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Published in:
2011 | OriginalPaper | Chapter
A General Efficient Framework for Pricing Options Using Exponential Time Integration Schemes
Authors : Yannick Desire Tangman, Ravindra Boojhawon, Ashvin Gopaul, Muddun Bhuruth
Published in: Financial Econometrics Modeling: Derivatives Pricing, Hedge Funds and Term Structure Models
Publisher: Palgrave Macmillan UK
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In numerical option pricing, spatial discretization of the pricing equation leads to semi-discrete systems of the form (4.1)<math display='block'> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mrow><mo>(</mo> <mi>τ</mi> <mo>)</mo></mrow><mo>=</mo><mi>A</mi><mi>V</mi><mrow><mo>(</mo> <mi>τ</mi> <mo>)</mo></mrow><mo>+</mo><mi>b</mi><mrow><mo>(</mo> <mi>τ</mi> <mo>)</mo></mrow><mo>,</mo> </mrow> </math>$${V}^{\prime}\left( \tau \right)=AV\left( \tau \right)+b\left( \tau \right),$$ where A ∊ ℜm×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 <math display='block'> <mrow> <mi>V</mi><mrow><mo>(</mo> <mrow> <msub> <mi>τ</mi> <mrow> <mi>j</mi><mo>+</mo><mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo></mrow><mo>=</mo><msup> <mi>e</mi> <mrow> <mi>A</mi><mi>Δ</mi><mi>τ</mi> </mrow> </msup> <mi>V</mi><mrow><mo>(</mo> <mrow> <msub> <mi>τ</mi> <mi>j</mi> </msub> </mrow> <mo>)</mo></mrow><mo>+</mo><msup> <mi>e</mi> <mrow> <mi>A</mi><msub> <mi>τ</mi> <mrow> <mi>j</mi><mo>+</mo><mn>1</mn> </mrow> </msub> </mrow> </msup> <mstyle displaystyle='true'> <mrow> <msubsup> <mo>∫</mo> <mrow> <msub> <mi>τ</mi> <mi>j</mi> </msub> </mrow> <mrow> <msub> <mi>τ</mi> <mrow> <mi>j</mi><mo>+</mo><mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo><mi>A</mi><mi>t</mi> </mrow> </msup> <mi>b</mi><mrow><mo>(</mo> <mi>t</mi> <mo>)</mo></mrow><mi>d</mi><mi>t</mi> </mrow> </mrow> </mstyle><mo>,</mo> </mrow> </math>$$V\left( {{{\tau }_{{j+1}}}} \right)={{e}^{{A\Delta \tau }}}V\left( {{{\tau }_{j}}} \right)+{{e}^{{A{{\tau }_{{j+1}}}}}}\int\nolimits_{{{{\tau }_{j}}}}^{{{{\tau }_{{j+1}}}}} {{{e}^{{-At}}}b\left( t \right)dt} ,$$ and approximation of the above equation by the exponential forward Euler method leads to the scheme (4.2)<math display='block'> <mrow> <msup> <mi>V</mi> <mrow> <mi>j</mi><mo>+</mo><mn>1</mn> </mrow> </msup> <mo>=</mo><msub> <mi>φ</mi> <mn>0</mn> </msub> <mrow><mo>(</mo> <mrow> <mi>A</mi><mi>Δ</mi><mi>τ</mi> </mrow> <mo>)</mo></mrow><msup> <mi>V</mi> <mi>j</mi> </msup> <mo>+</mo><mi>Δ</mi><mi>τ</mi><msub> <mi>φ</mi> <mn>1</mn> </msub> <mrow><mo>(</mo> <mrow> <mi>A</mi><mi>Δ</mi><mi>τ</mi> </mrow> <mo>)</mo></mrow><mi>b</mi><mrow><mo>(</mo> <mrow> <msub> <mi>τ</mi> <mi>j</mi> </msub> </mrow> <mo>)</mo></mrow><mo>,</mo> </mrow> </math>$${{V}^{{j+1}}}={{\varphi }_{0}}\left( {A\Delta \tau } \right){{V}^{j}}+\Delta \tau {{\varphi }_{1}}\left( {A\Delta \tau } \right)b\left( {{{\tau }_{j}}} \right),$$ where ρ0(z) = ez and ρ1(z)=(ez-1)/z.