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Pricing a Stock Appreciation Contract as a Scaled Call Option

Article Quant Q&A · Author: Wolfy
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Summary

The problem describes a one-year contract that pays a fixed fraction of any positive increase in a stock’s value. The attempted solution recognizes that the payoff has the shape of a call option: it is proportional to the positive part of the difference between the ending stock price and the starting stock price. This identifies the relevant Black–Scholes building block as a call struck at the initial stock price.

The question focuses on how the payment fraction enters the option price, since the cited solution appears to use that fraction divided by the initial stock price. The excerpt does not include the author’s solution or resolve this normalization issue. Thus it frames a useful pricing question about payoff units and scaling, but does not provide a complete derivation, state the exact pricing convention, or discuss discounting and assumptions beyond invoking Black–Scholes. Readers must ensure the option formula’s payoff matches the contract’s dollar payoff before applying any multiplier.

Key ideas

  • A payoff based on a positive stock price increase has the shape of a call payoff struck at the initial stock price.
  • The contract price should reflect the stated fraction of the option-like payoff.
  • Scaling depends on matching the Black–Scholes formula’s payoff units to the contract payoff.
  • The excerpt raises but does not resolve why the proposed solution divides the fraction by the initial stock price.

Tags

OptionsDerivatives pricingEquities
Full text
# Evaluating contract $D$ where the stock follows the Black Scholes assumption


# Evaluating contract $D$ where the stock follows the Black Scholes assumption












Ch.7 Mark Joshi Problem 14

> A contract, $D$, pays $30\%$ of the increase (if any) of a stock's value in a year. If $S_t$ follows Black-Scholes assumptions, give a formula in terms of the Black-Scholes formula for the price of $D$.

Attempted solution - Seems to me that we have $$0.3\times(S_1 - S_0)_{+}$$ so we are evaluating a call option struck at $S_0$ and we need to multiply by $0.3$ to get the price. Although, the solution by Mark Joshi does the same but divides $0.3$ by $S_0$, no idea why and where that came from.

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.