Pricing a European Option with a Quadratic Payoff

Summary

The document presents a pricing expression for a European option whose terminal payoff is the positive part of the underlying price multiplied by its excess over the strike. It gives a formula involving the initial underlying price, strike, rate, volatility, time to maturity, and normal distribution terms, with separate definitions for the terms d_i.

The writer asks how this contract's price compares with a standard European call and why a bank might be reluctant to sell it. No comparison, explanation, derivation, numerical example, or supporting evidence is supplied. The formula and questions frame a pricing topic, but the document itself does not resolve the key issues. Any conclusions about relative value or seller risk would require additional analysis of the payoff's exposure, assumptions, and hedging behavior.

Key ideas

• The contract pays the positive part of the underlying price times its excess over the strike at maturity.
• The document supplies a pricing expression but does not derive it.
• It asks for a comparison with a standard European call, without providing an answer.
• It raises seller reluctance as a question but gives no explanation.

Tags

# Pricing of an option


# Pricing of an option












I've priced a European option with payoff $\max\{S_T(S_T - K), 0\}$ and found

$S_0(S_0 \exp((r + \sigma^2)T) \mathcal{N}(d_3) - K\mathcal{N}(d_1))$

where $d_i = \frac{\ln(\frac{S_0}{K} + (r + \frac{i\sigma^2}{2})T}{\sigma\sqrt{T}}$

I'm now asked to compare the price of this option to that of a standard European Call and also to answer why would a bank be reluctant to sell such an option but I'm stumped on those two.

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