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Suppose an option with a payoff function

$$ \max((1+k)S_1,kS_2) $$ where $S_1, S_2$ are stock prices and $k>0$ is a constant value.

To value such an option, one would decompose this payoff function into plain vanilla options (and possibly bonds), as they can be valued using the Black-Scholes formula.

How can this payoff function be decomposed?

Thank you very much!

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1 Answer 1

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$$\max(S_1, S_2) = S_1 + \max(S_2-S_1, 0)$$ In the case of GBM, the second term in the RHS can be valued using Magrabe's formula. The scalars $k, 1+k$ don't change this significantly.

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    $\begingroup$ I got $ (1+k)S_1 + max(kS_2-(1+k)S_1,0) $. Is it correct to price the part $max(kS_2-(1+k)S_1,0)$ by directly utilizing the Magrabe's formula, i.e., by simply plugging $kS_2$ and $(1+k)S_1$ into the Magrabe's formula? Thanks! $\endgroup$
    – math4biz
    Commented Oct 20, 2023 at 14:38
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    $\begingroup$ Yes -- another kind of loose way of thinking of it is that in GBM, having scalar multiple of a stock (instead of just one unit) is sort of the same as having a different initial stock price $\endgroup$
    – Rylan
    Commented Oct 20, 2023 at 14:41
  • $\begingroup$ You mentioned that the formula assumes GBM, how about if we have the dynamic $$ S_{ik} = S_{i0}*e^{(\mu_i-\frac{1}{2}(\sigma^2_{i,1}+\sigma^2_{i,2}))k+\sigma_{i,1}Z_{i,1}+\sigma_{i,2}Z_{i,2}}$$, where $i=1,2$? $\endgroup$
    – math4biz
    Commented Oct 20, 2023 at 16:30
  • $\begingroup$ I might be missing something but that appears to be lognormal also and so could come from a GBM as well $\endgroup$
    – Rylan
    Commented Oct 22, 2023 at 8:20

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