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In Edmond Levy's 1992 paper, he introduced a moment-matching method to approximate the price of an Asian option assuming GBM for the underlying.

It suggested that, if some monitor points are already observed, and the average of these points are $A$, then in the pricing formula, the strike is adjusted to $K^*=K-\frac{m+1}{N+1}A$, where $m+1$ is the number of points observed, and $N+1$ is the total number of monitor points.

However, it is possible that $K^*$ is below $0$, which causes trouble when we try to log them in $d_1$ and $d_2$. Is it the method's own limitless or did I do something wrong here?

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    $\begingroup$ If $K*$ is below 0, you do not need to compute $d_1$ and $d_2$, as then there is no optionality. $\endgroup$
    – Gordon
    Oct 19 '15 at 19:08
  • $\begingroup$ I would suggest that this question be closed, as the OP has never commented on any of the answers. That is, answers to this question do not appear to be needed. $\endgroup$
    – Gordon
    Apr 11 '17 at 13:06
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When the adjusted strike goes to zero or negative, it can be proven that the call option will always be exercised, therefore the price of a call is given by the discounted of the underlying and strike (as also mentioned by Gordon). This is like a forward therefore there is no need to compute d1 and d2.

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    $\begingroup$ In agreement to the post above, think about the problem as an earnout where the target company is paid a certain percentage of revenue about a threshold. If that threshold is not met, the strike price is adjusted downward by whatever the revenue achieved is and the expected remainder of revenue is modeled over the remaining duration of the earnout and the achieved amount would be added back to the call option value at the end (since there is no risk to it). $\endgroup$
    – RandyF
    Mar 17 '16 at 19:55
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    $\begingroup$ If the amount is achieved (which would be calculated as a negative adjusted K), the value of the option would just be the discounted expected revenue (only the unknown part is discounted) minus the strike price. Similar to an Asian option where we are calculating the probability of the remaining data point being above a certain threshold, if the condition will be met regardless of how low the stock price goes, as said above, all optionality is removed, and we're only concerned with what the average strike price is in a risk-neutral framework minus the strike price. $\endgroup$
    – RandyF
    Mar 17 '16 at 20:00

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