Interest rate derivative pricing with stochastic volatility

One purpose of exotic derivative pricing models is to enable financial institutions to quantify and manage their financial risk, arising from large books of portfolios. These portfolios consist of many non-standard exotic financial products. Risk is managed by means of the evaluation of sensitivity parameters, i.e. the so-called Greeks, the deltas, vegas, gammas and also volgas, vannas, and others. In practice, practitioners do not expect an exotic derivative pricing model to be a high precision predictive model. What is important is a high precision replication of the hedging instruments, as well as efficient computation with the model. Plain vanilla interest rate options like swaptions and caps are liquidly traded instruments, serving as fundamental building blocks of hedging portfolios for exotic products. In the early twenty-first century, the so-called implied volatility skew and smile in the market became pronounced in the interest rate plain vanilla market. The stochastic alpha beta rho (SABR) model then became widely accepted as the market standard to model this implied volatility skew/smile. The model's popularity is due to the existence of an accurate analytic approximation for the implied volatilities, presented by Hagan et al.. This approximation formula is often used by practitioners to inter- and extrapolate the implied volatility surface. The application of the SABR model is so prevalent that one can even observe SABR-type implied volatility curves in the market nowadays (which means that the SABR model can perfectly resemble one set of market implied volatilities with different strike prices). This PhD thesis considers the SABR model as its basis for further extension, and focuses on the various problems arising from the application of the SABR model in both plain vanilla and exotic option pricing, from a modelling as well as numerical point of view. In Chapter 2, we present an analytic approximation to the convexity correction of Constant Maturity Swap (CMS) products under a two-factor SABR model by means of small time asymptotic expansion technique. In Chapter 3, we apply the small time asymptotic expansion differently, to a problem of approximating the first and second moments of the integrated variance of the log-normal volatility process in the context of defining a low-bias discretization scheme for the SABR model. With the approximated moment information, we can approximate the density of the integrated variance by means of a log-normal distribution with the first two moments matched to that information. The conditional SABR process turns out to be a squared Bessel process, given the terminal volatility level and the integrated variance. Based on the idea of mixing conditional distributions and a direct inversion of the noncentral chi-square distributions, we propose the low-bias SABR Monte Carlo scheme. The low-bias scheme can handle the asset price process in the vicinity of the zero boundary well. The scheme is stable and exhibits a superior convergence behaviour compared to the truncated Euler scheme. In Chapter 4, we extend the discretization scheme proposed in Chapter 3 towards a SABR model with stochastic interest rate in the form of a Hull-White short rate model, the SABR-HW model. The hybrid model is meant for pricing long-dated equity-interest-rate linked exotic options with exposure to both the interest rate and the equity price risk. To facilitate the calibration of the SABRHW model, we propose a projection formula, mapping the SABR-HW model parameters onto the parameters of the nearest SABR model. The numerical inversion of the projection formula can be used to calibrate the model. In Chapter 5, we focus on a version of the stochastic volatility LIBOR Market Model with time-dependent skew and volatility parameters. As a result of choosing time-dependent parameters, the model has the flexibility to match to the market quotes of an entire swaption cube (in terms of various combinations of expiry, tenor and strike), as observed in the current interest rate market. Thus, this model is in principle well-suited for managing the risk of a complete exotic option trading book in a financial institution, consisting of both exotic options and its plain vanilla hedge instruments. The calibration of the model to the swaption quotes relies on a model mapping procedure, which relates the model parameters (most often time-dependent) in a high-dimensional LMM model to swaption prices. The model mapping procedure maps the high-dimensional swap rate dynamics implied by the model onto a one-dimensional displaced diffusion process with time-dependent coefficient. Those time-dependent parameters are subsequently averaged to obtain the effective constant parameters of the projected model. Two known projection methods that are available in the literature, the freezing projection and the more involved Markov projection, have been compared within the calibration process. The basic freezing projection achieves a good accuracy at significantly less computational cost in our tests, and it is thus applied within the calibration purpose. A second advantage of the freezing projection formula is that it enables us to formulate the time-dependent skew calibration problem as a convex optimization problem. Our contribution in this chapter is the convex optimization formulation of the skew calibration problem. Based on the convex formulation, we are able to translate the calibration of a large number of free variables into a well-known quadratic programming problem formulation, for which efficient algorithms are available. The convexity of the formulated optimization problem guarantees the obtained solution to be a global optimum. The stability of the procedure can be beneficial for application in the day-to-day derivative trading practice, i.e. the daily re-calibration and hedging.

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