How Uniform Price Scaling Affects Derivative Values

Summary

The document raises questions about how derivative values change when the underlying stock price, strike, and other price-related inputs are all multiplied by the same constant. It distinguishes this transformation from using a constant as a numeraire, where the strike is not changed. It asks whether vanilla option prices scale proportionally under standard Black–Scholes and geometric Brownian motion assumptions, and whether exotic contracts can behave differently.

A variance swap is offered as a possible example of a derivative whose value may be affected differently. However, the document contains only the questions and assumptions; it provides no answer, derivation, pricing result, or evidence. It is therefore a useful prompt for studying pricing homogeneity and contract definitions, but it does not establish a general scaling rule. Any conclusion would depend on which inputs and payoff terms are scaled and on the precise derivative being priced.

Key ideas

• The questions concern scaling the underlying, strike, and related price inputs together.
• The document distinguishes input scaling from changing the numeraire.
• It asks whether vanilla option values scale under standard Black–Scholes assumptions.
• A variance swap is raised as a possible example of an exotic with different scaling behavior.
• The source provides questions but no derivation or conclusions.

Tags

# Scaling Stock Price and Strike etc. by a Constant


# Scaling Stock Price and Strike etc. by a Constant












Please provide steps to justify the below.

1) If the stock prices, strike and other price related parameters are scaled by the same constant, will the derivative price scale accordingly? I would think this is different from using a constant as a numeraire, since a numeriance does not affect the strike price. Please confirm.

2) Would scaling by a constant lead to any issues (or a different price), when pricing vanilla options?

3) Are there any derivatives (exotics) whose price is affected when scaling by a constant? One example, is a variance swap as suggested by user @Mats Lind

Related Question: Using a Constant as a Numeraire

The rest of standard Geometric Brownian Motion and Black Scholes assumptions apply.

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