A General Efficient Framework for Pricing Options Using Exponential Time Integration Schemes

In numerical option pricing, spatial discretization of the pricing equation leads to semi-discrete systems of the form (4.1) V ′ ( τ ) = A V ( τ ) + b ( τ ) , $${V}^{\prime}\left( \tau \right)=AV\left( \tau \right)+b\left( \tau \right),$$ where A ∊ ℜ m×m is in general a negative semi-definite matrix and b(τ) generally represents boundary condition implementations, a penalty term for American option or approximation of integral terms on an unbounded domain in models with jumps. With advances in the efficient computation of the matrix exponential (Schmelzer and Trefethen 2007), exponential time integration (Cox and Matthews 2002) is likely to be a method of choice for the solution of ODE systems of the form (4.1). Duhamel’s principle states that the exact integration of (4.1) over one time step gives V ( τ j + 1 ) = e A Δ τ V ( τ j ) + e A τ j + 1 ∫ τ j τ j + 1 e − A t b ( t ) d t , $$V\left( {{{\tau }_{{j+1}}}} \right)={{e}^{{A\Delta \tau }}}V\left( {{{\tau }_{j}}} \right)+{{e}^{{A{{\tau }_{{j+1}}}}}}\int\nolimits_{{{{\tau }_{j}}}}^{{{{\tau }_{{j+1}}}}} {{{e}^{{-At}}}b\left( t \right)dt} ,$$ and approximation of the above equation by the exponential forward Euler method leads to the scheme (4.2) V j + 1 = φ 0 ( A Δ τ ) V j + Δ τ φ 1 ( A Δ τ ) b ( τ j ) , $${{V}^{{j+1}}}={{\varphi }_{0}}\left( {A\Delta \tau } \right){{V}^{j}}+\Delta \tau {{\varphi }_{1}}\left( {A\Delta \tau } \right)b\left( {{{\tau }_{j}}} \right),$$ where ρ0(z) = e z and ρ1(z)=(e z -1)/z.