Pricing a European Power Call Option

This question is about valuing a European-style option whose payoff depends on a power of the underlying asset price rather than the price itself, creating a non-linear exposure. The setup lives squarely within continuous-time option pricing under the standard geometric Brownian motion assumption, but asks the candidate to adapt familiar machinery to a slightly exotic payoff. It is typical of quant interviews for derivatives or exotics desks, and for quantitative research roles working with extensions of the Black–Scholes framework, where comfort with non-plain-vanilla payoffs is essential.

To answer it, a candidate needs to recognize that a power of a lognormally distributed asset remains lognormal, albeit with altered parameters. That insight allows the payoff to be treated as a vanilla option written on a synthetic underlying. The derivation leans on stochastic calculus intuition, transformations of random variables, and careful handling of drifts and volatilities under the risk-neutral measure. Interviewers watch for clean identification of the transformed process, correct parameter adjustments, and the disciplined reuse of the standard pricing formula rather than ad hoc manipulation.

What it tests

When pricing options with non-linear payoffs, such as those involving $S^a$, the key is to recognize that the underlying process can be transformed so the payoff resembles a standard option on a new, synthetic asset. Specifically, if the original asset price $S$ follows geometric Brownian motion, then $S^a$ also follows a lognormal process, but with modified drift and volatility parameters. This transformation allows us to use the Black-Scholes framework by adjusting for the new drift ($m$) and volatility ($a \sigma$) that arise from the power function. The reason this works is that the log of $S^a$ is $a \ln S$, which preserves the normality of returns but scales both the mean and variance, so the pricing machinery remains applicable after suitable adjustments. This principle generalizes: whenever the payoff is a monotonic function of a lognormal variable, consider transforming the process to recast the problem into a standard form.