Computational Issues of Stochastic-Alpha-Beta-Rho (SABR) Model

The Stochastic-Alpha-Beta-Rho (SABR) stochastic volatility model, introduced by Hagan et al. (2002), is widely used in the foreign exchange and interest rate markets. It can yield a good fitting to implied volatility smile curves and capture the correct co-movements between the smile curve and the underlying level. It has better hedging performance than other models. This dissertation mainly studies the computational issues of the SABR model as well as probabilistic characterization.;In the first part, an innovative probabilistic approach is used to analyze the model and obtain the asymptotic joint transition density. The existing results do not assume the origin is the absorbing boundary, thus the SABR model could have arbitrage opportunity. We have considered the absorbing boundary to exclude the arbitrage opportunity through the principle of not feeling the boundary. This principle is essentially the estimation of the hitting probability around the origin, which claims that the hitting probability is exponentially negligible when initial volatility and volatility of volatility are small.;Applying this principle, we can approximate the SABR model with a time-changed Bessel process. Then, the asymptotics of the joint density is achieved by using a stochastic Taylor expansion, which inherits the spirit of small total volatility-of-volatility expansion. The numerical experiments show that our joint density of the SABR model outperforms the others such as Hagan, Lesniewski, and Woodward (2005) and Wu (2012).;In the second part, we work out the asymptotic formulae for the first passage time, the marginal and joint survival probability densities of the SABR model. To the best of the author's knowledge, this is the first time to obtain asymptotic solutions of problems with a lower absorbing boundary. These formulae have a wide range of applications such as the survival probability, pricing down-and-out barrier option under SABR model etc. Compared with Monte Carlo and finite difference methods, the survival densities not only provide fast pricing tools, but also can be used to compute the Greeks fast and accurately. Therefore, these asymptotic formulae play an important role in hedging the risks associated with options.;These problems are formulated in terms of solutions of backward Kolmogorov equations with terminal and boundary conditions. After the scaling procedure, combining the crucial coordinate transform we have performed, the first passage time and survival densities problems turn out to be partial differential equations (PDEs) with the total volatility-of-volatility as parameter, and the leading order operators of the PDEs are the infinitesimal generators of one or two dimensional Brownian motions. For the joint survival density, an additional de-correlation procedure is needed. With the help of the symmetric property of Brownian motions, the analytical solutions can be obtained after expanding these problems with respect to the total volatility-of-volatility. The numerical results show our analytic formulae are accurate and efficient.;Key words: Stochastic-Alpha-Beta-Rho (SABR) Model; Not Feeling the Boundary; Joint Transition Density; Time-Changed Bessel Process; First Passage Times; Survival Density; Barrier Options.