_{1}

^{*}

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.

Spread options, whose payoff is contingent upon the price difference of two underlying assets form the simplest type of multi-asset options, are very popular in markets as diverse as interest rate markets, currency and foreign exchange markets, commodity markets, and energy markets nowadays [

lognormal random variables. The simplest approach is to evaluate the expectation of the final payoff over the joint probability distribution of the two correlated lognormal underlyings by means of numerical integration. However, practitioners often prefer to use analytical approximations rather than numerical methods because of their computational ease. Among various analytical approximations, e.g. Carmona and Durrleman [

Accordingly, it is the aim of this paper to present a simple unified approach, namely the Lie-Trotter operator splitting method [

The price of a European call spread option obeys the two-dimensional Black-Scholes equation

with the final payoff condition

Here

Proposition 1:

The price of the two-asset spread option can be approximated by

where

Proof:

In terms of the two new variables

Equation (1) can be rewritten as follows:

where

The final payoff condition now becomes

Accordingly, the formal solution of Equation (9) is given by

Since the exponential operator

and obtain an approximation to the formal solution

for

It is not difficult to recognise that

with the initial condition:

where

As a result, we obtain

which is exactly the approximate price formula given in Equation (3). (Q.E.D.)

In terms of the spot asset prices, namely

where

It should be noted that for the Lie-Trotter operator splitting approximation to be valid,

To price a European

subject to the final payoff condition

where

Proposition 2:

The price of the

where

Proof:

Introducing the new variables:

where

The formal solution of Equation (36) is given by

Then we apply the Lie-Trotter operator splitting method [

where the relation

is utilized.

Next, in terms of the two new variables

we rewrite

where

It is clear that the exponential operator

Lie-Trotter operator splitting method [

result, we obtain

for

Since

where

It is obvious that this approximate solution is identical to the approximate price formula given in Equation (29). (Q.E.D.)

In terms of the spot asset prices, namely

where

Obviously, this approximate price formula resembles the price formula of Kirk’s approximation in the two-asset case very closely. In fact, by setting

In this section, illustrative numerical examples are presented to demonstrate the accuracy of the extended Kirk approximation for the three-asset spread options. Although most spread options involve two assets only, yet there is a growing demand for three-asset spread options which can be found in the models for power plants or their financial equivalents—tolling contracts. We examine a simple three-asset spread option with the final payoff

In this paper, we have proved that Kirk’s approximation for two-asset spread options can be rigorously derived

. Prices of a European three-asset call spread option. Other input parameters are: r = 0.05, s_{1} = s_{2} = s_{3} = 0.3, r_{12} = 0.4, r_{23} = 0.2, r_{13} = 0.8, S_{1} = 50, S_{2} = 60 and S_{3} = 150. Here “EK” refers to the extended Kirk approximation while “MC” denotes the Monte Carlo estimates with 900,000,000 replications. The relative errors of the “EK” option prices with respect to the “MC” estimates are also presented

K\T | 0.25 | 0.5 | 1 | 2 | |
---|---|---|---|---|---|

30 | 13.5410 | 16.4210 | 20.7761 | 27.1974 | EK |

-0.3% | -0.3% | -0.3% | -0.3% | error | |

13.5763 ± 0.0089 | 16.4735 ± 0.0142 | 20.8471 ± 0.0185 | 27.2841 ± 0.0264 | MC | |

35 | 10.3383 | 13.5024 | 18.1191 | 24.8196 | EK |

-0.2% | -0.2% | -0.2% | -0.2% | error | |

10.3576 ± 0.0086 | 13.5286 ± 0.0124 | 18.1541 ± 0.0176 | 24.8573 ± 0.0276 | MC | |

40 | 7.6613 | 10.9586 | 15.7231 | 22.6176 | EK |

0.0% | 0.0% | 0.0% | 0.1% | error | |

7.6608 ± 0.0068 | 10.9572 ± 0.0109 | 15.7197 ± 0.0161 | 22.6072 ± 0.0244 | MC | |

45 | 5.5097 | 8.7824 | 13.5805 | 20.5856 | EK |

0.4% | 0.3% | 0.3% | 0.3% | error | |

5.4903 ± 0.0051 | 8.7565 ± 0.0106 | 13.5421 ± 0.0141 | 20.5302 ± 0.0260 | MC | |

50 | 3.8470 | 6.9540 | 11.6795 | 18.7162 | EK |

0.9% | 0.7% | 0.6% | 0.5% | error | |

3.8140 ± 0.0040 | 6.9058 ± 0.0086 | 11.6109 ± 0.0136 | 18.6195 ± 0.0245 | MC |

. Prices of a European three-asset call spread option. Other input parameters are: r = 0.05, s_{1} = s_{2} = s_{3} = 0.6, r_{12} = 0.4, r_{23} = 0.2, r_{13} = 0.8, S_{1} = 50, S_{2} = 60 and S_{3} = 150. Here “EK” refers to the extended Kirk approximation while “MC” denotes the Monte Carlo estimates with 900,000,000 replications. The relative errors of the “EK” option prices with respect to the “MC” estimates are also presented

