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.
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
, 178-187. doi: 10.4236/jmf.2014.43016
 Carmona, R. and Durrleman, V. (2003) Pricing and Hedging Spread Options. SIAM Review, 45, 627-685.http://dx.doi.org/10.1137/S0036144503424798
 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
 Bjerksund, P. and Stensland, G. (2011) Closed Form Spread Option Valuation. Quantitative Finance, iFirst, 1-10.
 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
 Kirk, E. (1995) Correlation in the Energy Markets. Managing Energy Price Risk. Risk Publications and Enron, London, 71-78.
 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
 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
 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
 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
 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
 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
 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
 Hatano, N. and Suzuki, M. (2005) Finding Exponential Product Formulas of Higher Orders. Lecture Notes in Physics, 679, 37-68.
 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