Risk-Neutral Pricing Beyond Lognormal Underlying Models

Summary

The document asks whether derivative prices can be found by discounting expected payoffs under a risk-neutral assumption when the underlying security is not lognormal. It distinguishes the familiar Black–Scholes formula, which relies on assumptions such as lognormal prices and constant variance, from the broader idea of valuing payoffs under risk-neutral dynamics.

The answer says the lognormal assumption is not required for that broader method. It attributes the result to the Feynman–Kac relationship and replication: under suitable conditions, a delta-hedged portfolio must earn the risk-free rate, yielding a pricing equation that can be expressed as a risk-neutral expected payoff. The document offers this as a brief conceptual explanation rather than a worked derivation. It does not specify the assumptions or boundary conditions needed for replication, nor does it address market frictions, incomplete markets, or how to construct the risk-neutral dynamics for a particular non-lognormal security.

Key ideas

• The Black–Scholes formula relies on particular assumptions, including lognormal underlying prices.
• Risk-neutral expected-payoff valuation can apply beyond the lognormal setting.
• The explanation links risk-neutral valuation to replication and the growth rate of a delta-hedged portfolio.
• The document gives no detailed derivation or treatment of market frictions and incomplete markets.

Tags

# Can derivatives of non-lognormal securities be priced using risk-neutral evaluations assuming risk-free drift?


# Can derivatives of non-lognormal securities be priced using risk-neutral evaluations assuming risk-free drift?












The Black-Scholes model essentially says that, if we assume some things (lognormal, constant variance, etc.) then the following the fair price of a

$$C = N(d_1) S_t - N(d_2) K e^{-rt}$$

However, another interpretation of the Black-Scholes models is that if we make the above assumptions, then we can merely pretend the drift of the underlying security is the risk-free interest rate and then interpret a derivative's expected payout as equal to its fair price.

Clearly, the lognormal assumption is required for the standard Black-Scholes formula. However, I want to know if the lognormal assumption is required for the "pretend riskless drift and then compute expected-value" perspective.

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/80188

Not required. The result "pretend riskless drift and then compute expected-value" comes from feynman-kac, as soon as you write the condition for replication. The result is a reformulation of the fact that the delta hedged portfolio should grow at the risk free rate.

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