_{1}

^{*}

The aims of this paper are twofold. Firstly, we present an approximating formula for pricing basket and multi-asset spread options, which genuinely extends Caldana and Fusai’s (2013) two-asset spread options formula. Secondly, under the lognormal setting, we show that our formula becomes a Black and Scholes type formula, extending Bjerksund and Stensland’s (2011). Numerical experiments and comparison with Monte Carlo simulations and other methods available in the literature are discussed. The main contribution of this paper is to provide practitioners with a pricing formula, which can be used for pricing basket and multi-asset spread options, even under a non-Gaussian framework.

Multi-asset spread options (or basket-spread options) are options whose payoff at maturity is given by the difference (or so-called the spread) between two baskets of aggregated asset prices. For a standard European multi-asset spread call option, the payoff function reads:

where K is the strike price,

This embodies a general class of options, including the two-asset spread (M = N = 1), basket (N = 0) and single-asset vanilla (M = 1, N = 0) options.

Multi-asset spread options are prevalent in a variety of markets, including the fixed income, foreign exchange, commodity, energy and equity markets. They are useful financial tools for hedging a portfolio of long and short positions in the underlying assets. A simple, accurate and efficient method to price and hedge multi-asset spread options is therefore inevitable.

In most contributions from the literature on basket and multi-asset spread option pricing, the underlying asset prices are assumed to follow lognormal processes. However, the celebrated Black and Scholes (1973) formula in [

If we consider first the case of two-asset spread options, namely M = N = 1 in Equation (1), under the Black and Scholes framework, Carmona and Durrleman in [

If we consider then the case of multi-asset spread options, in a lognormal setting, besides to what it has been already said above, we can mention the work of Borovkova et al. in [

For the basket option pricing case (where

Few results are available in the literature concerning the pricing of multi-asset spread options under a non-Gaussian setting, as pointed out in [

A noticeable work on the pricing of two-asset spread options under the non-Gaussian framework is the one of Caldana and Fusai in [

Caldana et al. in [

The aims of this paper are twofold. Firstly, we present a general closed form approximation formula for the pricing of multi-asset spread options, which genuinely extends the one in [

The main contribution of this paper is to provide practitioners with a general closed form approximation pricing formula, which can be used for real-time pricing of multi-asset spread options, even under a non-Gaussian framework.

The rest of the paper is outlined as follows: in Section 2 we present the general closed form approximation pricing formula for multi-asset spread options. This is done via a procedure, which requires only a univariate Fourier inversion and it is applicable to models for which the joint characteristic function of the underlying assets is known in closed form. This approach has been proposed by Caldana and Fusai in [

In this section we present the general closed form approximation formula for the pricing of basket and multi-asset spread options. The approach extends the one in [

In particular, define the event A as follows:

where

and where the coefficients

In what follows, for sake of simplicity in the notation, we will drop the explicit dependency on

Let

We assume that the joint characteristic function of the

The main result is stated in the following proposition, for which a proof is reported in Appendix A.1.

Proposition 1 (Closed Form Approximation Pricing Formula for Multi-Asset Spread Options) The price

where for an opportune damping coefficient

where

Proof. See Appendix A.1.

Some comments on the above approximation formula are due.

First, if we look at Equation (6), in order to compute the price of the multi-asset spread call option, a univariate Fourier inversion is required. The damping coefficient

Second, if the characteristic function

Moreover, the a priori choice for

The approximation can also be applied to the Greeks computation. In particular, assuming that interchange of differentiation and integration is allowed, the formula for the first-order sensitivity of the multi-asset spread option price to a change in the spot price of a generic asset is given by

Similar formulas can be computed for the other Greeks but, as pointed out in [

We conclude this section by showing how the above approximation formula can be adapted for the pricing of basket (

In particular, if we assume

which is the well-known payoff function for a basket call option.

Then the following corollary holds.

Corollary 1 (Closed Form Approximation Pricing Formula for Basket Options). The price

where for an opportune damping coefficient

where

Proof. The result follows by repeating the proof in Proposition 1 assuming

If we assume

which is the well-known payoff function for a two-asset spread call option.

