Pricing European Options on Convex Transformations of an Asset

Summary

The document asks whether a European option can be priced in a Black–Scholes setting when its payoff depends on a general transformation of the underlying asset. It illustrates the question with a call payoff equal to the positive part of a transformed terminal asset value minus the strike. The central issue is what properties the transformation must satisfy, and whether convexity alone supports useful pricing conclusions.

The text contains no answer, derivation, pricing formula, or evidence. It therefore raises a theoretical question rather than presenting a valuation method. A complete analysis would need to specify the transformation’s domain and regularity and examine the resulting payoff under the assumed risk neutral distribution; the document does not establish that convexity is sufficient or discuss hedging or practical implementation.

Key ideas

• The proposed payoff applies a transformation to the terminal underlying value before comparing it with the strike.
• The question is framed within Black–Scholes assumptions for a European option.
• The author asks whether convexity of the transformation is enough for meaningful pricing results.
• No pricing method, assumptions, or conclusion is supplied.

Tags

# B&S pricing of option with convex transformation


# B&S pricing of option with convex transformation












Assuming B&S world, is it possible to price an (European) option on a general transformation $f(\cdot)$ of $X$? What kind of assumptions should we make on $f$? Is convexity sufficient to find some meaningful results?

For example for a call: $$ \max(0, f({X_T})-K) $$

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