Trading styles and long-run variance of asset prices

The variance of a financial asset is a proxy for how predictable the asset is after some given period. The following aims to quantify the change in variance of an instrument under the simplifying asssumption that the price dynamics are governed by either prevailing trend-following strategies or prevailing mean-reverting strategies. On the one hand, if the overall view of the market is momentum based then an increase in the price on any trading day is likely to increase the share price on the following day (Chan et al., 1996; Jegadeesh and Titman, 2001). Conversely, if the overall view of the market is mean reverting then an increase is likely to be followed by a successive decrease the following day. Importantly, the former mimics the typical scenario arising in passive investment, where for example additional positive investment follows initially favourable conditions thereby driving up the price of the asset. In the following, we investigate how each of these styles impacts the overall variance of an instrument.

We explore this probabilistically through examining a correlated random walk (Renshaw and Henderson, 1981) - a model such that the returns take values either -1 or +1 and where a day’s closing returns is equal to the previous day’s with probability $p$. Such a model touches on random walk theory for asset pricing. There exists a long history of using random walks in finance and econophysics (Fama, 1995; Sewell, 2011; Scalas, 2006b), a full review is beyond the scope of this article. Of note also, random walk models appear in the evaluation of the Efficient Market Hypothesis (Malkiel, 2003; Malkiel and Fama, 1970) where the predictability of a process is central to establishing whether markets are efficient or not.

In addition to the simplistic model, we consider two models applicable to financial assets over different return horizons. We consider firstly daily US stock data where after developing and validating an appropriate model we estimate the probability of each stock moving in the same direction on subsequent days. Exploiting this, we are able to quantify the change in variance relative to a random walk for each of the assets and thereby show that the variance is deflated in most of these stocks. We also test the prediction accuracy of a similar model on a week’s worth of high-frequency EUR/USD where we are able to conclude again that the net trading style is mean-reverting and to a more exagerated extent than the daily stock data.

We consider two discretet-time stochastic process models for asset prices $(X_{t})$ and $(Y_{t})$ for $t=0,1,...$. The first, (RW) is a random walk where $\mathds{P}[X_{t+1}=X_{t}+1]=\mathds{P}[X_{t+1}=X_{t}-1]=\frac{1}{2}$. The second, random walk (CRW) is a random walk with persistence in the direction it travels, i.e., $Y_{t}$ for $t=0,1,...$

1. 1.

   If $Y_{t}=Y_{t-1}+1$ (last move was up) then $Y_{t+1}=Y_{t}+1$ (next move will be up) with probability $p$

2. 2.

   If $Y_{t}=Y_{t-1}-1$ (last move was down) then $Y_{t+1}=Y_{t-1}-1$ (next move will be down) with probability $p$

We plot these two situations in Figure 1, simulating 1000 realisations of each of the two models over 1,000 units of time, setting $p=\frac{1}{4}$ and $p=\frac{3}{4}$ for CRW and where the initial directions were distributed randomly up and down with probability 1/2. We see that there appears to be a relative increase in the variance of the assets for the random walk with persistence compared to that without when $p=\frac{3}{4}$ and a relative decrease when $p=\frac{1}{4}$. As a result, we look at the standard deviation of each of the two models through time in Figure 2a where we see the standard deviation is close to double that of the random walk model when $p=\frac{3}{4}$. Indeed, in Figure 2b the variance (square of std. deviation.) is approximately linear in both cases.

Next, we examine the relative increase in variance between the two models as we vary $p$, the probability we move in the same direction as the previous day. We show $\mathds{V}[Y_{T}]/\mathds{V}[X_{T}]$ in Figure 2c where $T=200$ is the final day. Interestingly, we see that over a broad range of $p$, e.g. $p\in[0.2,0.8]$ then the relative increase in variance is approximately linear in the log-scale (suggesting it is exponential in $p$) and for $p$ close to $1$ the relative increase in volatiliy rises super-exponetially. We note, that the variance reduces for $p<0.5$ as in this case the model of CRW if the last move was down (for example) the next move will be up with high probability and so as $p\rightarrow 0$ then the asset price tends to something that alternates between $0$ and $+1$ or 0 and $-1$. Finally, assuming that in both cases the variance is approximately linear in time, we are able to take the ratio of the volatilities between the two to give some measure of the relative change in variance per unit of time as well.

We here introduce a semiparametric model for a price process, relaxing the assumption that the increments must be in $\{-1,+1\}$ to a more general class of return distributions. We consider discretely observed price processes, with consideration of continuous-time processes to follow.

