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Handling a Nonpositive Adjusted Strike in Levy's Asian Option Approximation

Article Quant Q&A · Author: sanliusinger
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Summary

The document describes a question about Levy's moment-matching approximation for Asian options under geometric Brownian motion. When some monitoring points have already been observed, the method adjusts the strike using the observed average and the share of monitoring points already fixed. The question is what to do when this adjusted strike becomes zero or negative, since the standard option formula uses its logarithm.

The answer says that for a call, a nonpositive adjusted strike implies exercise is certain within this approximation, so the value can be handled as a discounted forward-style payoff rather than by calculating the usual d1 and d2 terms. This addresses the logarithm problem with a special case. The note is brief and offers no derivation, numerical example, or treatment of puts; its conclusion is specifically about the call case described.

Key ideas

  • Levy's approximation adjusts the strike to account for monitoring observations already known.
  • A zero or negative adjusted strike makes the usual logarithmic terms in the formula unusable.
  • The answer treats a call with a nonpositive adjusted strike as certain to be exercised.
  • The note gives no derivation or guidance for puts and other cases.

Tags

OptionsDerivatives pricing
Full text
# Negative adjusted strike in Levy's Asian option approximation?


# Negative adjusted strike in Levy's Asian option approximation?












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?

## Answer by SmallChess (score 1)

https://quant.stackexchange.com/a/21794

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.