Pricing and Hedging Derivatives with Sharp Profiles Using Tuned High Resolution Finite Difference Schemes

We price and hedge different financial derivatives with sharp profiles by solving the corresponding advection-diffusion-reaction partial differential equation using new high resolution finite difference schemes, which show superior numerical advantages over standard finite difference methods. High order finite difference methods, which are commonly used techniques in the computational finance literature, fail to handle the discontinuities in the payoff functions of derivatives with discontinuous payoff functions, like digital options. Their numerical solutions produce spurious oscillations in the neighborhood of the discontinuities, which make the numerical derivatives prices and hedges impractical. Hence, we extend the linear finite difference methods to overcome these difficulties by developing high resolution non-linear schemes that resolve these discontinuities and facilitate pricing and hedging these options with higher accuracy. These approximations detect the discontinuous profiles automatically using non-linear functions, called limiters, and smooth discontinuities minimally and locally to produce non-oscillatory prices and Greeks with high resolution. These limiters are modified and more relaxed versions of standard limiting functions in fluid dynamics area to accommodate for the extra physical diffusion (volatility) in financial problems. We prove that this family of new schemes is total variation diminishing (TVD), which guarantees the non oscillatory solutions. Also, we deduce and illustrate the limiting functions ranges and characteristics that allow the TVD condition to hold. We test these methods to price and hedge financial derivatives with digital-like profiles under Black-Scholes-Merton (BSM), constant elasticity of variance (CEV) and Heath-Jarrow-Morton (HJM) models. More specifically, we price and hedge digital options under BSM and CEV models, and we price bonds under HJM model. Finally, we price supershare and gap options under the BSM model. Using the new limiters we developed show higher accuracy profiles (solutions) for the option prices and hedges than standard finite difference schemes or standard limiters, and guaranteed non-oscillatory solutions.