Scaling a Call Option Payoff into a Standard Call

Summary

The document considers an option whose terminal payoff is the positive part of a constant multiple of the underlying asset price minus a strike. It shows that when the multiplier is positive, the multiplier can be factored outside the positive-part operation, leaving a scaled standard call payoff with an adjusted strike.

This payoff identity means the instrument can be valued as the multiplier times a vanilla call with strike equal to the original strike divided by that multiplier, assuming the same underlying and expiry. The response offers an algebraic reduction rather than a new pricing model, and it does not discuss edge cases such as a zero or negative multiplier, contract terms, or model assumptions. Its main lesson is to simplify the payoff before seeking a specialized closed-form solution.

Key ideas

• For a positive constant multiplier, the payoff equals that multiplier times a standard call payoff.
• The equivalent call uses a strike scaled by dividing the original strike by the multiplier.
• The algebraic equivalence permits use of ordinary call valuation methods under matching contract assumptions.
• The response does not address zero or negative multipliers or additional exotic contract features.

Tags

# Exotic option pricing


# Exotic option pricing












I'm trying to price an option with payoff $\max\{a\cdot S_t - K,0\}$ where $a$ is a known constant. Ideally I'm looking for a closed form, continuous-time solution. Where should I begin?

## Answer by Alexey Kalmykov (score 8, accepted)

https://quant.stackexchange.com/a/3971

The payoff $\max\{a\cdot S_t - K,0\}$ can be re-written as $a\cdot\max\{S_t - K/a,0\}$. Therefore it can be priced as a regular call option with the strike $K/a$.

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.