Pricing a Squared-Stock-Payoff Claim Under Black–Scholes

Summary

The document poses a risk-neutral pricing problem for a derivative that pays the square of a non-dividend-paying stock price at maturity. It proposes discounting the expected terminal payoff, but the attempted expression does not satisfy the Black–Scholes partial differential equation. This flags a likely issue in how the terminal stock-price distribution or its second moment is being handled.

The prompt gives no completed derivation or corrected price. To resolve it, one must use the risk-neutral stock dynamics, calculate the conditional expectation of the squared terminal price, discount at the risk-free rate, and verify that the resulting value satisfies the PDE and terminal payoff. The stated attempt is incomplete: it does not clearly define its symbols or provide enough calculation to diagnose the precise error.

Key ideas

• The claim pays the square of the stock price at maturity.
• Risk-neutral pricing requires the discounted conditional expectation of that terminal payoff.
• The proposed calculation is reported not to satisfy the Black–Scholes PDE.
• The document provides no correction, so the moment calculation and PDE verification remain unresolved.

Tags

# Option pricing with risk-neutral approach


# Option pricing with risk-neutral approach












Problem Given $Y_t$ price of a stock (no-dividents), and a derivative paying $Y_T^2$ at maturity $T$, evaluate the price of the instrument now using risk-neutral approach and check that it satisfies Black&Scholes PDE.

My attempt $$V_0 = \exp(-rT)\cdot E(Y_T^2|F_0) = ... = \exp(-rT)\cdot(\sigma^2 + \mu^2)$$

Unfortunately, this result doesn't satisfy the PDE.

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