We have tested various combinations such as 20% - 80%, 10% - 90% and 5% - 95%. 10% - 90% so far produces the most accurate calibration and the result is therefore presented in this thesis. 41 To highlight the smoothness of parameters, obtained parameters and final spot volatilities are reported in Figure 5.3.3 and 5.3.4 below. 0.036 0.25 0.40 0.034 0.2 0.30 0.15 0.032 0.20 0.1 0.03 0.028 0.05 0.10 0 0.00 1 2 3 4 5 6 7 8 9 10 -0.94 1 2 3 4 5 6 7 8 9 10 . Right: Figure 5.3.3. Model parameters. Left: 1 1 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 1 3 5 7 0.8 0.6 0.4 0.2 0 1 9 . 3 5 7 -0.2 1 3 5 7 9 -0.4 -0.6 1 3 5 7 9 9 Figure 5.3.4. Model parameters. Left: 0 . Center left: -0.8 . Right: . Center right: 2.500% 2.000% 1.500% 1.000% 0.500% 0.000% 1 2 3 4 5 6 7 8 9 Figure 5.3.5. Calibrated YoY spot volatility. 10 . 42 Thus, by forcing the regularity of parameters and calibrating only against market quoted data, we can still achieve an accurate calibration and generate a reasonably shaped implied volatility surface as in Figure 5.3.5. Finally, plugging the parameters to price ZC options yields Table 5.3.3. 1 2 3 4 5 6 7 8 9 10 1% -1.82% -1.02% -0.54% -0.33% -0.29% 0.48% 0.95% 1.66% 2.15% 2.49% 2% 1.81% 1.09% 0.40% 0.12% -0.05% 1.21% 2.13% 3.63% 4.82% 5.76% 3% 0.02% 3.37% 1.15% -1.28% -2.96% -1.77% -0.60% 2.46% 5.24% 7.73% 4% -11.88% 4.06% 5.78% 1.61% -3.71% -5.46% -7.25% -5.47% -3.52% -1.59% 5% -29.45% -0.55% 11.63% 11.96% 6.78% 4.15% -0.57% -0.18% -0.52% -1.82% Table 5.3.3. Relative percentage error of ZC option prices with maturities from 1yr to 10 yr and strikes from 1% to 5%. We observe that the relative errors shown in Table 5.3.3 now are much reduced. However, this does not necessarily mean that we should absolutely favor this method over the previous. Should our goal be to arbitrage the two markets, interpolation-based calibration should still be employed with certain supplementary statistical analysis on the relative errors as we have recommended. On the other hand, if we were to price an exotic inflation-linked instrument, non-interpolationbased calibration is more appropriate. Another advantage of this approach is that as we keep all parameterized structures, we can easily extend the parameters to price more extreme options. This is contrary to interpolation-based calibration where we have to take all the options of interest upon calibration, some of which might be thinly traded, upon calibration. 43 With non-interpolation-based scheme, we only need to calibrate with respect to more liquid options and extend the calibrated parameters to price other options. In Table 5.3.2, we also present the relative error of YoY caps with strikes of 6% and maturity of 12 years. Note that they are not included in the calibration process. Readers may notice that should we extend the parameters, it is inevitable to change the total number of instruments M in formula (5) (from 10 to 12 in our case) and change the correlation structure. While this is true, it is observed that total error generally changes little or decreases when plugging in correlations from a new M. We believe that the correlation structure now contains more complete information than before. 44 6. Conclusions In this thesis, we have extended the inflation – linked stochastic volatility market model, proposed in Mercurio and Moreni (2005) and modified in Liang (2010), to two – factor regime. This extension provides greater flexibility in fitting volatility surface, while retaining the tractability of the original model. Literature on two - factor Heston model has shed light on its internal structure and why it works much better than single factor model. We still believe that it will be constructive to conduct principal component analysis on the implied volatility of inflation – linked market. It will tell us more on why two – factor model works better or not and how many factors make a good modeling, etc. Based on this extended model set – up, we have derived a theoretical hedging strategy, following which, YoY options can be hedged