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Apologies in advance if this problem is somewhat ill-posed. But I was thinking given the price of a call option can be formulated in terms of a implied probability density function at time $T$, would it be possible to price a call option given a non-lognormal distribution of daily returns?

My question would be how would one scale a daily returns distribution (one that is not lognormal) to match the tenor of the option to get the implied distribution of spot at $T$? I would imagine once we have that we can simply compute the below integral to get the price of the option?

$$ c = e^{-rT}*\int_{K}^{\infty}(S-K)p(S)dS $$

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Absolutely! You're on the right track. Option pricing can be understood in terms of an integral over the risk-neutral probability distribution of the underlying asset's payoff at expiration. When the distribution of returns is log-normal (as in the Black-Scholes model), the integral simplifies to the familiar Black-Scholes formula. However, if we know the distribution, we can always price the option using the general formula.

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