A barrier option is an option whose payoff depends on whether the price path of the underlying asset ever reaches certain predetermined price levels called the barriers. A single(double-) barrier option is a barrier option with one (two, respectively) barrier(s). No simple and exact closed-form pricing formula for double-barrier options has been reported in the literature. This paper proposes a...

We study pricing under the local volatility. Our research is mainly intended for pedagogical purposes. In the first part of our work we study the local volatility modeling. We derive the local volatility formula in terms of the European call prices and in terms of the market implied volatilities. We propose and calibrate to the DAX option data a functional form for the implied volatility which ...

Svetlozar T. Rachev Chair-Professor, Chair of Statistics, Econometrics and Mathematical Finance, School of Economics and Business Engineering, University of Karlsruhe and KIT, Kollegium am Schloss, Bau II, 20.12, R210, Postfach 6980, D-76128, Karlsruhe, Germany and Department of Statistics and Applied Probability, University of California, Santa Barbara, and Chief-Scientist, FinAnalytica Inc. E...

This paper develops a rigorous asymptotic expansion method with its numerical scheme for the Cauchy-Dirichlet problem in second order parabolic partial differential equations (PDEs). As an application, we propose a new approximation formula for pricing a barrier option under a certain type of stochastic volatility model including the log-normal SABR model.

Abstract. External barrier options are two-asset options with stochastic variables where the payoff depends on one underlying asset and the barrier depends on another state variable. The barrier state variable determines whether the option is knocked in or out when the value of the variable is above or below some prescribed barrier level. This paper derives the explicit analytic solution of the...

This paper considers the pricing of long-term options on assets such as housing, where either government intervention or economic nature asset is assumed to limit large falls in prices. The observed price modelled by a geometric Brownian motion (the 'notional price') reflected at lower barrier. resulting has standard dynamics but with localised barrier, which allows arbitrage interim losses; th...