Why Risk-Neutral Option Pricing Uses the Same Rate for Drift and Discounting

Summary

The document contrasts the standard Black–Scholes pricing equation, which uses the risk-free rate r in both the underlying-price drift term and the option discount term, with a modified equation that uses a separate rate r′ to discount the payoff. It invokes the Feynman–Kac representation to express the modified equation as a discounted expected terminal payoff under a measure where the underlying retains drift r.

The question asks whether this setup can arise from a no-arbitrage hedging argument, perhaps with two separate economies. It offers no derivation or resolution. The distinction raises a key modeling issue: the measure governing expected underlying returns and the rate used to discount cash flows are linked by the pricing framework and market assumptions. The document is therefore useful as a question about risk-neutral valuation, but it supplies no conditions under which the modified equation would be arbitrage-free.

Key ideas

• The standard European option pricing equation uses the risk-free rate in both its drift and discount terms.
• The modified equation separates the underlying drift rate from the payoff discount rate.
• Feynman–Kac expresses the modified equation as a discounted conditional expectation under a specified process.
• The document poses, but does not answer, whether the modified setup admits a no-arbitrage hedging derivation.

Tags

# Option price with underlying growth rate distinct from discount rate


# Option price with underlying growth rate distinct from discount rate












Consider a European style option.

The price equation is $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \tag1$$ where $S$ is the underlying stock price and $V(t,S)$ is the option price. This is derived using for example an arbitrage free hedging argument.

Consider a similar equation $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - r'V = 0. \tag2$$ Note: $r'\neq r$. Its solution by the Feynman-Kac formula is $$V(t,S)=\mathbf E^Q\Big[e^{-\int_s^T r'ds} V(T,S(T))\,\big|\,S_t=S\Big]$$ under the measure $\mathbf Q$ where $S(t)$ is an Ito process $$dS(t) = rdt+\sigma dB$$ with $B$ being a Brownian motion.

I suppose you can argue that $\mathbf Q$ in Equation (2) is a real world measure. Is there a no arbitrage hedging argument under certain conditions to derive Equation (2)? Perhaps it needs to be under two economies with interest rates $r$ and $r'$ but somehow unrelated? Maybe some kind of foreign exchange world? Let me know if this question makes sense at all.

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