The following corollary shows that in this particular case our formula coincides with the one in [

Corollary 2 (Closed Form Approximation Pricing Formula for Two-Asset Spread Options). The price

where for an opportune damping coefficient

where

Proof. The result follows by repeating the proof in Proposition 1 assuming

As mentioned in Section 1, a first attempt to extend the approach in [

In particular, the starting point of the authors is to consider the geometric average of the underlying prices

where no assumption on the sign of the

Then, they define a feasible but sub-optimal exercise strategy by looking at the set

for an opportune parameter

The lower bound

and the explicit computation is given in the following proposition, see [

Proposition 2 (Caldana et al. (2016) Lower Bound for Multi-Asset Spread Options). Let

where

Then, the price

where

where

and where the function

Proof. See [

As we can see from Equation (18), the computation of the lower bound in [

The second approximation formula discussed in [

This section discusses in more detail the geometric Brownian case. In particular, in what follows we will consider a multi-variate Black and Scholes model. The evolution of the underlying prices, under the risk-neutral measure

where r is the risk-free rate, q is the vector of dividend yields for each asset, 1 is a vector whose entries are all equal to one,

The risk-neutral joint characteristic function of the

where

Expression (23) can be used to compute the closed form approximation formula presented in Section 2.

However, under the Black and Scholes framework, all formulas can be explicitly computed. In particular, in what follows we will derive the so-called Extended Bjerksund and Stensland pricing formula for multi-asset spread options, via the conditional expectation method. We are aware of a different derivation of this pricing formula, only valid for the particular case

Before doing it, we give a bit of insight about the origins of this formula. If we consider the pricing of a two-asset spread option, then it can be proved that the Kirk’s formula in [

where

Bjerksund and Stensland in [

If we consider now the multi-asset spread options pricing problem, Kirk’s formula has been extended to deal with more than two underlyings by Lau and Lo in [

However, the same reasoning as the one above can be applied. Indeed, it can be verified that the Extended Kirk pricing formula proposed by Lau and Lo in [

where

and where the coefficients

Therefore, the idea behind the Extended Bjerksund and Stensland pricing formula is to use this feasible but non-optimal exercise strategy in order to price the multi-asset spread option. The final result is reported in the following proposition.

Proposition 3 (Bjerksund and Stensland Pricing Formula for Multi-Asset Spread Options). The price

where

with

and

where

Proof. See Appendix A.2.

As far as the Greeks computation is concerned, this can be done in a straightforward way, since it is based on the calculation of the derivatives of the formula in Equation (26).

In this section, we present two non-Gaussian price models on which we will analyze the performance of our approximation formula. For each model, we give a brief description and we provide the joint characteristic function of the assets log-returns

In [

Mathematically, under the risk-neutral measure

where

In addition,

are

The Huang and Kou model in [

which is a

Under the risk-neutral measure

Finally, the quantities

as reported in [

Then, the following proposition holds.

Proposition 4 (Caldana et al. (2016), Proposition 4). The joint characteristic function of the log-returns for the

where

Proof. As pointed out in [

The second model we present here is a mean-reverting jump diffusion model discussed in [

As pointed out by the authors, a distinctive feature of electricity markets is the formation of price spikes which are caused when the maximum supply and current demand are close, often when a generator or part of the distribution network fails unexpectedly.

In particular, for

where

As done in [

If we define the vector

and the matrix

and if we assume independence between the jump processes, then the following result holds.

Proposition 5 (Caldana et al. (2016), Proposition 5). The joint characteristic function of the log-returns for the

Proof. As pointed out in [

In this section, we discuss numerical examples for the pricing of basket and multi-asset spread options under the assumption that the underlying prices follow the stochastic dynamics introduced in Sections 3 and 4.

In this section we deal with the pricing of basket options under the assumption of lognormality for the underlying asset prices. In particular, we compare our approximating formula in Equation (26) with the six different methods discussed in [

・ Levy (1992),

・ Gentle (1993),

・ Milevsky and Posner (1998a),

・ Milevsky and Posner (1998b),

・ Beisser (1999), and

・ Ju (2002)

As benchmark values, the authors in [

Input parameters are as in [

The authors in [

In what follows, we will show that, using the same parameter setting, our approximating formula is as accurate as the best methods compared in [^{2}. Besides that, our approximation is as accurate as the one proposed in [

In this section we deal with the pricing of multi-asset spread options under the assumption of lognormality for the underlying asset prices. In particular, we compare our approximating formula in Equation (26) with the four different methods discussed in [

・ Improved Comonotonic Upper Bound (ICUB) as in Section 2.2 in [

Spread | K | Beisser (1999) | Gentle (1993) | Ju (2002) | Caldana et al. (2016) | EBS | MC | S.E. |
---|---|---|---|---|---|---|---|---|