K\T | 0.25 | 0.5 | 1 | 2 | |
---|---|---|---|---|---|

30 | 20.1436 | 26.0640 | 34.5186 | 46.3820 | EK |

-0.3% | -0.3% | -0.1% | 0.3% | error | |

20.2066 ± 0.0168 | 26.1269 ± 0.0278 | 34.5402 ± 0.0425 | 46.2242 ± 0.0716 | MC | |

35 | 17.4529 | 23.5976 | 32.2944 | 44.4495 | EK |

-0.1% | 0.0% | 0.1% | 0.5% | error | |

17.4778 ± 0.0172 | 23.6076 ± 0.0268 | 32.2508 ± 0.0390 | 44.2275 ± 0.0787 | MC | |

40 | 15.0417 | 21.3320 | 30.2150 | 42.6221 | EK |

0.1% | 0.2% | 0.4% | 0.7% | error | |

15.0290 ± 0.0171 | 21.2938 ± 0.0266 | 30.1079 ± 0.0362 | 42.3425 ± 0.0656 | MC | |

45 | 12.9002 | 19.2587 | 28.2733 | 40.8938 | EK |

0.4% | 0.4% | 0.6% | 0.9% | error | |

12.8527 ± 0.0182 | 19.1753 ± 0.0234 | 28.1113 ± 0.0381 | 40.5466 ± 0.0628 | MC | |

50 | 11.0139 | 17.3676 | 26.4620 | 39.2590 | EK |

0.7% | 0.7% | 0.8% | 1.0% | error | |

10.9347 ± 0.0151 | 17.2410 ± 0.0236 | 26.2513 ± 0.0345 | 38.8658 ±0.0637 | MC |

. Prices of a European three-asset call spread option. Other input parameters are: r = 0.05, s_{1} = 0.5, s_{2} = 0.4, s_{3} = 0.3, r_{12} = 0.4, r_{23} = 0.2, r_{13} = 0.8, S_{1} = 50, S_{2} = 60 and S_{3} = 150. Here “EK” refers to the extended Kirk approximation while “MC” denotes the Monte Carlo estimates with 900,000,000 replications. The relative errors of the “EK” option prices with respect to the “MC” estimates are also presented

K\T | 0.25 | 0.5 | 1 | 2 | |
---|---|---|---|---|---|

30 | 13.8987 | 16.9731 | 21.6091 | 28.4620 | EK |

-0.5% | -0.6% | -0.8% | -1.2% | error | |

13.9635 ± 0.0092 | 17.0801 ± 0.0119 | 21.7922 ± 0.0191 | 28.8014 ± 0.0242 | MC | |

35 | 10.6503 | 13.9613 | 18.8011 | 25.8630 | EK |

-0.4% | -0.5% | -0.7% | -1.1% | error | |

10.6905 ± 0.0078 | 14.0301 ± 0.0118 | 18.9316 ± 0.0163 | 26.1380 ± 0.0238 | MC | |

40 | 7.9011 | 11.3085 | 16.2480 | 23.4424 | EK |

-0.1% | -0.2% | -0.4% | -0.9% | error | |

7.9101 ± 0.0074 | 11.3342 ± 0.0104 | 16.3206 ± 0.0172 | 23.6452 ± 0.0223 | MC | |

45 | 5.6664 | 9.0191 | 13.9501 | 21.1989 | EK |

0.4% | 0.2% | -0.1% | -0.6% | error | |

5.6426 ± 0.0064 | 9.0004 ± 0.0092 | 13.9627 ± 0.0145 | 21.3286 ± 0.0210 | MC | |

50 | 3.9253 | 7.0838 | 11.9023 | 19.1291 | EK |

1.3% | 0.8% | 0.4% | -0.3% | error | |

3.8758 ± 0.0058 | 7.0242 ± 0.0081 | 11.8587 ± 0.0147 | 19.1882 ± 0.0216 | MC |

by applying the Lie-Trotter operator splitting approximation to the Black-Scholes equation, and that for cases with vanishing strike prices, the Lie-Trotter splitting approximation becomes exact and Margrabe’s formula is recovered. Our derivation also shows that Kirk’s approximation is more favourable for those cases with two positively correlated assets. Moreover, we apply the Lie-Trotter operator splitting method to obtain a genera- lisation of Kirk’s approximation for multi-asset spread options in a straightforward manner. The derived price formula for the multi-asset spread option closely resembles Kirk’s approximation in the two-asset case. By setting the total number of assets to be two, we can recover Kirk’s approximation readily. Thus, the ge- neralisation possesses the same nice features as Kirk’s approximation. For instance, as shown by the illustrative examples, the generalization is found to be very accurate and efficient in pricing the multi-asset spread options. All in all, our approach is able to provide a new perspective on Kirk’s approximation and the generalisation; that is, they are simply equivalent to the Lie-Trotter operator splitting approximation to the Black-Scholes equation.