What impact does arbitrage have on realised volatility estimates?

Question

Doing some research modeling/estimating volatility in the bitcoin market. There is quite a bit of scope for arbitrage within crypto-currency markets. Wonder if this has any impact on my volatility estimates? does it impact statistical significance or out of sample estimation accuracy?

Answer 1

If you look at it from a mathematical point of view - presence of arbitrage should not matter for volatility estimates.

Absence of arbitrage can be associated with the existence of an equivalent martingale measure for the bank account numeraire. (first fundamental theorem of asset pricing)

Let's assume the real world process is something like $dS_t=\mu(t,S_t)dt+\sigma(t,X_t)dW_t$. If we can get rid of the drift $\mu(t,S_t)dt$ by deviding by a suitable numeraire $N_t$ the process $S_t/N_t$ will be a martingale. By the change of numeraire approach we could also immideately derive an equivalent martingale measure for the bank account $B(t)$.

Now assume there is arbitrage in the market. It follows that we can't make our process driftless under any numeraire. Thus the dynamics of our market asset is such that absence of arbitrage can't be guaranteed.

Still it won't matter for we are only concerned with the properties of the drift and not with the properties of the volatility term. Thus the drift could be seen to be primarily responsible for arbitrage in the setting shown abovel. For the drift does not matter when estimating volatility, arbitrage should not have an effect on vol.-estimates.

Still this is purely theoretic - perhaps someone working with HF-data can contribute another perspective :)

Answer 2

Perhaps not the most encouraging answer, but: I would think that it is contingent upon the specific implementation, magnitude, regularity, and transiency of arbitrage available as well as the volatility estimate time-scale.

In a very simple case, the existence of arbitrage opportunities would likely result in larger fraction of informed traders (relative to liquidity traders). This larger fraction should result in market makers that quote wider spreads than they would otherwise to protect themselves from being the counterparty of an arbitrageur rather than the desired liquidity trader. However, the existence of arbitrage opportunities may also imply that we observe a decreased influence of liquidity traders relative to the informed traders, who trade uni-directionally. This may an offsetting effect, depending on the magnitude of the arbitrage.

Effectively, as your volatility time-scale gets more granular, the more it is "polluted" by a bid-ask bounce, artificially raising volatility. There are ways to attempt to correct for this and other higher frequency-based issues, e.g. Roll (1984), Zhang et al. (2005), Barndorff-Nielsen and Shephard (2004), and Andersen et al. (2010).