A Simple Generalisation of Kirk’s Approximation for Multi-Asset Spread Options by the Lie-Trotter Operator Splitting Method

Affiliation(s)

Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Hong Kong, China.

Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Hong Kong, China.

ABSTRACT

In this paper, by means of the Lie-Trotter operator splitting method, we have presented a new unified approach not only to rigorously derive Kirk’s approximation but also to obtain a generalisation for multi-asset spread options in a straightforward manner. The derived price formula for the multi-asset spread option bears a great resemblance to Kirk’s approximation in the two-asset case. More importantly, our approach is able to provide a new perspective on Kirk’s approximation and the generalization; that is, they are simply equivalent to the Lie-Trotter operator splitting approximation to the Black-Scholes equation.

KEYWORDS

Lognormal Random Variables, Black-Scholes Equation, Spread Options, Kirk’s Approximation, Lie-Trotter Operator Splitting Method

Lognormal Random Variables, Black-Scholes Equation, Spread Options, Kirk’s Approximation, Lie-Trotter Operator Splitting Method

Cite this paper

Lo, C. (2014) A Simple Generalisation of Kirk’s Approximation for Multi-Asset Spread Options by the Lie-Trotter Operator Splitting Method.*Journal of Mathematical Finance*, **4**, 178-187. doi: 10.4236/jmf.2014.43016.

Lo, C. (2014) A Simple Generalisation of Kirk’s Approximation for Multi-Asset Spread Options by the Lie-Trotter Operator Splitting Method.

References

[1] Carmona, R. and Durrleman, V. (2003) Pricing and Hedging Spread Options. SIAM Review, 45, 627-685.

http://dx.doi.org/10.1137/S0036144503424798

[2] Deng, S.J., Li, M. and Zhou, J. (2008) Closed-Form Approximation for Spread Option Prices and Greeks. Journal of Derivatives, 15, 58-80.

http://dx.doi.org/10.3905/jod.2008.702506

[3] Bjerksund, P. and Stensland, G. (2011) Closed Form Spread Option Valuation. Quantitative Finance, iFirst, 1-10.

[4] Venkatramana, A. and Alexander, C. (2011) Closed form Approximation for Spread Options. Applied Mathematical Finance, 18, 447-472.

http://dx.doi.org/10.1080/1350486X.2011.567120

[5] Kirk, E. (1995) Correlation in the Energy Markets. Managing Energy Price Risk. Risk Publications and Enron, London, 71-78.

[6] Margrabe, W. (1978) The Value of an Option to Exchange One Asset for Another. Journal of Finance, 33, 177-186.

http://dx.doi.org/10.1111/j.1540-6261.1978.tb03397.x

[7] Lo, C.F. (2013) A Simple Derivation of Kirk’s Approximation for Spread Options. Applied Mathematics Letters, 26, 904-907.

http://dx.doi.org/10.1016/j.aml.2013.04.004

[8] Trotter, H.F. (1958) Approximation of Semi-Groups of Operators. Pacific Journal of Mathematics, 8, 887-919.

http://dx.doi.org/10.2140/pjm.1958.8.887

[9] Li, M., Zhou, J. and Deng, S.J. (2010) Multi-Asset Spread Option Pricing and Hedging. Quantitative Finance, 10, 305-324.

http://dx.doi.org/10.1080/14697680802626323

[10] Trotter, H.F. (1959) On the Product of Semi-Groups of Operators. Proceedings of the American Mathematical Society, 10, 545-551.

http://dx.doi.org/10.1090/S0002-9939-1959-0108732-6

[11] Suzuki, M. (1985) Decomposition Formulas of Exponential Operators and Lie Exponentials with Some Applications to Quantum Mechanics and Statistical Physics. Journal of Mathematical Physics, 26, 601-612.

http://dx.doi.org/10.1063/1.526596

[12] Drozdov, A.N. and Brey, J.J. (1998) Operator Expansions in Stochastic Dynamics. Physical Review E, 57, 1284-1289.

http://dx.doi.org/10.1103/PhysRevE.57.1284

[13] Hatano, N. and Suzuki, M. (2005) Finding Exponential Product Formulas of Higher Orders. Lecture Notes in Physics, 679, 37-68.

[14] Blanes, S., Casas, F., Chartier, P. and Murua, A. (2013) Optimized Higher-Order Splitting Methods for Some Classes of Parabolic Equations. Mathematics of Computation, 82, 1559-1576.

http://dx.doi.org/10.1090/S0025-5718-2012-02657-3

[1] Carmona, R. and Durrleman, V. (2003) Pricing and Hedging Spread Options. SIAM Review, 45, 627-685.

http://dx.doi.org/10.1137/S0036144503424798

[2] Deng, S.J., Li, M. and Zhou, J. (2008) Closed-Form Approximation for Spread Option Prices and Greeks. Journal of Derivatives, 15, 58-80.

http://dx.doi.org/10.3905/jod.2008.702506

[3] Bjerksund, P. and Stensland, G. (2011) Closed Form Spread Option Valuation. Quantitative Finance, iFirst, 1-10.

[4] Venkatramana, A. and Alexander, C. (2011) Closed form Approximation for Spread Options. Applied Mathematical Finance, 18, 447-472.

http://dx.doi.org/10.1080/1350486X.2011.567120

[5] Kirk, E. (1995) Correlation in the Energy Markets. Managing Energy Price Risk. Risk Publications and Enron, London, 71-78.

[6] Margrabe, W. (1978) The Value of an Option to Exchange One Asset for Another. Journal of Finance, 33, 177-186.

http://dx.doi.org/10.1111/j.1540-6261.1978.tb03397.x

[7] Lo, C.F. (2013) A Simple Derivation of Kirk’s Approximation for Spread Options. Applied Mathematics Letters, 26, 904-907.

http://dx.doi.org/10.1016/j.aml.2013.04.004

[8] Trotter, H.F. (1958) Approximation of Semi-Groups of Operators. Pacific Journal of Mathematics, 8, 887-919.

http://dx.doi.org/10.2140/pjm.1958.8.887

[9] Li, M., Zhou, J. and Deng, S.J. (2010) Multi-Asset Spread Option Pricing and Hedging. Quantitative Finance, 10, 305-324.

http://dx.doi.org/10.1080/14697680802626323

[10] Trotter, H.F. (1959) On the Product of Semi-Groups of Operators. Proceedings of the American Mathematical Society, 10, 545-551.

http://dx.doi.org/10.1090/S0002-9939-1959-0108732-6

[11] Suzuki, M. (1985) Decomposition Formulas of Exponential Operators and Lie Exponentials with Some Applications to Quantum Mechanics and Statistical Physics. Journal of Mathematical Physics, 26, 601-612.

http://dx.doi.org/10.1063/1.526596

[12] Drozdov, A.N. and Brey, J.J. (1998) Operator Expansions in Stochastic Dynamics. Physical Review E, 57, 1284-1289.

http://dx.doi.org/10.1103/PhysRevE.57.1284

[13] Hatano, N. and Suzuki, M. (2005) Finding Exponential Product Formulas of Higher Orders. Lecture Notes in Physics, 679, 37-68.

[14] Blanes, S., Casas, F., Chartier, P. and Murua, A. (2013) Optimized Higher-Order Splitting Methods for Some Classes of Parabolic Equations. Mathematics of Computation, 82, 1559-1576.

http://dx.doi.org/10.1090/S0025-5718-2012-02657-3