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Stochastic Volatility Models and Hybrid Derivatives Stochastic volatility models and hybrid derivatives Claudio Albanese Department of Mathematics / Imperial College London ??? Presented at Bloomberg and at the Courant Institute, New York University New York, September 22nd, 2005 Co-authors. • Oliver Chen (National University of Singapore) • Antonio Dalessandro (Imperial College London) • Manlio Trovato (Merrill Lynch London) available at: www.imperial.ac.uk/mathfin 1 Contents. • Part I. A stochastic volatility term structure model. • Part II. Credit barrier models with functional lattices. • Part III. Estimations under P and under Q. • Part IV. Pricing credit-equity hybrids. • PART V. Credit correlation modeling and synthetic CDOs. • Part VI. Pricing credit-interest rate hybrids. 2 PART I. A stochastic volatility term structure model It is widely recognized that fixed income exotics should be priced by means of a stochastic volatility model. Callable constant maturity swaps (CMS) are a particularly interesting case due to the sensitivity of swap rates to implied swaption volatilities for very deep out of the money strikes. In this paper we introduce a stochastic volatility term structure model based on a continuous time lattice which allows for a numerically stable and quite efficient methodology to price fixed income exotics and evaluate hedge ratios. 3 Introduction The history of interest rate models is characterized by a long series of turns. The Black formula for caplets and swaptions was designed to take as underlying a single forward rate under the appropriate forward measure, see (Joshi & Rebonato 2003). This has the advantage to lead to a simple pricing formula for European options but also the limitation of not being extendable to callable contracts. To have a more consistent model, short rate models where introduced in (Cox, Ingersoll & Ross 1985), (Vasicek 1977), (Black & Karasinski 1991) and (Hull & White 1993). These models are distinguished by the exact specification of the spot rate dynamics through time, in particular the form of the diffusion process, and hence the underlying distribution of the spot rate. 4 LMM models Next came LIBOR market models, also known as correlation models. First introduced in (Brace, Gatarek & Musiela 1996) and (Jamshidian 1997), forward LIBOR models affirmed themselves as a mainstream methodology and are now discussed in textbooks such as for instance (Brigo & Mercurio 2001). Various extensions of forward LIBOR mod- els that attempt to incorporate volatility smiles of interest rates have been proposed. Local volatility type extensions were pioneered in (Andersen & Andreasen 2000). A stochastic volatility extension is proposed in (Andersen & Brotherton-Ratcliffe 2001), and is further extended in (Andersen & Andreasen 2002). A different approach to stochastic volatility forward LIBOR models is described in (Joshi & Rebonato 2003). Jump-diffusion forward LIBOR models are treated in (Glasserman & Merener 2001), (Glasserman & Kou 1999). A cali- bration framework is proposed in (Piterbarg 2003). 5 Stochastic volatility models Modeling stochastic volatility within LIBOR market models is a chal- lenging task from an implementation viewpoint. In fact, Monte Carlo methods tend to be slow and inefficient in the presence of a large num- ber of factors. In a strive to achieve a more reliable pricing framework, in recent years, we witnessed a move away from correlation models and the emergence and recognition as market standard of the SABR model by (S.Hagan, Kumar, S.Lesniewski & E.Woodward 2002) in the fixed income domain. 6 SABR SABR however is unlikely to be the definitive solution and is perhaps rather just yet another stepping stone in a long chain. In fact, like- wise to the Black formula approach, SABR takes up a single forward rate under the corresponding forward measure as underlier. As a con- sequence, within this framework it is not possible to price callable swaps and Bermuda swaptions. There are also calibration inconsis- tencies. Implied swaption volatilities with very large strikes are probed by constant maturity swaps (CMS), structures which receive fixed, or LIBOR plus a spread, and pay the equilibrium swap rate of a given maturity. The asymptotic behavior of implied volatilities for very large strikes turns out to flatten out to a constant, as opposed to diverging rapidly as SABR would predict. Finally, some pricing inconsistencies may emerge with SABR due to the fact that the model is solved by means of asymptotic expansions with a finite, irreducible error. 7 Stochastic volatility term structure models In this article we attempt to go beyond SABR by introducing a stochas- tic volatility short rate model which has the correct asymptotic behav- ior for implied swaption volatilities and can be used for callable swaps and Bermuda swaptions. Our model is solved efficiently by means of numerical linear algebra routines and is based on continuous time lat- tices of a new type. No calculation requires the use of Montecarlo or asymptotic methods and prices and hedge ratios are very stable, even for extreme values of moneyness. 