We assume, as before, that the sign of the increment is governed by a correlated random walk model, with increments taking values in $\{-1,1\}$, which we denote as $(S_{n})$. Conditional on $S_{n-1}$, then $S_{n}=S_{n-1}$ with probability $p$ and $S_{n}=-S_{n-1}$ with probability $1-p$. Coupled with this, we also introduce random variables representing the magnitude of the change in price at each iteration, which we denote as $(\delta_{n})$. The overall price process may then be expressed as

where $\delta_{i}\geq 0$ are random variables used to model an increment of an arbitrary price process. We assume that conditional on a realisation of $(S_{i})_{i=0}^{n}$ the increments are conditionally independent, with $\delta_{i}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}\nu$ for some distribution $\nu$.

Proof is in Appendix A.1. Previous expressions for limiting variance of a correlated random walk model can be found in (Renshaw and Henderson, 1981; Mauldin et al., 1996; Guo et al., 2017), though without the additional complexity arising from a random magnitude distribution $\nu$.

Considering Equation (2), we see that for a magnitude distribution $\nu$ concentrated on a single (strictly positive) point $x$, then the $\mathds{E}_{\nu}[\delta^{2}]=\mathds{E}_{\nu}[\delta]^{2}$ and the limiting variance is

thereby capturing the previous analysis as a special case. Indeed, we see that in this case for $p>\frac{1}{2}$ then the the variance is increased and for $p<\frac{1}{2}$ the reverse is true, indeed the variance is a montonically increasing function of $p$. We compare the change in variance as a function of $p$ with the previous simulation estimates, overlaying the analytical solution in Figure 3. We see that $p/1-p$ fits the experiment exceptionally well for $T=200$. Additionally we see that for $T=20$ the fit is less good in the extremal values of $p$ though still permissible. Such a result suggests that the result is indeed asymptotic (i.e. the change in variance is not necessarily $p/1-p$ over a small period) though in any case we expect the long run behaviour to better capture the current dynamics of an asset price. Finally, we also see according to Proposition 1 that a random walk model can be recovered through setting $p=\frac{1}{2}$, in which case $\lim_{n\rightarrow\infty}\frac{1}{n}\mathds{V}[Y_{n}]=\mathds{E}_{\nu}[\delta^{2}]$.

In addition to the discrete-time analysis in the previous section we also develop methodology capable of learning $p$ in a flexible model of high-frequency data. Such a model may also be applicable to highly illiquid assets, exhibiting only occasional changes in price.

As a result, rather than considering trades occuring at regular intervals we assume the times at which a price changes is itself a stochastic process. The resulting model falls under the class of Markov renewal processes (Pyke, 1961). They are closely related to the continuous time random walks of (Kotulski, 1995; Tunaley, 1974; Scalas, 2006a). In the physics literature continuous time random walks have found applications in econophysics and finance, see (Scalas, 2006a, b) for some summarising results illustrating the deep connectiom between continuous time random walks and fractional calculus. We show in Appendix A.2 that much of the previous analysis in discrete time generalises to this setting under the critical assumption that the arrival times between events are independent of the sign and magnitude process.

We have explored using a probabilistic model for investment styles to show that the variance of a financial asset is directly dependent on the probability of moving in the same direction on successive days. The theoretical analysis shows that variance may actually be reduced through reversal strategies - capturing the case that the asset is more likely to move in opposing directions on subsequent days. We have applied a simple model to US stock data, showing that such a regime is indeed prevalent in 97 of the largest stocks and thereby proven that relative to a random walk the variance of these stocks is actually reduced as a result of this frenetic behaviour. Indeed such a result suggests that these stocks are actually more predictable than a random walk due to this artefacct. A similar result is also shown for higher-frequency data in the case of the EUR/USD to a more exagerated extent. Similar ideas have been explored from a different perspective with the identification of a Hurst exponent of a stochastic process, quantifying the asymptotic behaviour of the range of a stochastic processs (Qian and Rasheed, 2004), though in general describe the behaviour only qualitatively.

The models considered above allow for many more flexible generalisations. In particular, (LeBaron, 1992) remarks that period of high volatility are typically occur during periods of low autocorrelation (and vice verser). It would be interesting to allow for variable volatility in the above model, exploring dynamically changing correlation parameters $p$. Alternatively, it would also be interesting to consider the sign process for multiple assets simultanesously rather than focussing - as is the case here - on a single asset. Similarly, it may be fruiful to consider a sign process where subsequent values depend on more than one lag of the process, thereby providing a still more flexible model. Finally, a principled incorporation of processes exhibiting long-term drift would also provide additional insight into asset price behaviour.

More generally, that autocorrelation contributes to larger variance is well known within the computational statistics literature. In particular, the autocorrelation in a class of algorithms referred to as Markov chain Monte Carlo algorithms typically serves to increase the variance arising from the Monte Carlo noise (Robert and Casella, 2013) in a directly analagous fashion to the increase in variance as defined above.