50 | 50 | 54.16 | 51.99 | 54.31 | 54.16 | 54.16 | 54.28 | 0.0383 |

−0.22% | −4.22% | 0.06% | −0.22% | −0.22% | ||||

40 | 60 | 47.27 | 44.43 | 47.48 | 47.27 | 47.27 | 47.45 | 0.0375 |

−0.38% | −6.36% | 0.06% | −0.38% | −0.38% | ||||

30 | 70 | 41.26 | 37.93 | 41.52 | 41.26 | 41.26 | 41.50 | 0.0369 |

−0.58% | −8.60% | 0.05% | −0.58% | −0.58% | ||||

20 | 80 | 36.04 | 32.40 | 36.36 | 36.04 | 36.04 | 36.52 | 0.0363 |

−1.31% | −11.28% | −0.44% | −1.31% | −1.31% | ||||

10 | 90 | 31.53 | 27.73 | 31.88 | 31.53 | 31.53 | 31.85 | 0.0356 |

−1.00% | −12.94% | 0.09% | −1.00% | −1.00% | ||||

0 | 100 | 27.63 | 23.78 | 28.01 | 27.63 | 27.63 | 27.98 | 0.0350 |

−1.25% | −15.01% | 0.11% | −1.25% | −1.25% | ||||

−10 | 110 | 24.27 | 20.46 | 24.67 | 24.27 | 24.27 | 24.63 | 0.0344 |

−1.46% | −16.93% | 0.16% | −1.46% | −1.46% | ||||

−20 | 120 | 21.36 | 17.65 | 21.77 | 21.36 | 21.36 | 21.74 | 0.0338 |

−1.75% | −18.81% | 0.14% | −1.75% | −1.75% | ||||

−30 | 130 | 18.84 | 15.27 | 19.26 | 18.84 | 18.84 | 19.22 | 0.0332 |

−1.98% | −20.55% | 0.21% | −1.98% | −1.98% | ||||

−40 | 140 | 16.65 | 13.25 | 17.07 | 16.65 | 16.65 | 17.05 | 0.0326 |

−2.35% | −22.29% | 0.12% | −2.35% | −2.35% | ||||

−50 | 150 | 14.75 | 11.53 | 15.17 | 14.75 | 14.75 | 15.15 | 0.0320 |

−2.64% | −23.89% | 0.13% | −2.64% | −2.64% |

・ Shifted Log-Normal Approximation (SLN) as in [

・ Hybrid Moment Matching with Improved Comonotonic Upper Bound (HMMICUB) as in Section 3.1 in [

・ Hybrid Moment Matching with Deng et al. (2008) Spread Approximation (HMMDLZ) as in Section 3.1 in [

As benchmark values, the authors in [

In what follows, we will show that, using the same parameter setting, our approximating formula outperforms the methods compared in [

The results reported in

In both scenarios, our approximating formula outperforms the best methods considered in [

In this section we deal with the pricing of basket and multi-asset spread options where the underlying asset prices follow the stochastic dynamics reported in Sections 4. In particular, we compare our approximating formula in Equation (6) with the the two approaches discussed in [

・

・

Input parameters are as in [

Spread | K | ICUB | SLN | HMMICUB | HMMDLZ | EK | EBS | MC | S.E. |
---|---|---|---|---|---|---|---|---|---|

15 | 15 | 19.9819 | 19.6925 | 19.5251 | 19.5231 | 19.5848 | 19.6816 | 19.6849 | 0.0009 |

1.51% | 0.04% | −0.81% | −0.82% | −0.51% | −0.02% | ||||

10 | 20 | 17.0143 | 16.7345 | 16.5693 | 16.5673 | 16.6366 | 16.7009 | 16.7051 | 0.0008 |

1.85% | 0.18% | −0.81% | −0.82% | −0.41% | −0.03% | ||||

5 | 25 | 14.4105 | 14.1460 | 13.9964 | 13.9944 | 14.0731 | 14.0961 | 14.1010 | 0.0008 |

2.19% | 0.32% | −0.74% | −0.76% | −0.20% | −0.03% | ||||

0 | 30 | 12.1523 | 11.9059 | 11.7811 | 11.7790 | 11.8687 | 11.8466 | 11.8519 | 0.0007 |