with ZC options. This new perspective enables maximum leverage of a trading book containing both options and saves transaction cost. Under the same setting, the convexity adjustment of YoY swap rate is also approximated. We detail in the thesis the process of how to capture the convexity through concrete trading activities. The structure of inflation market where two product structures co-exist makes calibration unique and few articles specifically tackle this issue. In this thesis, we fill the gap by proposing and testing two methods: interpolation and non – interpolation – based calibration. The interpolation – based calibration conforms more to the orthodox methodology. We devised a simple market model where volatility is constant and introduced the notion of implied correlation to interpolate YoY cap prices. In this way, the information of ZC option market is incorporated into YoY cap prices. It is observed that the resulted implied volatility surface is smoother than that resulted from a naïve interpolation of flat volatilities. This method produces a more accurate 45 calibration of YoY option prices though that of ZC options is reasonable only for short maturities and ATM options. On the other hand, non – interpolation based calibration is introduced so that no arbitrary/false information is produced as a result of interpolation process. We assign different weight to the calibration of ZC and YoY options, with the former certainly smaller than latter. Though we can optimize the weights attached more systematically, our calibration experience shows that 10% 90% works the best so far. The advantage of this methodology is that firstly, it ensures a smooth parameter set; secondly, we obtain a more acceptable calibration of ZC options at the expense of a slightly worse calibration of YoY options and finally the parameters can be extended to price more extreme options. Interesting and revealing, it is yet beyond the scope of the thesis to conclude which method is superior or more applicable. In further research, we believe that more practical tests have to be conducted. Areas of consideration can include: 1. Models can be tested across a period to further assess the quality of calibration. 2. Calibrated parameters should be tested across a period to assess its stability. 46 Bibliography 1. Albrecher, H., Mayer, P., Schoutens, W and Tistaert, J. The Little Heston Trap. Wilmott Magazine, January Issue, 83-92, 2006. 2. Andersen L. and Rotherton – Ratchliffe R. Extended Libor Market Models with Stochastic Volatility, SSRN Working Paper; Retrieved on 12 Oct 2009 from World Wide Web: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=294853 3. Bates, D. Post – 87 Crash Fears in S&P 500 Futures Options. Journal of Econometrics, 94, 181-238, 2000. 4. Belgrade, N., Benhamou E. and Koehler E. A Market Model for Inflation. SSRN Working Paper; Retrieved on 11 Nov 2009 from World Wide Web: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=576081 5. Bergomi, L. (2004). Smile Dynamics, Risk September, 117 – 123. 6. Brace, A., Gatarek, D. and Musiele, D. The Market Model of Interest Rate Dynamics, Mathematical Finance 7, 127 – 155, 1997. 7. Brigo, D. and Mercurio, F. Interest rate models: theory and practice, Springer, Berlin, 2006. 8. Carr, P. and Madan, D. Option Valuation using the Fast Fourier Transform, Journal of Computational Finance , 2, 6 1- 73, 1999. 9. D. Heath, R. A. Jarrow, and A. Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology,” Econometrica, 60, 1(1992): 77 - 105. 10. Da Fonseca, J., Grasseli, M., and Tebaldi, C. (2008). A Multifactor Volatility Heston Model. Quantitative Finance, 8 (6): 591 – 604. 11. Deacon, M., Derry, A. and Mirfendereski, D. Inflation – indexed securities: bonds, swaps and other derivatives, 2nd edition, John Wiley & Sons, N.J., 2004. 47 12. Gatheral, J. The volatility surface: a practitioner‟s guide, John Wiley & Sons, N.J., 2006. 13. Geman, H., EL Karoui, N. and Rochet, J.C. Changes of Numeraire, Changes of Probability Measure and Option Pricing, Journal of Applied Probability, 32, 443 – 458, 1995. 