Pricing Calls from a Risk-Neutral Underlying Distribution

Summary

The document describes a general way to value a call option when the underlying’s terminal price distribution is known. The option value is the discounted expected payoff under a risk-neutral probability distribution: integrate the payoff above the strike against the terminal spot density, then discount at the risk-free rate. The familiar Black–Scholes result is presented as a special case associated with lognormally distributed returns; the integral itself can accommodate other distributions.

The question also asks how to scale a non-lognormal daily return distribution to the option’s maturity. The answer confirms the general distribution-based pricing principle but does not explain how to aggregate daily returns, construct the terminal distribution, or ensure it is risk-neutral. Those are substantive requirements: a historical or physical return distribution alone does not establish a no-arbitrage option price. The document gives a conceptual framework, not a complete procedure or empirical pricing example.

Key ideas

• A call can be valued as the discounted risk-neutral expected payoff at expiration.
• The payoff integral uses the terminal spot distribution above the strike.
• The Black–Scholes formula is a special case of distribution-based option valuation.
• A non-lognormal distribution can be used in the framework if it is the appropriate risk-neutral terminal distribution.
• The answer does not specify how to derive that distribution from daily returns.

Tags

# Is it possible to price a call option given a daily underlying returns distribution?


# Is it possible to price a call option given a daily underlying returns distribution?












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?

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

## Answer by Amit Kumar Jha (score 1)

https://quant.stackexchange.com/a/76859

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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