2.53% | 0.46% | −0.60% | −0.62% | 0.14% | −0.04% | ||||

−5 | 35 | 10.2123 | 9.9851 | 9.8898 | 9.8876 | 9.9890 | 9.9226 | 9.9281 | 0.0007 |

2.86% | 0.57% | −0.39% | −0.41% | 0.61% | −0.06% | ||||

−10 | 40 | 8.5588 | 8.3506 | 8.2860 | 8.2837 | 8.3962 | 8.2897 | 8.2951 | 0.0006 |

3.18% | 0.67% | −0.11% | −0.14% | 1.22% | −0.07% | ||||

−15 | 45 | 7.1581 | 6.9683 | 6.9330 | 6.9305 | 7.0528 | 6.9122 | 6.9174 | 0.0006 |

3.48% | 0.74% | 0.23% | 0.19% | 1.96% | −0.07% |

Spread | K | ICUB | SLN | HMMICUB | HMMDLZ | EK | EBS | MC | S.E. |
---|---|---|---|---|---|---|---|---|---|

22.50 | 2.50 | 24.6617 | 23.1681 | 23.5137 | 23.5138 | 23.4535 | 23.5493 | 23.5925 | 0.0009 |

4.53% | −1.80% | −0.33% | −0.33% | −0.59% | −0.18% | ||||

15.00 | 10.00 | 18.5944 | 16.8591 | 17.1363 | 17.1373 | 17.0131 | 17.1596 | 17.2049 | 0.0008 |

8.08% | −2.01% | −0.40% | −0.39% | −1.11% | −0.26% | ||||

7.50 | 17.50 | 13.0945 | 11.3394 | 11.3854 | 11.3873 | 11.2434 | 11.3588 | 11.4099 | 0.0007 |

14.76% | −0.62% | −0.21% | −0.20% | −1.46% | −0.45% | ||||

0 | 25.00 | 8.4135 | 6.9203 | 6.6579 | 6.6584 | 6.5548 | 6.5418 | 6.6009 | 0.0006 |

27.46% | 4.84% | 0.86% | 0.87% | −0.70% | −0.89% | ||||

−7.50 | 32.50 | 4.064 | 3.7629 | 3.3226 | 3.3147 | 3.2670 | 3.1203 | 3.1872 | 0.0004 |

50.80% | 18.06% | 4.25% | 4.00% | 2.50% | −2.10% | ||||

−15.00 | 40.00 | 2.3929 | 1.7925 | 1.3950 | 1.3853 | 1.3636 | 1.1852 | 1.2518 | 0.0003 |

91.16% | 43.19% | 11.44% | 10.66% | 8.93% | −5.32% | ||||

−22.50 | 47.50 | 1.0323 | 0.7369 | 0.4861 | 0.4913 | 0.4758 | 0.3537 | 0.4026 | 0.0001 |

156.41% | 83.04% | 20.74% | 22.03% | 18.19% | −12.15% |

In

As pointed out in [

In order to deal with the curse of dimensionality, we are aware of a method discussed in [

