Why the Black–Scholes Hedged Portfolio Is Locally Riskless

Summary

The document examines a step in Black–Scholes reasoning: delta hedging removes the diffusion term from a portfolio’s dynamics, leaving an integral over time of terms involving derivatives of the option value and the underlying price. It asks why this integral can be called locally riskless even though its integrand depends on a random process, and why right-endpoint Riemann sums appear to require future information at each step.

The key distinction is between the limiting time integral and the finite approximations used to express it. The integral has no diffusion component in its dynamics; randomness in its integrand does not itself create an instantaneous Brownian risk term. Right-endpoint sums are mathematical approximations that use future sample values within each interval, so they need not represent implementable adapted trading strategies. Their limit can represent the pathwise time integral even when individual sums are not locally riskless trading processes. The question provides no formal derivation or resolution of technical assumptions behind this interpretation.

Key ideas

• Delta hedging removes the diffusion term from the Black–Scholes portfolio dynamics.
• A random time integrand does not by itself imply a diffusion risk term in the integral.
• Right-endpoint Riemann sums can depend on future values relative to the interval start.
• Mathematical approximations to an integral need not themselves represent admissible trading strategies.
• The integral’s local risk property concerns its dynamics, not whether each finite sum is implementable.

Tags

# Locally riskless


# Locally riskless












Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:

$$ \int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\partial^2 f}{\partial x^2}(S(t),t)\cdot\sigma^2\cdot S(t)^2\right) \,dt $$

The process is still random because it has stochastic processes as integrands, but the fact that the diffusion term is zero makes it locally riskless.

On the other hand, this is not an Itô integral, but a pathwise Riemann integral. Therefore, nothing precludes me from expressing the integral as right Riemann Sums:

$$ \lim_{n\to\infty} \sum_{i=1}^{n} \left(\frac{\partial f}{\partial \tau}(S(t_i^n),t_i^n)+\frac{1}{2}\cdot\frac{\partial^2 f}{\partial x^2}(S(t_i^n),t_i^n)\cdot\sigma^2\cdot S(t_i^n)^2\right) \,*(t_i^n-t_{i-1}^n) $$

The stochastic process whose integrals are of the form within the limit operation are clearly not locally riskless since at every instance, knowledge about the future is required. How to reconcile this?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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