14. Hagan, P. and Lesniewski, A. LIBOR Market Model with SABR Style Stochastic Volatility, Retrieved on 12 Oct 2009 from World Wide Web: http://www.lesniewski.us/papers/working/SABRLMM.pdf 15. Heston, S. A Closed – Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6, 327 – 343, 1993. 16. Hull, J. Futures, Options and Other Derivatives, 6th edition, Prentice Hall, N.J., 2006. 17. Jarrow, R. and Yildirim, Y. Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model, Journal of Financial and Quantitative Analysis, 38(2), 409 – 430, 2003. 18. Kahl, C. and Jäckel, P. Not – so – complex Logarithms in the Heston Model. Wilmott, 94 – 103, 2005. 19. Kazziha, S. Interest Rate Models, Inflation – based Derivatives, Trigger Notes and Cross – Currency Swaptions, PhD Thesis, Imperial College of Science, Technology and Medicine, London,1999. 20. Liang, L. Inflation-Linked Option Pricing, Final Year Project, National University of Singapore, 2010. 21. Lin, S. Finite Difference Schemes for Heston Model, Graduate project, The University of Oxford, 2008. 48 22. Mercurio, F. and Moreni, N. A Multi-Factor SABR Model for Forward Inflation Rates, Bloomberg Portfolio Research Paper No. 2009 – 08, FRONTIERS. 23. Mercurio, F. and Moreni, N. Pricing Inflation – Indexed Options with Stochastic Volatility, Internal report , Banca IMI, Milan. Retrieved on 12 Oct 2009 from World Wide Web: www.fabiomercurio.it/stochinf.pdf 24. Mercurio, F. Pricing Inflation – Indexed Derivatives, Internal report, Banca IMI, Milan. Retrieved on 12 Oct 2009 from World Wide Web: http://www.fabiomercurio.it/Inflation.pdf 25. Mercurio, F. Pricing Inflation – indexed Derivatives. Quantitative Finance, 5(3), 289 – 302, 2005. 26. Peter, C., Heston, S and Kris Jacobs. The shape and term structure of index option smirk: Why multifactor stochastic volatility models work so well, Working paper, McGill University and University of Maryland, 2007. 27. Rough, F. D. and Vainberg, G. Option pricing models and volatility using Excel – VBA, John Wiley & Sons, N.J., 2007. 28. Schöbel, R. and Zhu, J. Stochastic Volatility with an Ornstein Uhlenbeck Process: An Extension. European Finance Review, 3, 23 – 46, 1999. 29. Ungari, S. Inflation Market Handbook. Société Générale, 2008. 30. Zhang, H. and Rangaiah, G. P. A Hybrid Global Optimization Algorithm. The 5th International Symposium on Design, Operation and Control of Chemical Process. 31. Zhu, J. An Extended Libor Market Model with Nested Stochastic Volatility Dynamics. 2007. Available online at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=955352 49 Appendices Appendix I Derivation of YoY caplet price under one factor stochastic volatility Recall the following set-up as in section 2: And the stochastic process of volatility V is With j = 1 & 2 and k = i & i – 1, And 50 with Freezing the stochastic drift to time 0, we have Let Then by Itô‟s lemma, We define further that Note that 51 By standard option pricing theory, we have the price of a YoY caplet of period [ strike k under - measure: Where we define We then have By we define So we will evaluate firstly and 52 Let , then We note that We assume that where we define With terminal conditions as By Feynman – Kac formula, we have 53 Substituting the assumed form, we obtain Putting everything together, we obtain 54 Or equivalently for j = 1 and 2, So the partial differential equation is satisfied if the follows hold, for j = 1 and 2 Thus More specifically, setting , we have for j = 1 and 2 55 Thus, So we have at this stage Where we recall that We simplify further via the same drift – freezing technique, Define the characteristic function as follows: So by Feynman-Kac formula, we get 56 We suppose that the characteristic function takes the following form, Where we have as terminal conditions Substituting this into the original PDE, we obtain further that 57 Combining everything back together, Or equivalently, for j = 1 and 2, Thus, the differential equations satisfied by and are 58 So we can solve explicitly, By