If we look at

Spread | K | ECF | MC | C.I. Length | ||
---|---|---|---|---|---|---|

50 | 50 | 51.3843 | 51.0643 | 51.3769 | 51.4609 | 3.6686 ´ 10^{−3} |

−0.15% | −0.77% | −0.16% | ||||

40 | 60 | 42.6134 | 41.9998 | 42.6132 | 42.7487 | 5.2246 ´ 10^{−3} |

−0.32% | −1.75% | −0.32% | ||||

30 | 70 | 34.7380 | 33.6724 | 34.7329 | 34.9309 | 6.5806 ´ 10^{−3} |

−0.55% | −3.60% | −0.57% | ||||

20 | 80 | 27.9956 | 26.3980 | 27.9769 | 28.2380 | 7.9691 ´ 10^{−3} |

−0.86% | −6.52% | −0.92% | ||||

10 | 90 | 22.4645 | 20.3784 | 22.4336 | 22.7460 | 9.1516 ´ 10^{−3} |

−1.24% | −10.41% | −1.37% | ||||

0 | 100 | 18.0726 | 15.6289 | 18.0371 | 18.3801 | 1.0277 ´ 10^{−2} |

−1.67% | −14.97% | −1.87% | ||||

−10 | 110 | 14.6576 | 12.0121 | 14.6257 | 14.9774 | 1.1022 ´ 10^{−2} |

−2.14% | −19.80% | −2.35% | ||||

−20 | 120 | 12.0282 | 9.3174 | 12.0047 | 12.3481 | 1.1549 ´ 10^{−2} |

−2.59% | −24.54% | −2.78% | ||||

−30 | 130 | 10.0046 | 7.3281 | 9.9907 | 10.3229 | 1.2029 ´ 10^{−2} |

−3.08% | −29.01% | −3.22% | ||||

−40 | 140 | 8.4370 | 5.8579 | 8.4310 | 8.7468 | 1.2316 ´ 10^{−2} |

−3.54% | −33.03% | −3.61% | ||||

−50 | 150 | 7.2090 | 4.7614 | 7.2076 | 7.5032 | 1.2477 ´ 10^{−2} |

−3.92% | −36.54% | −3.94% |

Spread | K | ECF | MC | C.I. Length | ||
---|---|---|---|---|---|---|

20 | 5 | 26.5940 | 26.5936 | 26.5910 | 26.5942 | 1.4267 ´ 10^{−4} |

0.00% | 0.00% | −0.01% | ||||

15 | 10 | 21.6486 | 21.6104 | 21.5761 | 21.6547 | 1.4270 ´ 10^{−3} |

−0.03% | −0.20% | −0.36% | ||||

10 | 15 | 17.0447 | 16.8630 | 16.8108 | 17.0682 | 3.3615 ´ 10^{−3} |

−0.14% | −1.20% | −1.51% | ||||

5 | 20 | 13.1066 | 12.7435 | 12.7100 | 13.1508 | 3.3615 ´ 10^{−3} |

−0.34% | −3.10% | −3.35% | ||||

0 | 25 | 9.9462 | 9.4451 | 9.4431 | 10.0063 | 6.6941 ´ 10^{−3} |

−0.60% | −5.61% | −5.63% | ||||

−5 | 30 | 7.5059 | 6.9338 | 6.9588 | 7.5795 | 8.5450 ´ 10^{−3} |

−0.97% | −8.52% | −8.19% | ||||

−10 | 35 | 5.6608 | 5.0748 | 5.1170 | 5.7401 | 9.4264 ´ 10^{−3} |

−1.38% | −11.59% | −10.86% | ||||

−15 | 40 | 4.2798 | 3.7181 | 3.7688 | 4.3585 | 1.0472 ´ 10^{−2} |

−1.81% | −14.69% | −13.53% | ||||

−20 | 45 | 3.2497 | 2.7337 | 2.7865 | 3.3310 | 1.1202 ´ 10^{−2} |

−2.44% | −17.93% | −16.35% |

Results for the mean-reverting jump-diffusion model in

In

Spread | K | ECF | MC | C.I. Length | ||
---|---|---|---|---|---|---|

50 | 50 | 59.7381 | 59.9191 | 60.0094 | 60.2354 | 1.7226´ 10^{−2} |

−0.83% | −0.53% | −0.38% | ||||

40 | 60 | 52.4154 | 52.4967 | 52.5980 | 52.8421 | 1.5666 ´ 10^{−2} |

−0.81% | −0.65% | −0.46% | ||||

30 | 70 | 45.6244 | 45.6139 | 45.7291 | 45.9875 | 1.4105 ´ 10^{−2} |

−0.79% | −0.81% | −0.56% | ||||

20 | 80 | 39.4076 | 39.3213 | 39.4528 | 39.7336 | 1.3311 ´ 10^{−2} |

−0.82% | −1.04% | −0.71% | ||||

10 | 90 | 33.7918 | 33.6515 | 33.8009 | 34.0946 | 1.2744 ´ 10^{−2} |

−0.89% | −1.30% | −0.86% | ||||

0 | 100 | 28.7861 | 28.6166 | 28.7845 | 29.0848 | 1.2179 ´ 10^{−2} |

−1.03% | −1.61% | −1.03% | ||||

−10 | 110 | 24.3817 | 24.2075 | 24.3932 | 24.6916 | 1.2049 ´ 10^{−2} |

−1.26% | −1.96% | −1.21% | ||||

−20 | 120 | 20.5538 | 20.3964 | 20.5980 | 20.8951 | 1.2273 ´ 10^{−2} |

−1.63% | −2.39% | −1.42% | ||||

−30 | 130 | 17.2644 | 17.1400 | 17.3549 | 17.6510 | 1.2885 ´ 10^{−2} |

−2.19% | −2.90% | −1.68% | ||||

−40 | 140 | 14.4663 | 14.3852 | 14.6100 | 14.8944 | 1.3328 ´ 10^{−2} |

−2.87% | −3.42% | −1.91% | ||||

−50 | 150 | 12.1067 | 12.0735 | 12.3047 | 12.5775 | 1.4203 ´ 10^{−2} |

−3.74% | −4.01% | −2.17% |

