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I am looking at the pricing of a two asset multi strike option in the Black Scholes framework but I am struggling with coming up with a pricing formula.

The payoff of the option at maturity is \begin{equation} \max(S_1-K_1,S_2-K_2,0) \end{equation} I have tried reviewing some literature but I haven't found anything about this particular option. Any idea?

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  • $\begingroup$ There is no closed form formula, but you can use numerical quadrature to compute $E[\text{payoff}]$ $\endgroup$ – Antoine Conze Feb 27 '18 at 14:16
  • $\begingroup$ It looks like the solution would be the maximum value of two vanilla options. $\endgroup$ – David Addison Feb 27 '18 at 15:14
  • $\begingroup$ @AntoineConze thanks for your reply, I know how to solve the problem numerically but I would like to find an analytical solution, do you think that is not possible? I have tried using the change of numeraire approach used in Pricing Rainbow Options (Ouwehand & West, 2006) but due to the fact that the two strike prices are different I wasn't able to solve the problem $\endgroup$ – Nik345 Feb 28 '18 at 0:06
  • $\begingroup$ @DavidAddison Yes at maturity the payoff is the same as a choice between two call options, but I don't think that implies that the price at t=0 would be the max of the two calls $\endgroup$ – Nik345 Feb 28 '18 at 0:07
  • $\begingroup$ @Nik345 I don't think there is an analytical solution because even after change of numeraire you end up with quantities such as $P(S_1 > K_1, S_2>S_1-K_1+K_2)$ which cannot be computed analytically in a 2 dimensional GBM settings. At best you can obtain a semi-analytical solution by noting that conditional on $S_1$ the payoff is a combination of vanilla options on $S_2$. $\endgroup$ – Antoine Conze Feb 28 '18 at 7:44

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