Key takeaways
  • Two pricing models fold in market noise
  • Drift and volatility can both shift
  • A binomial tree extends Grossman-Stiglitz
  • Market completeness gives one pricing map

Tiny trading frictions can bend the prices you see on a screen, even when the underlying asset is supposed to look smooth. This paper tackles that problem by building microstructure noise into models for assets and European options, rather than treating it as an afterthought. The authors present two market-complete frameworks: one in continuous time based on Black–Scholes–Merton, and one in discrete time as a binomial tree extension of the Grossman–Stiglitz model. Because both models are market complete, they produce a unique equivalent martingale measure, giving a one-to-one map between the risk-neutral and real-world price dynamics. The authors also work through empirical examples to estimate the coefficients that capture the noise’s influence on prices. In the discrete model, they can separate the noise effect on both volatility and the drift coefficient. Their examples provide evidence for the primary microstructure noise they believe the data capture.

A stock ticker can jump by a cent for reasons that have nothing to do with news. Those jumps come from market microstructure noise, the small drag from bids, asks, and trade timing. This work treats that drag as part of the price model. It does not sweep it aside as clutter. That matters because the clean efficient price is never seen directly. The efficient price means the value you would see if trading frictions vanished. If you use the wrong model, you may blame the market when the real culprit is the trading floor. The same problem shows up in European options, which pay on one future date. A small bias can change the quote you trust.

Two ways to price a noisy market

The abstract gives two routes. The first lives in Black-Scholes-Merton, the classic smooth-price model for options. The second lives in a binomial tree. That is a step-by-step ladder of prices. It extends Grossman-Stiglitz, a classic noisy-market model. Both models are market complete. That means one rule prices every payoff. This also gives a unique equivalent martingale measure, a risk-neutral lens that turns one price story into another. In plain terms, one set of settings links the world you see with the world you model. The empirical examples then fit the coefficients that measure the noise. The discrete model goes further. It pulls noise apart from volatility, the size of swings, and drift, the average push up or down.

  • The continuous model folds noise into Black-Scholes-Merton pricing.
  • The binomial tree extends Grossman-Stiglitz in discrete steps.
  • Both models stay market complete and give one pricing map.
  • The discrete model splits noise between volatility and drift.

“We present two models for incorporating the total effect of market microstructure noise into dynamic pricing of assets and European options.”

From the abstract

How the model turns noise into numbers

The method starts with a simple idea. Do not treat noise as leftover error. Put it inside the price model. In the smooth-time version, that means starting from Black-Scholes-Merton and adding the noise effect. In the step-by-step version, it means building the noise into the tree itself. Calibration is the tuning step that makes the model match observed prices. Once the model is complete, the fit reveals the coefficients. The key gain is separation. Volatility tells you how wide prices swing. Drift tells you whether the path leans up or down. The discrete setup can pull both apart.

Why that split matters

Most market tools treat microstructure noise like static on a radio. This work says the static can be priced. That changes how you read a chart. A move may not be pure news. Part of it may come from the way trades are posted and measured. The discrete model is important here. It can say whether a shift sits in volatility or in drift. That matters for assets and European options, because both depend on the path shape. It also matters for anyone trying to infer a hidden efficient price from messy trade data. The model gives that search a cleaner frame.

Where the surprise points next

The surprise is not just that noise widens swings. It can also tug on drift in the discrete model. That makes the average push in a price path less naive than many traders assume. It also means a pricing model can sort signal from trading grit with more care. The clearest next check is whether this drift split stays visible in new empirical examples. If it does, the model becomes a sharper test of when a price move is real and when it is noise. That is a small change in wording. It is a big change in how you read a quote screen.