setting we obtain 59 Appendix II Riccati equation Let A and B be solutions of the following general Riccati equation: By setting and by rewriting (1) now as We can then solve the above easily as A can then be solved via integration , where 60 With and , so that finally we get Appendix III Structural deficiency of one factor model Recall that is introduced in the following manner: Looking instead at the transformed volatility process: As such, we see that the coefficient acts as a scaling factor, increasing both the volatility of volatility and the long term variance. We see that the two have counter – effects against each other. While increasing volatility of volatility increases the curvature of the smile, a greater long term variance flattens it, while moving it parallel upwards. Furthermore, the correlation is not affected by the factor loading. The implication of this is that beside the volatility of volatility we do not possess much flexibility in terms of fitting the curvature of the smile. We can, thus, anticipate a case where the model will not generate an accurate calibration. If the volatility surface consists of skewed as well as curved smiles, the one-factor model will find it hard to accommodate the two different regimes. 61 Appendix IV Interpolation based on flat volatilities Shown in Table 7.4.1 is the market price of EUR HICP YoY caps on 13 Jan 2010. In this appendix, we illustrate an interpolation scheme with this starting point. 1% 0.02387 0.07115 0.10236 0.14642 2 5 7 10 2% 0.013 0.04082 0.05972 0.08691 3% 0.00677 0.02152 0.03177 0.04679 4% 0.00376 0.0116 0.01705 0.02513 5% 0.00231 0.00691 0.01007 0.01477 Table 7.4.1. EUR HICP YoY Cap prices Flat volatilities of 2, 5, 7 and 10 yrs are derived from the market quoted cap prices. We then linearly interpolate and extrapolate the flat volatilities to obtain the flat volatility surface as presented in table 7.4.2 and figure 7.4.1. 1 2 3 4 5 6 7 8 9 10 1% 1.79% 1.65% 1.51% 1.36% 1.22% 1.15% 1.08% 1.03% 0.99% 0.94% 2% 1.74% 1.59% 1.44% 1.30% 1.15% 1.08% 1.01% 0.97% 0.93% 0.88% 3% 1.80% 1.63% 1.46% 1.30% 1.13% 1.09% 0.99% 0.94% 0.90% 0.86% 4% 1.94% 1.75% 1.56% 1.37% 1.18% 1.10% 1.02% 0.97% 0.93% 0.88% 5% 2.13% 1.92% 1.71% 1.49% 1.28% 1.19% 1.10% 1.05% 1.00% 0.95% Table 7.4.2. EUR HICP YoY Cap flat vol. Bolded are quoted and others are linearly interpolated 62 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Figure 7.4.1. EUR HICP YoY Cap flat vol surface Next, we compute the other YoY cap prices from the interpolated flat volatilities. The full matrix of YoY cap prices is shown in table 7.4.3. Finally, the spot volatility surface, i.e. figure 7.4.2 can be generated from the caplet prices. 1% 1 0.009105 2 0.02387 3 0.040026 4 0.056173 5 0.07115 6 0.087174 7 0.10236 8 0.117924 9 0.13252 10 0.14642 2% 0.004287 0.013 0.022852 0.032492 0.04082 0.050638 0.05972 0.069483 0.078478 0.08691 3% 0.001864 0.00677 0.012484 0.017723 0.02152 0.026982 0.03177 0.037337 0.042298 0.04679 4% 0.000866 0.00376 0.007232 0.010109 0.0116 0.014628 0.01705 0.020187 0.022852 0.02513 5% 0.000451 0.00231 0.004624 0.006366 0.00691 0.008753 0.01007 0.011988 0.013538 0.01477 Table 7.4.3. EUR HICP YoY Cap prices. Bolded are quoted and others are from interpolated flat vols. 63 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 1 2 3 4 5 6 7 8 9 10 Figure 7.4.2. EUR HICP YoY Cap resulted spot vol [...]... against actual data to measure the accuracy In this subsection, we improve on Belgrade et al (2004) to design a more accurate and robust interpolation scheme and, consequently, a calibration algorithm based on the interpolated prices Based on market model approach, we present in the following parts an interpolation scheme that will achieve theoretical consistency across two markets and at the same time taking... the inflation- linked stochastic volatility market model under a multi - factor setting 2 The hedging analysis is conducted from a new perspective that ZC options are used to hedge YoY options – a strategy of great practical interest 9 3 While most