Spread | K | ECF | MC | C.I. Length | ||
---|---|---|---|---|---|---|

20 | 5 | 27.7176 | 27.4510 | 29.0320 | 28.6779 | 1.0476 ´ 10^{−1} |

−3.35% | −4.28% | 1.23% | ||||

15 | 10 | 24.2968 | 23.9533 | 25.4614 | 25.0422 | 8.5187 ´ 10^{−2} |

−2.98% | −4.35% | 1.67% | ||||

10 | 15 | 21.2185 | 20.8709 | 22.2513 | 21.8059 | 7.1737 ´ 10^{−2} |

−2.69% | −4.29% | 2.04% | ||||

5 | 20 | 18.4747 | 18.1838 | 19.4079 | 18.9539 | 6.4985 ´ 10^{−2} |

−2.53% | −4.06% | 2.40% | ||||

0 | 25 | 16.0482 | 15.8554 | 16.9166 | 16.4551 | 5.6703 ´ 10^{−2} |

−2.47% | −3.64% | 2.80% | ||||

−5 | 30 | 13.9159 | 13.8434 | 14.7496 | 14.3020 | 5.2007 ´ 10^{−2} |

−2.70% | −3.21% | 3.13% | ||||

−10 | 35 | 12.0514 | 12.1065 | 12.8735 | 12.4293 | 4.8954 ´ 10^{−2} |

−3.04% | −2.60% | 3.57% | ||||

−15 | 40 | 10.4277 | 10.6068 | 11.2534 | 10.8309 | 4.5859 ´ 10^{−2} |

−3.72% | −2.07% | 3.90% | ||||

−20 | 45 | 9.0178 | 9.3106 | 9.8558 | 9.4376 | 4.3508 ´ 10^{−2} |

−4.45% | −1.35% | 4.43% |

non-Gaussian models of Section 4. If we look at the results in

Results in

This paper presents an approximating formula for the pricing of basket and multi-asset spread options, which genuinely extends the one in [

Pellegrino, T. (2016) A General Closed Form Approximation Pricing Formula for Basket and Multi-Asset Spread Options. Journal of Mathematical Finance, 6, 944-974. http://dx.doi.org/10.4236/jmf.2016.65063

Proof. We observe that

and the same holds for

Therefore, we can re-write the set A defined in Equation (2) as

i.e.

Following [

by an exponentially decaying term, tuned by a parameter

Then, we apply the Fourier transform to this modified call option price for the multi-asset spread option pricing problem, as follows:

Now, from Equation (31) and by applying Fubini’s theorem, we can re-write the above expression as

where

Now, the inner integral in

Therefore,

i.e.

Finally, from the definition of the characteristic function for the vector

where

This concludes the proof.

A.2. Proof of Proposition 3Proof. In the multi-variate Black and Scholes model, see [

We denote the joint distribution of the multi-variate random vector

as

and

Define

Straightforward calculations show that

Then, we set

with

where

Therefore, we can re-write the set A as follows:

where

If we define

Stensland lower bound can be equivalently re-written as

The following result will be used.

Proposition 6 (Distribution of a Multivariate Normal Distribution). Let

and accordingly

where

Then, the distribution of

where

and

Note that the covariance does not depend on the value

Proof. See [

If we look at the definition of the random variable Z in Equation (33), this can be re-written in a more convenient way as follows:

where

and from the property related to affine transformations of multivariate normal distribution, we know that, if

This holds in particular for

Besides that, we have

Therefore, the vector

where

and

By applying the results in Proposition 6, we conclude that the distribution of the vector

where

and therefore, it follows that

and

We can now compute the approximated payoff expectation in Equation (34) as follows:

^{3}This property states that for a log-normal random variable X with parameters

By using the partial expectation property of the lognormal distribution^{3} and discounting, the above expectation gives us the Extended Bjerksund and Stensland pricing formula:

where

The coefficients

This concludes the proof.