Pricing Barrier Options with One-Factor Interest Rate Models

Abstract

A numerical integration method may be used to price barrier options using one-factor interest rate models when the transition distribution function of the underlying rate is known but explicit pricing formulas are not available.For the Hull and White model, barriers on bonds are transformed to smooth time-dependent barriers on the short rate. For the swap market model, time-dependent barriers are imposed on forward swap rates. The first passage time densities of the underlying interest rate are solved for numerically from integral equations.Comparison to barrier options on equity where there are analytical formulas confirms that the method is faster and more accurate than a Monte Carlo method.Unlike lattice methods, the numerical integration method does not require finding optimal positioning of lattice nodes. The method has advantages over Monte Carlo and lattice methods when the barriers are time-dependent and the underlying process is a more general diffusion process.T he pricing of barrier options is often discussed. Single-barrier and doublebarrier options are the most commonly investigated. When the underlying asset is assumed to be lognormally distributed as in the Black and Scholes [1973] model, pricing formulas are available for both single-and double-barrier options. The pricing of options with time-dependent barriers in the interest rate market, where the underlying state variables are usually assumed to follow more general diffusion processes, is less well established.We focus on the problem of pricing barrier options on zero-coupon bonds and barrier swaptions in one-factor interest rate models. Computational examples are given for the Hull and White model [1990] and the swap market model (see Jamshidian [1997]). Both models take initial term structures as inputs. By including the current forward rate curve or bond price curve in the models, other interest rate options can be priced consistently with initial term structures. This is an essential practical requirement in the interest rate markets to avoid arbitrage and to permit consistent hedging.In the Hull and White model, the short rate is the only factor, and there are explicit expressions for zero-coupon bonds. These features allow us to transform barriers on the underlying zero-coupon bond or forward swap rate to time-dependent barriers on the short rate.The lognormal swap market model is widely used because it allows easy calibration to swaption prices through Black's formula. As we are interested in cases for which there are no analytical pricing formulas, we will price swaptions with time-dependent barriers in the swap market model.In general, the first passage time densities of interest rates to time-dependent barriers can be solved only numerically. We use the algorithm suggested by Park and Schuurmann [1976, 1980] to solve for first passage time densities. Given these densities, we are able to value single

Analysis of quadrature methods for pricing discrete barrier options

“…A mathematically oriented discussion of the barrier option pricing problem is contained in Rich (1994). In a nutshell, there are several approaches to barrier option pricing: (a) the probabilistic method, see Kunitomo and Ikeda (1992); (b) the Laplace Transform technique, see Jamshidian (1997), Geman and Yor (1996), Sbuelz (1999), Pelsser (2000), Fusai (2001); (c) the Black-Scholes PDE, which can be solved using separation of variables, see Hui (1996) and Hui et al (2000) or finite difference schemes, see Boyle and Tian (1998) or Zvan et al (2000); (d) binomial and trinomial trees see Boyle and Lau (1994), Ritchken (1995), Heynen and Kat (1997), Tian (1999); (e) Monte Carlo simulations with various enhancements, see Andersen and Brotherton-Ratcliffe (1996), Baldi et al (1998), Beaglehole et al (1997), Kuan and Webber (2003).…”

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Pricing American Options under the Constant Elasticity of Variance Model and Subject to Bankruptcy

“…Following Kuan and Webber (2003), the next proposition shows that such first passage time density can be efficiently computed, for any exercise boundary specification, through the standard partition method proposed by Park and Schuurmann (1976).…”