8 Nearly stationary calibration Our model is made consistent with the term structure of interest rates and the term structure of implied at-the-money volatilities by means of a calibration procedure that greatly reduces the degree of time dependency of coefficients. As a consequence, the model is nearly stationary over time horizons in excess of 30 years. 9 Applications In this presentation, I discuss the model by reviewing in detail an implementation example. While leaving it up to the interested reader to pursue the many conceivable variations and extensions, we describe the salient features of our modeling by focusing in detail to the problem of pricing and finding hedge ratios for Bermuda swaptions and callable CMS swaps. 10 The model Our model is built upon a specification of a short rate process rt which combines local volatility, stochastic volatility and jumps. We make our best effort to calibrate the model to a stationary process and introduce the least possible degree of explicit time dependence in such a way to refine fits of the term structure of interest rates and of selected at-the-money swaption volatilites. The model is specified in a largely non-parametric fashion within a functional analysis formalism and expressed in terms of continuous time lattice models. A sequence of several steps is required to specify the short rate process rt. 11 The conditional local volatility processes We introduce M states of volatility. The process conditioned to stay in one of such states α ∈ {1, ...M} is related to the solution rαt of the following equation: βα drαt = κα(θα − rαt)dt + σαrαt dW. (1) 12 Short rate volatilities for each of the four volatility states 13 The functional analysis formalism In the functional analysis formalism we use, these SDEs are associated to the Markov generators 2 2βα 2 r ∂ σαrαt ∂ L = κα(θα − rαt) + . (2) α ∂r 2 ∂r2 14 Functional lattices To build a continuous time lattice (also called functional lattice), we discretize the short rate variable and constrain it to belong to a finite lattice Ω containing N + 1 points r(x) ≥ 0, where x = 0, ...N, r(0) = 0 and the following ones are in increasing order. The discretized Markov r generator LΩα is defined as the operator represented by a tridiagonal matrix such that X r LΩα(x, y) = 0 y X r LΩα(x, y)(y − x) = κα(θα − r(x)) y X r 2 2 2βα LΩα(x, y)(y − x) = σαr(x) . y 15 Model parameters In our example, we select an inhomogeneous grid of N = 70 points spanning short rates from 0% to 50%. We also choose to work with M = 4 states for volatility and make the following parameter choices: α σα βα θα κα 0 31% 30% 2.10% .17 1 46% 40% 5.50% .18 2 75% 50% 8.50% .23 3 100% 60% 12.00% .24 16 How to solve in the special case of a local volatility model (M=1) and without jumps Although our model is more complex than a simple local volatility process, it is convenient to describe our resolution method in the r specific case of the operator LΩα with constant α. This method can then be generalized and is at the basis of other extensions such as the introduction of jumps (see below). 17 Spectral analysis We start by considering the following pair of eigenvalue problems: r rT LΩαun = λnun LΩαvn = λnvn T where the superscript denotes matrix transposition, un and vn are r the right and left eigenvectors of LΩα, respectively, whereas λn are the corresponding eigenvalues. Except for the simplest cases, the Markov r generator LΩα is not a symmetric matrix, hence un and vn are different. 18 Spectral analysis Also, in general the eigenvalues are not real. We are only guaranteed that their real part is non-positive Reλn ≤ 0 and that complex eigen- values occur in complex conjugate pairs, in the sense that if λn is an eigenvalue then also λ¯n is an eigenvalue. We set boundary conditions in such a way that there is absorption at the endpoints, and in par- ticular at the point corresponding to zero rates r(0) = 0. With this choice, we are also guaranteed that there exists a zero eigenvalue. 19 Spectral analysis There is no guarantee, in the most general case, that there exists a complete set of eigenfunctions. However, the chance that such a complete set does not exist for a Markov generator specified non- parametrically is zero, so we can safely assume that this is the case. In the unlikely case that this assumption is not correct, the numerical linear algebra routines needed to solve our lattice model will identify the problem and a small perturbation of the given operator will suffice to rectify the situation. Assuming completeness, the diagonalization problem can be rewritten in the following matrix form: r −1 LΩα = UΛU (3) where U is the matrix having as columns the right eigenvectors and Λ is the diagonal matrix having the eigenvalues λi as elements.
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