Semi-Closed Form Prices of Barrier Options in the Time-Dependent CEV and CIR Models

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

“…This is similar to the idea of the generalized integral transform (GIT) method that we use in this paper, while our setting is more general. Indeed, we allow any diffusion model with timedependent coefficients and time-dependent barriers and rebates at hit subject to the condition that the pricing partial differential equation (PDE) can be reduced to the heat equation (or, as shown in [5] to the Bessel equation). It can also be checked that the pricing PDE in [30] by a simple change of the spatial variable can be transformed to the heat equation.…”

“…Our approach advocated in this paper further extends the technique we elaborated in a series of papers which dealt with a similar problem for single barrier options. In [4] we developed semi-analytic solutions for the barrier (perhaps, timedependent) and American options where the underlying stock is driven by a timedependent OU process with a lognormal drift. This model is equivalent to the familiar Hull-White model in Fixed Income that was separately considered in [19].…”

“…The GIT method is also known in physics, see [20,21] where some problems related to so-called Stefan problem were solved. But it was unknown in finance until the first use in [4]. It is important, that the solutions were constructed only for problems where the underlying is defined at the domain S ∈ [0, y(t)] with S being the stock price, and y(t) being the time-dependent barrier, however, for other domains the solution was yet unknown, even in physics.…”

“…Latter this technique was elaborated also for the CIR and CEV models, [5], and the Black-Karasinski model, [16]. In particular, in [5] the HP method was further generalized to be capable to solving not just the heat but also the Bessel equations, and was called the Bessel potential (BP) method. In [16] the PDE is also of a special kind.…”

“…Therefore, we think that "semi-analytic" is an appropriate term. Also, in some situations Ψ(t) can be found analytically, see e.g., [4,19].…”