articles treat the subject of calibration as if it is no different from that of interest rate models, it is actually more delicate in an inflation- linked. .. literatures detailed above and addresses related issues such as hedging, convexity adjustment and calibration As a whole, we strive to build a comprehensive framework under which an inflation- linked pricing model can be calibrated and applied Works on multi – factor stochastic volatility model, such as Bates (2000) and Christoffersen (2007) have inspired us to extend the model in Liang (2010) to two-factor... Mercurio and Moreni (2010) built a more complete model with SABR stochastic volatility process Details of SABR model can be found in Hagan (2002) In Liang (2010), a variation of Mercurio and Moreni (2005) model is built All forward CPIs are assumed to share one common volatility process - differentiated only by respective factor loadings The model is then implemented and calibrated with a boot-strapping algorithm... structures are defined as With j = 1 and 2, k = i and i – 1 and < , > denotes the quadratic co-variation of stochastic processes The technical aspect of quadratic variation can be found in Brigo and Mercurio (2006) And at the same time, we also have, in the nominal market, with By applying the same drift-freezing technique and fast Fourier transformation as in Liang (2010) and Mercurio and Moreni (2005)1,... yielded satisfactory result though equation (6) produces an even more accurate calibration and is therefore adopted This set of parameterization has a few repercussions: first and foremost, we reduce the total number of parameters from 84 to 24; secondly, the smoothness of the parameter sets is guaranteed A calibration with smooth and regular parameter sets is more reliable than that with an arbitrary and... volume has increased from almost zero in 2001 to $110 billion in 2007 Trading of inflation- linked options is also picking up gradually Most inflation models so far have been derived from interest rate models Currently, the pricing of inflation- linked options is addressed by resorting to a foreign currency analogy In Jarrow and Yildrim (2003), the dynamics of nominal and real rates are modeled by one-factor... designed a number of schemes, which can be broadly categorized as interpolation- and non-interpolation-based schemes Variable reduction is important in global optimization as we seek to avoid over-parameterization as well as to increase the efficiency of optimization To this end, we have adopted various parameterization functions for different parameter sets For example, the parameterization of correlations... in chapter 5 on calibration 20 4 Convexity Adjustment A peculiar feature of inflation market is that there are two structures co-existing in both swap and option markets, which brings us to the important notion of convexity adjustment When calculating YoY swap and YoY option, the main problem lies in calculating the YoY forward value A simplistic view would be to calculate the YoY forward ratio as the... Mercurio and Moreni (2005) and Liang (2010)2 Suppose we have already interpolated market quoted prices and a matrix of prices with no missing maturities is available3 The first year caplet prices are used to obtain Next, with the first six parameters fixed, the second year caplet prices are inputted to obtain The next four parameters are obtained similarly with third year caplet prices and so on The advantage ... of parameters and calibrating only against market quoted data, we can still achieve an accurate calibration and generate a reasonably shaped implied volatility surface as in Figure 5.3.5 Finally,... generates volatility skew at the expense of over – parameterization as remarked in Ungari (2008) As such, alternative approaches are gaining popularity For example, Kazziha (1999), Belgrade et al (2004)... spreads ZC option market is rather illiquid compared to YoY option market and bid-offer spreads are much larger An accurate calibration against mid-prices might not be more significant compared