Continuous Trading, Path Information, and Parameter Observability

# Does continuous trading make drift and volatility observable?


# Does continuous trading make drift and volatility observable?












In a two-page article published in 2026 and titled “Continuous Trading Reveals Expected Returns”, we (Mink & De Weert) show that the assumption of continuous trading - which is standard since Black and Scholes (1973) - implies that traders can directly observe both the drift and volatility of any traded security from market prices alone.

The core of the argument is straightforward. Consider a security $A$ of which the price $S_{A,t}$ follows a geometric Brownian motion (GBM) with drift $\mu_A$ and volatility $\sigma_A$. For simplicity, let the risk-free interest rate be zero. In a continuous-trading model, a self-financing portfolio constructed from risk-free bonds and $1/S_{A,t}$ units of a security $A$ then evolves as: \begin{equation} P_{A,t} = \int_0^t \frac{1}{S_{A,s}} dS_{A,s} = \mu_A t + \sigma_A W_{A,t}. \end{equation} Hence, we can use the solution of the GBM to write the price of $A$ as $S_{A,t}=S_{A,0}e^\left(P_{A,t}-0.5\sigma_{A}^{2}t\right)$, and can rearrange this expression to write $\sigma_A$ as a function of observed market prices (i.e., as a function of $P_{A,t}$, $S_{A,t}$ and $S_{A,0}$). The assumption that continuous trading is possible thus allows traders in the model to directly observe the volatility $\sigma_{A}$ from market prices without the need to somehow estimate this parameter. In a similar way, using a second portfolio across two securities together with a no-arbitrage condition, the two-pager shows that $\mu_A$ is observable from market prices too.

Edit 20-3-2026: After some comments below, we add this clarification of how traders in the model can also observe $\mu_{A}$. Consider a security $B$ of which the GBM contains the same Wiener process as security $A$, so that $W_{B,t}=W_{A,t}$. In real-world financial markets there may not exist two securities with perfectly correlated excess returns, but in the theoretical continuous-trading model there is nothing that rules out this possibility. In this model there simply exist $K<\infty$ securities and there are no assumptions added to prevent some pairs of securities from being perfectly correlated (e.g., a call option and its underlying stock are perfectly correlated in the model, although the former does not follow a GBM of course). Shreve (2004) zooms in on such a pair of perfectly correlated securities and defines the following trading strategy: \begin{equation} P_{AB,t}=\int_{0}^{t}\frac{1}{\sigma_{A}S_{A,s}}dS_{A,s}-\frac{1}{\sigma_{B}S_{B,s}}dS_{B,s}=\left(\frac{\mu_{A}}{\sigma_{A}}-\frac{\mu_{B}}{\sigma_{B}}\right)t+W_{A,t}-W_{B,t}, \end{equation} He concludes from this trading strategy that the no-arbitrage condition implies that if $W_{B,t}=W_{A,t}$ then $\mu_{A}/\sigma_{A}=\mu_{B}/\sigma_{B}$ so that $ P_{AB,t}=0$. In the two-pager, we point out that traders in the model can use this trading strategy of Shreve to identify a pair of securities $A$ and $B$ for which $P_{AB,t}=0$ (as $P_{AB,t}$ is an observed market price). They then have four equations (the two GBM’s for $A$ and $B$, as well as $\mu_{A}/\sigma_{A}=\mu_{B}/\sigma_{B}$ and $W_{B,t}=W_{A,t}$) and four unknowns ($\mu_{A}$, $\mu_{B}$, $W_{A,t}$, and $W_{B,t}$), which can be solved to recover the drifts $\mu_{A}$ and $\mu_{B}$. The assumption that continuous trading is possible thus allows traders in the model to observe the drifts as well.

If correct, this argument shows that if one assumes that continuous trading is possible, one thereby also assumes that traders can directly observe the drift and volatility of traded securities. As a result, for example, the option pricing model of Black and Scholes (1973) implicitly assumes that traders know the expected return on the underlying stock -- which is precisely the assumption its authors believed they had escaped. We therefore wonder whether the above argument is mathematically correct, or whether there is a mistake somewhere?

## Answer by Rylan (score 1)

https://quant.stackexchange.com/a/85529

My responses are a bit too long for a comment.

I may be misunderstanding, and I might be focusing too much on practical points, so feel free to ignore, but:

- It would seem that the argument is that given $P_{A, t}, S_{A, t},S_{A, 0}$, we can observe the volatility. I think this is correct, but as far as I can tell, $P_{A, t}$ essentially requires "viewing" the whole path of $S_{A, t}$, and if we can do that, we can compute the quadratic variation of $S_{A, t}$ and therefore the volatility. I appreciate that tracking $P_{A, t}$ specifically can be a useful "compression" of the path for these purposes.

- It looks like Shreve's argument is $W_A = W_B \to \frac{\mu_a}{\sigma_A} = \frac{\mu_B}{\sigma_B}$, but then the paper says therefore the traders can find $P_{AB, t} = (\frac{\mu_a}{\sigma_A} - \frac{\mu_B}{\sigma_B})t + W_{A, t} - W_{B, t} = 0$. This seems like a special case where two stocks are driven by the exact same Brownian motion process, as opposed to a typical GBM case where stocks are driven by correlated Brownian motions. Even still, I am failing to see how this lets one read off the drift. I think we get basically that the stocks are driven by eg $X_t = W_{A, t} + (\mu/\sigma) t$, which is observable given prices and $\sigma$s, but given $X_t = W_{A, t} + (\mu / \sigma)t = x$, this dosn't seem to reveal what $W_{A, t}$ or $\mu$ is.